tag:blogger.com,1999:blog-18256101275711144102018-03-06T12:29:27.159-08:00Doubting ThomasMusings of an Open MindOur Frankhttp://www.blogger.com/profile/02765024367933720371noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-1825610127571114410.post-67445163096768206622011-12-21T07:59:00.000-08:002012-02-10T02:07:48.049-08:00Musings on Euler's number and natural logarithmsIn other words: a proof that <br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\int_{1}^{x}\frac{1}{x}dx=\log_{e}x" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\int_{1}^{x}\frac{1}{x}dx=\log_{e}x" title="\int_{1}^{x}\frac{1}{x}dx=\log_{e}x" /></a> <br /><br />where <i>e</i>=2.718281828... (<a href="http://en.wikipedia.org/wiki/E_(mathematical_constant)">Euler's number</a>)<br /><br />Mainly a note to self, just so I don't have to remember it again... <br /><br />Prompted by a couple of posts by Adrian McMenamin on his <a href="http://cartesianproduct.wordpress.com/">blog</a><br /><br />Not entirely sure this is entirely robust or complete – but I think it’s basically the way I learned it and have remembered and rehearsed it in the past. Any comments/corrections from those more in the know welcome.<br /><br />Define the function <i>f</i>(<i>x</i>) as follows:<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=f(x)=\int_{1}^{x}\frac{1}{x}dx" target="_blank"><img src="http://latex.codecogs.com/gif.latex?f(x)=\int_{1}^{x}\frac{1}{x}dx" title="f(x)=\int_{1}^{x}\frac{1}{x}dx" /></a><br /><br />Then <i>f</i>(<i>x</i>) is uniquely defined (for all real non-zero positive values of <i>x</i>) and<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\frac{d(f(x))}{dx}=\frac{1}{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{d(f(x))}{dx}=\frac{1}{x}" title="\frac{d(f(x))}{dx}=\frac{1}{x}" /></a><br /><br />and<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=f(1)=0" target="_blank"><img src="http://latex.codecogs.com/gif.latex?f(1)=0" title="f(1)=0" /></a><br /><br />Also, for a variable <i>y</i> (independent of <i>x</i> but on the same domain)<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\frac{d(f(xy)))}{dx}=\frac{1}{xy}\frac{d(xy))}{dx}=\frac{1}{xy}y=\frac{1}{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{d(f(xy))}{dx}=\frac{1}{xy}\frac{d(xy)}{dx}=\frac{1}{xy}y=\frac{1}{x}" title="\frac{d(f(xy)))}{dx}=\frac{1}{xy}\frac{d(xy))}{dx}=\frac{1}{xy}y=\frac{1}{x}" /></a><br /><br />and<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\frac{d(f(x)@plus;f(y))}{dx}=\frac{d(f(x))}{dx}@plus;\frac{d(f(y))}{dx}=\frac{d(f(x))}{dx}=\frac{1}{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{d(f(x)+f(y))}{dx}=\frac{d(f(x))}{dx}+\frac{d(f(y))}{dx}=\frac{d(f(x))}{dx}=\frac{1}{x}" title="\frac{d(f(x)+f(y))}{dx}=\frac{d(f(x))}{dx}+\frac{d(f(y))}{dx}=\frac{d(f(x))}{dx}=\frac{1}{x}" /></a><br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\therefore \frac{d(f(xy))}{dx}=\frac{d(f(x)@plus;f(y))}{dx}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\therefore \frac{d(f(xy))}{dx}=\frac{d(f(x)+f(y))}{dx}" title="\therefore \frac{d(f(xy))}{dx}=\frac{d(f(x)+f(y))}{dx}" /></a><br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\therefore f(xy)=f(x)@plus;f(y)@plus;C" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\therefore f(xy)=f(x)+f(y)+C" title="\therefore f(xy)=f(x)+f(y)+C" /></a><br /><br />where <i>C</i> is some constant value.<br /><br />To find the value of <i>C</i> let <i>x</i> = 1. Then, since <i>f</i>(1) = 0<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=f(y)=f(y)@plus;C\Rightarrow C=0" target="_blank"><img src="http://latex.codecogs.com/gif.latex?f(y)=f(y)+C\Rightarrow C=0" title="f(y)=f(y)+C\Rightarrow C=0" /></a><br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\therefore f(xy)=f(x)@plus;f(y)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\therefore f(xy)=f(x)+f(y)" title="\therefore f(xy)=f(x)+f(y)" /></a><br /><br />Now define the function <i>g</i>(<i>x</i>) to be the inverse of the function <i>f</i>(<i>x</i>). i.e.<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=f(g(x))=x" target="_blank"><img src="http://latex.codecogs.com/gif.latex?f(g(x))=x" title="f(g(x))=x" /></a><br /><br />and<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=f(g(x))=x" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g(f(x))=x" title="g(f(x))=x" /></a><br /><br />So<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=x@plus;y=f(g(x@plus;y))" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x+y=f(g(x+y))" title="x+y=f(g(x+y))" /></a><br /><br />And from the previous result that<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\therefore f(xy)=f(x)@plus;f(y)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?f(xy)=f(x)+f(y)" title="\therefore f(xy)=f(x)+f(y)" /></a><br /><br />we get<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=x@plus;y=f(g(x))@plus;f(g(y))=f(g(x)g(y))" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x+y=f(g(x))+f(g(y))=f(g(x)g(y))" title="x+y=f(g(x))+f(g(y))=f(g(x)g(y))" /></a><br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\therefore f(g(x@plus;y))=f(g(x)g(y))" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\therefore f(g(x+y))=f(g(x)g(y))" title="\therefore f(g(x+y))=f(g(x)g(y))" /></a><br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\therefore g(x@plus;y)=g(x)g(y)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\therefore g(x+y)=g(x)g(y)" title="\therefore g(x+y)=g(x)g(y)" /></a><br /><br />i.e. the function <i>g</i> satisfies the basic exponential identity<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=a^{m@plus;n}=a^{m}a^{n}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?a^{m+n}=a^{m}a^{n}" title="a^{m+n}=a^{m}a^{n}" /></a><br /><br />So, <i>g</i>(<i>x</i>) can be expressed as some number (go on, let's call it <i>e</i>) raised to the power of <i>x</i>.<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g(x)=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g(x)=e^{x}" title="g(x)=e^{x}" /></a><br /><br />Since <i>f</i>(<i>x</i>) is the inverse of <i>g</i>(<i>x</i>), <i>f</i>(<i>x</i>) can be expressed as the logarithm to the base <i>e</i> of <i>x</i>. i.e.<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=f(x)=\log_{e}(x)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?f(x)=\log_{e}(x)" title="f(x)=\log_{e}(x)" /></a><br /><br />So<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\log_{e}(x)=f(x)=\int_{1}^{x}\frac{1}{x}dx" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\log_{e}(x)=f(x)=\int_{1}^{x}\frac{1}{x}dx" title="\log_{e}(x)=f(x)=\int_{1}^{x}\frac{1}{x}dx" /></a><br /><br />That's the first bit done (I think)...<br /><br /><b>...but what's the value of <i>e</i>?</b><br /><br />First, look at the properties of the derivative of the function*<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g(x)=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g(x)=e^{x}" title="g(x)=e^{x}" /></a><br /><br />i.e.<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\frac{d(g(x))}{dx}=\frac{d(e^{x})}{dx}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{d(g(x))}{dx}=\frac{d(e^{x})}{dx}" title="\frac{d(g(x))}{dx}=\frac{d(e^{x})}{dx}" /></a><br /><br />Remember (from above)<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=x=f(g(x))" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x=f(g(x))" title="x=f(g(x))" /></a><br /><br />So<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\frac{dx}{d(g(x))}=\frac{d(f(g(x)))}{d(g(x))}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{dx}{d(g(x))}=\frac{d(f(g(x)))}{d(g(x))}" title="\frac{dx}{d(g(x))}=\frac{d(f(g(x)))}{d(g(x))}" /></a><br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\therefore \frac{d(e^{x})}{dx}=\frac{d(g(x))}{dx}=\frac{1}{(\frac{dx}{d(g(x))})}=\frac{1}{(\frac{d(f(g(x)))}{d(g(x))})}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\therefore \frac{d(e^{x})}{dx}=\frac{d(g(x))}{dx}=\frac{1}{(\frac{dx}{d(g(x))})}=\frac{1}{(\frac{d(f(g(x)))}{d(g(x))})}" title="\therefore \frac{d(e^{x})}{dx}=\frac{d(g(x))}{dx}=\frac{1}{(\frac{dx}{d(g(x))})}=\frac{1}{(\frac{d(f(g(x)))}{d(g(x))})}" /></a><br /><br />Remember too that<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\frac{d(f(x))}{dx}=\frac{1}{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{d(f(x))}{dx}=\frac{1}{x}" title="\frac{d(f(x))}{dx}=\frac{1}{x}" /></a><br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\therefore \frac{d(e^{x})}{dx}=\frac{1}{(\frac{d(f(g(x)))}{d(g(x))})}=\frac{1}{(\frac{1}{g(x)})}=g(x)=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\therefore \frac{d(e^{x})}{dx}=\frac{1}{(\frac{d(f(g(x)))}{d(g(x))})}=\frac{1}{(\frac{1}{g(x)})}=g(x)=e^{x}" title="\therefore \frac{d(e^{x})}{dx}=\frac{1}{(\frac{d(f(g(x)))}{d(g(x))})}=\frac{1}{(\frac{1}{g(x)})}=g(x)=e^{x}" /></a><br /><br />[This result allows for the derivation of both the sum of a series and limit definitions of Euler's number. What follows here is the derivation of the sum of a series definition - the limit definition is derived <a href="#limdef">below</a>]<br /><br />It follows directly from the result<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\therefore \frac{d(e^{x})}{dx}=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{d(e^{x})}{dx}=e^{x}" title="\therefore \frac{d(e^{x})}{dx}=e^{x}" /></a> <br /><br />that all higher derivatives of <br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g(x)=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g(x)=e^{x}" title="g(x)=e^{x}" /></a> <br /><br />can be expressed as follows:<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\frac{d^{n}(e^{x})}{dx^{n}}=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{d^{n}(e^{x})}{dx^{n}}=e^{x}" title="\frac{d^{n}(e^{x})}{dx^{n}}=e^{x}" /></a><br /><br />for any positive integer <i>n</i>.<br /><br />So <br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g(x)=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g(x)=e^{x}" title="g(x)=e^{x}" /></a> <br /><br />is infinitely differentiable and <br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g^{(n)}(0)=e^{0}=1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g^{(n)}(0)=e^{0}=1" title="g^{(n)}(0)=e^{0}=1" /></a> <br /><br />where <br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g^{(n)}(0)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g^{(n)}(0)" title="g^{(n)}(0)" /></a><br /><br />represents the value of the nth derivative of <br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g(x)=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g(x)=e^{x}" title="g(x)=e^{x}" /></a><br /><br />with respect to <i>x</i> at <i>x</i> = 0. <br /><br />So <br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g(x)=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g(x)=e^{x}" title="g(x)=e^{x}" /></a><br /><br />can be expressed as a <a href="http://mathworld.wolfram.com/MaclaurinSeries.html">Maclaurin series</a>:<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g(x)=e^{x}=\sum_{n=0}^{\infty }\frac{g^{(n)}(0)}{n!}x^{n}=\sum_{n=0}^{\infty }\frac{1}{n!}x^{n}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g(x)=e^{x}=\sum_{n=0}^{\infty }\frac{g^{(n)}(0)}{n!}x^{n}=\sum_{n=0}^{\infty }\frac{1}{n!}x^{n}" title="g(x)=e^{x}=\sum_{n=0}^{\infty }\frac{g^{(n)}(0)}{n!}x^{n}=\sum_{n=0}^{\infty }\frac{1}{n!}x^{n}" /></a><br /><br />So set <i>x</i> = 1 to calculate <i>e</i> to arbitrary precision using<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=e=e^{1}=\sum_{n=0}^{\infty }\frac{1}{n!}1^{n}=\sum_{n=0}^{\infty }\frac{1}{n!}\approx 2.718281828..." target="_blank"><img src="http://latex.codecogs.com/gif.latex?e=e^{1}=\sum_{n=0}^{\infty }\frac{1}{n!}1^{n}=\sum_{n=0}^{\infty }\frac{1}{n!}= 2.718281828..." title="e=e^{1}=\sum_{n=0}^{\infty }\frac{1}{n!}1^{n}=\sum_{n=0}^{\infty }\frac{1}{n!}= 2.718281828..." /></a><br /><br /><a name="limdef"><u>LIMIT DEFINITION OF EULER'S NUMBER</u></a><br /><br />Consider the limit<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\lim_{y \to 0} \frac{y}{e^{y}-1}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\lim_{y \to 0} \frac{y}{e^{y}-1}" title="\lim_{y \to 0} \frac{y}{e^{y}-1}" /></a><br /><br />Now<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\lim_{y \to 0} y=0" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\lim_{y \to 0} y=0" title="\lim_{y \to 0} y=0" /></a><br /><br />and<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\lim_{y \to 0} e^{y}-1=0" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\lim_{y \to 0} e^{y}-1=0" title="\lim_{y \to 0} e^{y}-1=0" /></a><br /><br />and<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\frac{d(e^{y}-1)}{dy}=e^{y}\neq 0" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{d(e^{y}-1)}{dy}=e^{y}\neq 0" title="\frac{d(e^{y}-1)}{dy}=e^{y}\neq 0" /></a><br /><br />for any <i>y</i> and<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\lim_{y \to 0} (\frac{d(y)}{dy}}/{\frac{d(e^{y}-1)}{dy})=\lim_{y \to 0}\frac{1}{e^{y}}=1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\lim_{y \to 0} (\frac{d(y)}{dy}}/{\frac{d(e^{y}-1)}{dy})=\lim_{y \to 0}\frac{1}{e^{y}}=1" title="\lim_{y \to 0} (\frac{d(y)}{dy}}/{\frac{d(e^{y}-1)}{dy})=\lim_{y \to 0}\frac{1}{e^{y}}=1" /></a><br /><br />is defined, therefore <a href="http://mathworld.wolfram.com/LHospitalsRule.html">L'Hopital's rule</a> applies. i.e.<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\lim_{y \to 0} \frac{y}{e^{y}-1}=\lim_{y \to 0}(\frac{d(y)}{dy}}/{\frac{d(e^{y}-1)}{dy})=\lim_{y \to 0}\frac{1}{e^{y}}=1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\lim_{y \to 0} \frac{y}{e^{y}-1}=\lim_{y \to 0}(\frac{d(y)}{dy}}/{\frac{d(e^{y}-1)}{dy})=\lim_{y \to 0}\frac{1}{e^{y}}=1" title="\lim_{y \to 0} \frac{y}{e^{y}-1}=\lim_{y \to 0}(\frac{d(y)}{dy}}/{\frac{d(e^{y}-1)}{dy})=\lim_{y \to 0}\frac{1}{e^{y}}=1" /></a><br /><br />Substitute<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=y=\log_{e}(x@plus;1)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y=\log_{e}(x+1)" title="y=\log_{e}(x+1)" /></a><br /><br />Then<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\frac{y}{e^{y}-1}=\frac{\log_{e}(x@plus;1)}{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{y}{e^{y}-1}=\frac{\log_{e}(x+1)}{x}" title="\frac{y}{e^{y}-1}=\frac{\log_{e}(x+1)}{x}" /></a><br /><br />and as<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=y \to 0, x \to 0" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y \to 0, x \to 0" title="y \to 0, x \to 0" /></a><br /><br />Therefore<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\lim_{x \to 0}\frac{\log_{e}(x@plus;1)}{x}=\lim_{y \to 0}\frac{y}{e^{y}-1}=1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\lim_{x \to 0}\frac{\log_{e}(x+1)}{x}=\lim_{y \to 0}\frac{y}{e^{y}-1}=1" title="\lim_{x \to 0}\frac{\log_{e}(x+1)}{x}=\lim_{y \to 0}\frac{y}{e^{y}-1}=1" /></a><br /><br />Now<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\lim_{x \to 0}\frac{\log_{e}(x@plus;1)}{x}=\lim_{x \to 0}\log_{e}((x@plus;1)^{1/x})=1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\lim_{x \to 0}\frac{\log_{e}(x+1)}{x}=\lim_{x \to 0}\log_{e}((x+1)^{1/x})=1" title="\lim_{x \to 0}\frac{\log_{e}(x+1)}{x}=\lim_{x \to 0}\log_{e}((x+1)^{1/x})=1" /></a><br /><br />therefore (since the exponential function is continuous*)<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\lim_{x \to 0}(x@plus;1)^{1/x}=e" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\lim_{x \to 0}(x+1)^{1/x}=e" title="\lim_{x \to 0}(x+1)^{1/x}=e" /></a><br /><br />Or alternatively if<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=n=\frac{1}{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?n=\frac{1}{x}" title="n=\frac{1}{x}" /></a><br /><br />then<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\lim_{n \to \infty }(1@plus;\frac{1}{n})^{n}=e" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\lim_{n \to \infty }(1+\frac{1}{n})^{n}=e" title="\lim_{n \to \infty }(1+\frac{1}{n})^{n}=e" /></a><br /><br /><u>*POSTSCRIPT</u><br />I've used <i>g</i>(<i>x</i>) to denote the exponential function here to try and avoid preconceptions but it's obviously more commonly denoted by <br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=exp(x)=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?exp(x)=e^{x}" title="exp(x)=e^{x}" /></a><br /><br />I think I'm right in saying that, <i>without</i> a definition of the exponential function:<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=a^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?a^{x}" title="a^{x}" /></a><br /><br />and exponential identities such as <br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=a^{x@plus;y}=a^xa^y" target="_blank"><img src="http://latex.codecogs.com/gif.latex?a^{x+y}=a^xa^y" title="a^{x+y}=a^xa^y" /></a> <br /><br />can only be defined where <i>x</i> (and <i>y</i>) is rational.<br /><br />However, because<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=\log_{e}x=f(x)=\int_{1}^{x}\frac{1}{x}dx" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\log_{e}x=f(x)=\int_{1}^{x}\frac{1}{x}dx" title="\log_{e}x=f(x)=\int_{1}^{x}\frac{1}{x}dx" /></a><br /><br />is (by definition) continuous and differentiable (with non-zero derivative) at all points on its domain of all non-zero positive numbers (range is all real numbers), its inverse<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=g(x)=exp(x)=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?g(x)=exp(x)=e^{x}" title="g(x)=exp(x)=e^{x}" /></a><br /><br />is also continuous and differentiable at all points on its domain of all real numbers (its range is all non-zero positive numbers). Therefore <i>e</i> can be raised to the power of <i>any</i> real number, not just rationals. This leads on to the definition of values for non-zero positive real numbers raised to the power of <i>any</i> real number - and eventually can be completely extended to include negative and complex numbers. A few steps beyond that and ultimately comes the famous and beautiful (and immensely useful) Euler function and identity:<br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=e^{i\theta }=\cos \theta @plus;i\sin \theta" target="_blank"><img src="http://latex.codecogs.com/gif.latex?e^{i\theta }=\cos \theta +i\sin \theta" title="e^{i\theta }=\cos \theta +i\sin \theta" /></a><br /><br /><a href="http://www.codecogs.com/eqnedit.php?latex=e^{i\pi }@plus;1=0" target="_blank"><img src="http://latex.codecogs.com/gif.latex?e^{i\pi }+1=0" title="e^{i\pi }+1=0" /></a><br /><br />I think... I might post a proof of that too at some point. Weather, time and memory permitting.Our Frankhttp://www.blogger.com/profile/02765024367933720371noreply@blogger.com0tag:blogger.com,1999:blog-1825610127571114410.post-78113130161587852592011-06-15T13:51:00.000-07:002011-06-15T15:24:53.169-07:00Distracted from distraction by distractionIn which:<br /><br />Distracted = Intrigued<br /><br />distraction = !#MathsPuzzle<br /><br />distraction = #MathsPuzzle<br /><br />So (Intrigued - !#MathsPuzzle) ÷ #MathsPuzzle = the following... <br /><br />...which is on a completely different subject to my few previous posts and is probably of interest to even fewer people...<br /><br />This was prompted by a maths puzzle posted on twitter by Matt Parker @standupmaths. <br />See #MathsPuzzle if you’re into this kind of thing and fancy more. Some of us are believe it or not...<br /><br />The original puzzle was as follows:<br /><br /><em>"QUICK! Make 10 using all and only {4, 6, 7, 9} and any selections from {+, −, ×, ÷, ^} with all the brackets you fancy."</em><br /><br />Matt soon after gave his favourite answer (come on, everyone has one!):<br /><br /><em>"My favourite solution is "9+(7−6)^4" where the 4 is hidden up on a shelf, out of harm's way."</em><br /><br />and followed up with this <strong>much</strong> more intriguing problem:<br /><br /><em><strong>"Harder meta-#MathsPuzzle: What is the smallest number of randomly selected single digits where there must exist such a way to generate 10?"</strong></em><br /><br /><em>"The meta-#MathsPuzzle could be with or without repeated digits. With has some nice cases to consider but without feels like a nicer proof."</em><br /><br /><span style="font-size: large;"><span style="font-size: small;">So, the question posed is:</span></span><br /><br /><span style="font-size: large;"><strong>What is the smallest number of randomly selected <em>distinct</em> single digits where there must exist a way to generate 10 using the operators {+, −, ×, ÷, ^} with all the brackets you fancy?</strong></span><br /><br />Mmm... An everyday kind of question. After a lunchtime’s scribbling/head scratching I posted the following:<br /><br /><em>"<strong>Five</strong> with no repetition. 2 at least will sum to 9 and am fairly sure 1 can always be got from any 3 digits left #MathsPuzzle"</em><br /><br />Fairly soon after I realised that this was ... not quite right, though it did at least look to be possibly along the right lines of enquiry. I waited for the nerdosphere to spot my *ahem* deliberate mistakes and very gratifyingly it did – courtesy of Richard Snape @snapey1979 and Neil Jackson @neiljackson111. Brilliant! If twitter is micro-blogging this was functioning micro peer review in action!<br /><br />@snapey1979 said<em>"think you may be right with answer not sure of logic. Take 1,2,3,5,9. can't sum to 9 and get 1 but 9/3 +5+(2/1)=10 #mathspuzzle"</em><br /><br />@neiljackson111 said "<em>468,479,579 don't give 1 @Our_Frank: @standupmaths and am fairly sure 1 can always be got from any 3 digits left #iknowyouvemovedonbut"</em><br /><br />Both great contributions to work in progress and below is, I think, I hope, a more complete proof for five digits – without the limitations of 140 characters.<br /><br />While mulling all this over I began to suspect that the minimum number of digits needed may well in fact be four – if only because I’d not come across a set of four digits (from an admittedly very small sample) from which I couldn’t generate 10. <br /><br />Wonderfully, in the meantime @snapey1979 chucked a computer at the problem and found that all sets of five - and then all sets of four digits will indeed generate 1. See:<br /><br /><u><span style="color: #810081;"><a href="http://www.iesd.dmu.ac.uk/~cascade/mathspuzzle.txt">http://www.iesd.dmu.ac.uk/~cascade/mathspuzzle.txt</a></span></u><br />and<br /><a href="http://www.iesd.dmu.ac.uk/~cascade/mathspuzzle4.txt">http://www.iesd.dmu.ac.uk/~cascade/mathspuzzle4.txt</a><br /><br />I tweeted @snapey1979 to say I thought I might have a "proof" (for five digits).<br /><br />He said "love to see it if you get there. Elegance trumps brute force every time :-)"<br /><br /><br />We are children of the iterative age after all and @snapey1979's "brute force" results definitely have elegance - don't they? <br /> <br />I'll leave it up to the reader to decide whether the following is elegant even in the slightest. <br /> <br />Having said that, I would genuinely love to see a "non-brute force" proof that it can be done with four digits. <br /><br /><strong><u>THE PROOF</u></strong><br />In summary, it works along the following lines:<br /><br /><u>THEOREM 1</u>: You can always generate 1 from any four distinct single digits using the operators {+, −, ×, ÷, ^} with all the brackets you fancy.<br /><br />So, if your set of five digits contains a 9 you can generate 1 from the remaining 4 digits and simply add it to the 9 to get 10.<br /><br /><u>THEOREM 2</u>: In any set of five digits which does not contain a 9 there exists at least one pair of digits that will sum to 9.<br /><br /><u>THEOREM 3</u>: You can generate 1 from any three distinct single digits using the operators {+, −, ×, ÷, ^} with all the brackets you fancy EXCEPT the sets {4,6,8}, {4,7,9} and {5,7,9}<br /><br />So, if your set of five digits doesn’t contain a 9 you can generate 9 from the sum of two of the digits and generate 1 from the remaining three digits (and add to get 10) UNLESS the remaining three digits are {4,6,8} (they can’t be {5,7,9} or {4,7,9} because the original five digit set didn’t contain a 9).<br /><br />If the remaining digits are {4,6,8} the original set must have been {2,4,6,7,8} because {4,6,8} is what you’re left with after having removed two digits that sum to 9 and {2,7} is the only possibility.<br /><br />4+6=10 and {2,7,8} (what we’re left with) is obviously neither {4,6,8}, {4,7,9} nor {5,7,9} so you must be able to generate 1 from {2,7,8} (to get 10*1=10). Put explicitly: (4+6)*((8-7)^2)=10<br /><br /><u>PROOF OF THEOREM 3</u>: <em>You can generate 1 from any three distinct single digits using the operators {+, −, ×, ÷, ^} with all the brackets you fancy EXCEPT the sets {4,6,8}, {4,7,9} and {5,7,9}</em><br /><br />Back to front I know - but the proof of theorem 3 leads more or less directly to the proof of theorem 1...<br /><br /><strong>The key fact here is that 1^n=1 for any n i.e. if you can generate 1 from a pair of digits you can generate 1 from any set of 3 digits containing that pair.</strong><br /><br />1^(y+z)=1 so if x=1 you can generate 1 regardless of the values of y and z.<br /><br />If x=2:<br /> and y=3 then y-x=1 and (y-x)^z=1<br /> and y=4<br /> and z=5 then z-y=1 and (z-y)^x=1<br /> and z=6 then (z-y)/x=1<br /> and z=7 then (z-y)-x=1<br /> and z=8 then z/(x*y)=1<br /> and z=9 then z-(x*y)=1<br /> and y=5<br /> by similar logic if z is 6, 7 or 8 then you can generate 1<br /> (z differs from y by either 1,2 or 3 which either differs from x by 1 or is equal to x) <br /> if z=9 then (x*y)-z=1<br /><br />If x=3:<br /> Again, using the same logic if y=4 you can generate 1<br /> If y=5, you can generate 1 as z=6, 7, 8 or 9 (z-y is either 1, 2, 3 or 4 and x=3)<br /> If y=6, you can generate 1 as z=7, 8 or 9 (similar logic to above)<br /> If y=7, you can generate 1 as z=8 or 9 <br /> Etc.<br /><br />If x=4:<br /> and y=6<br /> and z=8 then... I can’t see a way of generating 1.<br /><br /><strong>I don’t think you can generate 1 from {4,6,8}</strong><br /><br /><strong> </strong>If z=9 then x+y-z=1<br /><br /> and y=7<br /> and z=9 then... I can’t see a way of generating 1.<br /><br /><strong>I don’t think you can generate 1 from {4,7,9}</strong><br /><br /> and y=8 the z=9 and y-z=1<br /><br />If x=5:<br /> and y=7<br /> and z=9 then... I can’t see a way of generating 1.<br /><br /><strong>I don’t think you can generate 1 from {5,7,9}</strong><br /><br /> and y=8 then z=9 and z-y=1<br /><br />If x=6:<br /> and y=8 then z=9 and z-y=1<br /> Etc.<br /><br />So... the only sets of 3 digits from which 1 cannot be generated are:<br /><strong>{4,6,8}, {4,7,9} and {5,7,9}</strong><br /><br /><u>PROOF OF THEOREM 1</u>: <em>You can always generate 1 from any four distinct single digits using the operators {+, −, ×, ÷, ^} with all the brackets you fancy.</em><br /><br />Any set of four digits must contain at least one subset of three digits from which 1 can be generated. If you generate a four digit set by adding a digit to any of the three sets {4,6,8}, {4,7,9} or {5,7,9} there will inevitably exist a three digit subset of it which is none of these sets. From this three digit subset you can generate 1 (by theorem 3) and raise it to the power of the remaining digit.<br /><br /><u>PROOF OF THEOREM 2</u>: <em>In any set of five digits which does not contain a 9 there exists at least one pair of digits that will sum to 9.</em><br /><br />Here are the two digit sets which sum to 9: {1,8}, {2,7}, {3,6} and {4,5}<br /><br />Each pair has one odd and one even number and no digits are repeated.<br /><br />If in the five digit set there are all 4 even digits then the remaining digit must be odd and will sum with one of the even digits to 9.<br /><br />If in the five digit set there are 3 even digits the remaining two must be odd. There is only one odd number that will not sum with any of the even digits to 9 so one of the two must.<br /><br />If in the five digit set there are 2 even digits the remaining three must be odd. There are only two odd numbers that will not sum with any of the even digits to 9 so one of the three must.<br /><br />If in the five digit set there is 1 even digit the remaining four must be odd. There are only three odd numbers that will not sum with any of the even digits to 9 so one of the four must.<br /><br />QED?Our Frankhttp://www.blogger.com/profile/02765024367933720371noreply@blogger.com0tag:blogger.com,1999:blog-1825610127571114410.post-38250851335610876872011-05-19T04:01:00.000-07:002011-05-19T04:01:19.636-07:00The Science of MMS (part 2)<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">A look at the claim that chlorine dioxide only destroys "bad" bacteria while leaving "good" bacteria alone. See <a href="http://ourfrank.blogspot.com/2011/05/science-of-mms-part-1.html">The Science of MMS (part 1)</a> for background. The following was sent on along with the stuff posted there.</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><b style="mso-bidi-font-weight: normal;"><u><span style="font-family: Calibri;">“Good” and “bad” bacteria</span></u></b></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">I contacted a friend who is a professional postdoctoral microbiologist so far more qualified on this sort of thing than I am.<span style="mso-spacerun: yes;"> </span>She says:</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">“I don't know to what degree this process can really be quantified in the human body, but the presence of any chlorine dioxide in the gut should, at the very least, kill off all the person's microbial flora and leave them at the mercy of whatever pathogenic bacteria turn up. This is especially a bad thing for young kids, as their gut flora is still getting established as a defensive structure anyway. That's one reason antibiotics aren't generally given to babies - internal flora gets destroyed.”</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">Just to be clear – what she means by “all the person’s microbial flora” is all <i style="mso-bidi-font-style: normal;">types </i>of microbes (not literally all microbes – if that happened I guess you could probably be very ill indeed).<span style="mso-spacerun: yes;"> </span>What she’s saying is that chlorine dioxide is <i style="mso-bidi-font-style: normal;">indiscriminate</i>.<span style="mso-spacerun: yes;"> </span>It’s not true that it kills the “bad” bacteria and leaves the “good” bacteria alone.<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">I’m certainly no microbiologist but I‘m pretty sure that you can’t generally define bacteria as “good” or “bad”.<span style="mso-spacerun: yes;"> </span>Bacteria can be harmless, helpful, necessary, pathogenic (disease causing) and often <i style="mso-bidi-font-style: normal;">conditionally </i>pathogenic<i style="mso-bidi-font-style: normal;"> – </i>i.e. a part of the normal flora of the body (usually) but pathogenic in certain circumstances – e.g. Streptococcus lives up your nose all the time but can also cause all kinds of infection if it ends up in the wrong place or in the wrong amount.<span style="mso-spacerun: yes;"> </span>E. coli has a bad name but it’s only certain strains that can cause food poisoning, many other strains are a natural part of the gut.<span style="mso-spacerun: yes;"> </span>So the same bacteria can be both “good” and “bad” depending on the circumstances and/or the strain.<span style="mso-spacerun: yes;"> </span>The argument that chlorine dioxide, as an oxidant, picks off only “bad” bacteria is hard to accept in that context.<span style="mso-spacerun: yes;"> </span>It would have to be a very sophisticated oxidant indeed that could distinguish not only types of bacteria but also whether their current mode of action could be said to be a good or a bad one.<span style="mso-spacerun: yes;"> </span>Chlorine dioxide is not at all sophisticated – it’s a very basic molecule.</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">There seems to be a suggestion on some of the web forums I’ve seen (although I don’t know whether this is actually a claim made by Jim Humble) that “good” bacteria are aerobic and “bad” bacteria are anaerobic.<span style="mso-spacerun: yes;"> </span>The claim appears to be that, as chlorine dioxide is an “oxidant” it disrupts anaerobic bacteria by producing oxygen, which anaerobic bacteria cannot “breathe”.<span style="mso-spacerun: yes;"> </span>Again the good/bad bit isn’t really evident: Bordatella (whooping cough) is aerobic, Chlamydia is aerobic, C. botulinum (botulism) is anaerobic, the vast majority of the natural gut bacteria (flora) are anaerobic.<span style="mso-spacerun: yes;"> </span>Again, it doesn’t matter anyway because chlorine dioxide can kill them all. </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">Chlorine dioxide seems to react, as an oxidant, with the proteins, amino acids and RNA within cells.<span style="mso-spacerun: yes;"> </span>It can’t distinguish between human, “good”, “bad”, viral or bacterial amino acids or proteins because chemically speaking (as far as chlorine dioxide and its oxidizing action is concerned) there really isn’t one.<span style="mso-spacerun: yes;"> </span>It is capable of harming all cells (not just bacteria) and the bits the make up the cells.<span style="mso-spacerun: yes;"> </span>I’m sure there are some differences between exactly how badly it affects different cells depending on the detail of the protein or whatever and where the damage is done in the cell etc. - I’ve seen a suggestion that chlorine dioxide is less effective against rotavirus (“gastric flu”) and E. coli (which can be both “good” and “bad” as described above) than other organisms, for example.<span style="mso-spacerun: yes;"> </span>That’s not to say it still doesn’t kill them at all though, it just does so less efficiently, and is certainly not to say that it can “select” between “good” and “bad” bacteria, whatever they might be.<span style="mso-spacerun: yes;"> </span><span style="color: black; font-family: "Verdana", "sans-serif";"></span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">Gut flora are not just a few bacteria that happen to be hanging around.<span style="mso-spacerun: yes;"> </span>They constitute a whole, highly developed “ecosystem”, that some people have likened to an organ of the body in itself.<span style="mso-spacerun: yes;"> </span>Obviously drugs can be developed which are a bit more discriminating than chlorine dioxide, to try and target specific groups of pathogenic bacteria and protect gut flora during treatment – “narrow spectrum” antibiotics in particular.<span style="mso-spacerun: yes;"> </span>Most prescribed antibiotics, in normal circumstances are narrow spectrum, I believe, and they still can cause some side effects in the stomach because they can affect the gut flora.<span style="mso-spacerun: yes;"> </span>You could in theory engineer drugs specifically to target particular pathogens but that’s the realm of high level chemistry, bio-engineering and even genetic modification – i.e. highly complex.<span style="mso-spacerun: yes;"> </span>Chlorine dioxide is a very simple molecule with a very simple action (oxidation) that disrupts all cellular and microbial activity.<span style="mso-spacerun: yes;"> </span>This is obviously the reason it’s used industrially as a biocide.<span style="mso-spacerun: yes;"> </span>Beyond water and food treatment it’s also used to treat the (liquid) water in cooling towers in power stations.<span style="mso-spacerun: yes;"> </span>I would expect this is more to prevent the build up of any organic material that might clog up the pumps etc. than because of any public health concern.<span style="mso-spacerun: yes;"> </span>It kills all the bacteria in water – not just those that might be harmful to people. <span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">So basically chlorine dioxide behaves like a completely indiscriminate antibiotic in the gut.<span style="mso-spacerun: yes;"> </span>The side effects of nausea, vomiting, stomach cramps, diarrhoea etc. you can get as you increase the dose of MMS are consistent with this as they are the same side effects you can get with antibiotics.<span style="mso-spacerun: yes;"> </span>These symptoms of sickness aren’t a sign of any “detox”</span><span style="font-family: Calibri;">, they’re a sign that you’re disrupting your gut flora (and with chlorine dioxide, possibly the cells of your stomach lining themselves too).<span style="mso-spacerun: yes;"> </span>One potential effect then is, as with antibiotics, to disrupt your immune system (particularly in young children, where it’s not been fully developed) and leave you open to further infection.<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">In terms of its action at low doses, chlorine dioxide is more like a broad spectrum antibiotic than a narrow spectrum one.<span style="mso-spacerun: yes;"> </span>Broad spectrum antibiotics are typically used in serious situations where you don’t know what the pathogen is (e.g. meningitis) and where the benefits (i.e. potential survival) outweigh the downsides (i.e. potential disruption of all microbial activity in the body) – or in serious infections where there are lots of different pathogenic bacteria and you need to kill them all - and are not usually delivered orally but generally intravenously I think.<span style="mso-spacerun: yes;"> </span>I would hope that most doses of MMS are low enough to not disrupt anything too much.<span style="mso-spacerun: yes;"> </span>The point is, that even if there are any benefits to MMS, they are likely to be outweighed by the downsides.<span style="mso-spacerun: yes;"> </span>There’s certainly no need to take even a low dose of a broad spectrum antibiotic for a relatively mild infection.</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">An analogy might be spraying your entire garden with weed killer.<span style="mso-spacerun: yes;"> </span>Obviously, just because it’s called weed killer doesn’t mean it only kills weeds.<span style="mso-spacerun: yes;"> </span>What’s a “weed” anyway?<span style="mso-spacerun: yes;"> </span>Spraying would affect your whole garden, with the severity of the effect depending on how much weed killer you sprayed.<span style="mso-spacerun: yes;"> </span>If you carried on doing this every day, even with a small amount, it would likely have a fairly significant effect on your whole garden eventually.<span style="mso-spacerun: yes;"> </span>The places where plants (weeds or not) were killed would be open to accept whichever seeds happened to turn up next – a weed probably, or maybe something you actually wanted to grow there, if you were lucky - but you would be leaving that entirely to chance.<span style="mso-spacerun: yes;"> </span>The overall effect is likely to be a much less healthy and “weedier” garden.<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">It’s not a perfect analogy because your gut flora can act against pathogens themselves and weeds aren’t harmful to your garden (or yourself) necessarily, they just change the way you’d like it to look.<span style="mso-spacerun: yes;"> </span>The fact that gut flora can act against pathogens obviously doesn’t mean that you never get stomach bugs – just that your gut can often deal with them to some degree itself – I guess, because of this, there are many “bugs” you’ll never know you’ve had (thank you natural gut flora!).<span style="mso-spacerun: yes;"> </span>It would all be a bit like your garden being able to regulate weeds by itself, which it clearly can’t, but hopefully you see the point.<span style="mso-spacerun: yes;"> </span>I suppose you could argue that MMS might not affect “good” bacteria so much because the gut flora are quite effective at righting any imbalance themselves, but as far as I understand it that’s why antibiotics don’t do as much damage as, in theory, they otherwise could, and anyway, is a completely different argument than is used in the marketing of MMS.<span style="mso-spacerun: yes;"> </span>This is probably why MMS isn’t marketed for stomach upsets (even though most of the effect of it is probably in the gut).<span style="mso-spacerun: yes;"> </span>In stomach upsets your gut flora are obviously already disrupted to some degree.<span style="mso-spacerun: yes;"> </span>Using MMS would likely only disrupt them further, which would mean that you would feel worse and it would take you longer to get better.</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">I don’t know anywhere near enough biology to know to what extent chlorine dioxide will be transported around the rest of the body.<span style="mso-spacerun: yes;"> </span>I would imagine that at the level of dose we’re talking about it would be mostly used up in the gut but, if we give MMS the benefit of the doubt and assume these small amounts can end up anywhere, then there’s good reason to believe it would affect cold viruses, malaria protozoa, cancer cells, HIV retrovirus etc. – because it can affect absolutely everything in the body (which includes the cells of the body itself, not just flora) – but probably to a fairly small degree because everything’s (hopefully) only affected only to a small degree.<span style="mso-spacerun: yes;"> </span>Any effects at all on pathogens etc. would basically be “side-effects” of you harming pretty much everything in your body (i.e. poisoning yourself).</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">One significant area of drug research is “targeted delivery” – i.e. developing ways of getting the active ingredient in the drug right to the specific location (or to the pathogen) where it is required to act.<span style="mso-spacerun: yes;"> </span>All drugs will have more than one effect on probably more than one system, organ etc. within the body – some good, some bad, and hopefully many fewer and not particularly serious bad ones than good ones.<span style="mso-spacerun: yes;"> </span>The whole point of medicine is to make you generally better not generally worse and the point of targeted delivery is to maximise the desired effects and minimise the undesired ones.<span style="mso-spacerun: yes;"> </span>This is especially important with highly toxic drugs, like chemotherapy for example.<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 6pt;"><span style="font-family: Calibri;">You probably couldn’t get a less targeted drug than MMS.</span></div>Our Frankhttp://www.blogger.com/profile/02765024367933720371noreply@blogger.com2tag:blogger.com,1999:blog-1825610127571114410.post-65979887713618422982011-05-18T07:30:00.000-07:002011-05-19T00:38:09.841-07:00The Science of MMS (part 1)<div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">If you have absolutely no interest in the alternative therapy MMS, look away now and go out and do something less boring instead.<span style="mso-spacerun: yes;"> </span>This is a pretty lengthy post.<span style="mso-spacerun: yes;"> </span>As the saying goes – it would have been shorter if I’d spent more time on it...</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">This is all a result of someone I know well getting in touch a few months ago about concerns they had for a friend of theirs who was using (the alternative therapy formerly known as) Miracle Mineral Solution (MMS), “discovered” by Bishop Jim Humble of the Genesis II Church of Health and Healing </span><a href="http://jimhumble.biz/"><span style="font-family: Calibri;">http://jimhumble.biz/</span></a><span style="font-family: Calibri;">.<span style="mso-spacerun: yes;"> </span>It appears that it’s now marketed as Master Mineral Solution.<span style="mso-spacerun: yes;"> </span>MMS is 28% sodium chlorite solution.<span style="mso-spacerun: yes;"> </span>Sodium chlorite is a pretty powerful bleaching agent.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">On the face of it, a bleaching agent is not exactly the sort of thing you’d necessarily expect to have healing properties.<span style="mso-spacerun: yes;"> </span>It’s certainly not the sort of thing that you’d expect to have been found to be a previously unrecognised panacea.<span style="mso-spacerun: yes;"> </span>It is claimed that MMS can be used to successfully treat pretty much anything: cancer, HIV/AIDS, malaria, the common cold – you name it, there’s probably a claim made for the successful treatment of it with MMS.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">I think it’s fair to say that as a natural scientist by nature, education and trade (some of the time), and a fan of Ben Goldacre’s book/column/blog “Bad Science”, I was naturally – erm – a little bit skeptical about these claims to say the least.<span style="mso-spacerun: yes;"> </span>I had, perhaps somewhat naively, assumed that a quick google would elicit some key facts and figures that could be built up into a decent understanding of this stuff.<span style="mso-spacerun: yes;"> </span>But the more I googled, beyond FDA warnings against its use </span><a href="http://1.usa.gov/d410UU"><span style="font-family: Calibri;">http://1.usa.gov/d410UU</span></a><span style="font-family: Calibri;"> etc., <span style="mso-spacerun: yes;"> </span>the less I found I really knew about it.<span style="mso-spacerun: yes;"> </span>The online conversation was largely dominated by two highly polarised camps: staunch advocates and staunch opponents and never the twain did meet.<span style="mso-spacerun: yes;"> </span>Except in (sometimes pretty vicious) forum flame wars.<span style="mso-spacerun: yes;"> </span>See, for example, what happened to Rhys Morgan when he posted on a Crohn’s disease forum last year after he’d seen the FDA warning </span><a href="http://thewelshboyo.wordpress.com/2010/08/10/bleachgate/"><span style="font-family: Calibri;">http://thewelshboyo.wordpress.com/2010/08/10/bleachgate/</span></a></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">Other than the spectacularly unfairly victimised (and brilliant) Rhys and a number of other questioning and thoughtful bloggers much of what I first read was basically either biased, unhelpful, or uninformative.<span style="mso-spacerun: yes;"> </span>Where there was no flame war there was really only preaching to the converted – on both sides.<span style="mso-spacerun: yes;"> </span>I knew that arguments like “it’s just so obviously a scam, there’s no point going into the science” (though I agreed completely with the first part of that statement) wouldn’t be enough for us.<span style="mso-spacerun: yes;"> </span>I was just trying to garner some trustable and verifiable information and knowledge. <span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">To prepare MMS you mix it with citric acid which produces chlorine dioxide solution.<span style="mso-spacerun: yes;"> </span>What I wanted to know was: what exactly is the amount of chlorine dioxide you end up taking with MMS?;<span style="mso-spacerun: yes;"> </span>what are the known effects of drinking these amounts?; are there any studies (other than claims made by Jim Humble etc.) of the health benefits of chlorine dioxide?<span style="mso-spacerun: yes;"> </span>Chlorine dioxide is used to treat drinking water in some places so clearly there must be “safe” levels (though obviously if MMS turned out to be no different from drinking water it would still be a scam). <span style="mso-spacerun: yes;"> </span>I couldn’t find the answers anywhere.<span style="mso-spacerun: yes;"> </span>I was lucky in that I wasn’t ill myself and the person we had our concerns for wasn’t particularly ill either – they were just using it to treat everyday coughs and colds etc.<span style="mso-spacerun: yes;"> </span>But I really did feel for people who were perhaps more seriously ill and were trying to find decent information on this stuff.<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">As much as it was true that arguments for MMS were built largely on unverifiable anecdotes of personal “success” with it, bold unfounded claims for its efficacy, unreferenced “studies”, conspiracy theories and pseudo-science etc. it was as frustrating, if not more so, that some of the supposedly “rational” opponents of MMS unfortunately just came across as little more than uninformed scientistic trolls.<span style="mso-spacerun: yes;"> </span>“Scientistic” isn’t a typo by the way - if you’re unfamiliar with the term see </span><a href="http://en.wikipedia.org/wiki/Scientism"><span style="font-family: Calibri;">http://en.wikipedia.org/wiki/Scientism</span></a><span style="font-family: Calibri;"> and/or </span><a href="http://bit.ly/juxw00"><span style="font-family: Calibri;">http://bit.ly/juxw00</span></a><span style="font-family: Calibri;"> - “scientism ... a kind of over-enthusiastic and uncritically deferential attitude towards science”.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">Labelling someone “delusional”, for example, is pretty insensitive and isn’t a particularly rational thing to do (unless you just happen to enjoy meaningless conflict and wasting bandwidth) as it’s hardly likely to win anyone over to your side of any argument. <span style="mso-spacerun: yes;"> </span>If you are concerned, as many are, because you believe (or know) MMS to be dangerous to health, then it would be much more helpful if you told us why you believe that or how you know.<span style="mso-spacerun: yes;"> </span>Insulting users of MMS (no matter how ridiculous you might believe their position to be) is only likely to be counterproductive to your aim of stopping its use.<span style="mso-spacerun: yes;"> </span>Those with opposite positions are only going to become entrenched against you and furthermore, the image of science and its healthily skeptical, questioning and thorough investigative nature is completely misrepresented and demeaned.<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">There are many and varied reasons for people seeking alternative therapies – e.g. mistrust of big pharma, conventional treatments not having worked etc.<span style="mso-spacerun: yes;"> </span>It’s hardly surprising that people choose to use MMS when it’s so cleverly promoted and so hard to research properly, because there is so much meaningless argument about it. <span style="mso-spacerun: yes;"> </span>I found the whole conversation, if you can call it that, well… just a little undignified at times, lacking in perspective and inconsiderate.<span style="mso-spacerun: yes;"> </span>It was not particularly informative and didn’t advance my own understanding or knowledge much, if it all.<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">Anyway, interest was sparked, so I went on a bit of a fact finding odyssey, which included working through some chemistry for the first time in about 20 years! <span style="mso-spacerun: yes;"> </span>The intention was to try and understand MMS as well as I could for myself and present, not an argument for or against it necessarily, but rather the facts as I understood them and as fair minded as possible a view of what was going on, so that the person we had our concerns for would be able to make as informed a decision as possible for themselves on whether they should continue to use it or not.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">It took some time but it seems that our efforts came to be appreciated and this person has now stopped using MMS and is advising others, including those who recommended it in the first place, to do the same - using some of the information given here.<span style="mso-spacerun: yes;"> </span>The text below is part of what was originally sent on to them.<span style="mso-spacerun: yes;"> </span>I’ll follow up with the rest on another post.<span style="mso-spacerun: yes;"> </span>I know it’s not much but I do wonder how many of the flamers and trolls, on either side of this “debate”, can claim to have influenced any change in even one person’s mind.<span style="mso-spacerun: yes;"> </span>I’m not naturally a blogger but am posting now as I just thought this all might be useful to others too.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">I’m basically a meteorologist – i.e. not a chemist, a biologist, a doctor, a toxicologist or whatever so not at all qualified in this field and this is therefore probably riddled with errors and misinterpretations.<span style="mso-spacerun: yes;"> </span>If anyone does ever happen to read this, and is more qualified than I am and spots anything that needs correcting then I’d be glad for your comments.<span style="mso-spacerun: yes;"> </span>This isn’t definitive by any means and though lengthy, is not comprehensive or particularly detailed either.<span style="mso-spacerun: yes;"> </span>It’s just meant to be factual, comprehensible (digestible, dare I say it) and as fair as is possible.<span style="mso-spacerun: yes;"> </span>I hope I’ve managed that.<span style="mso-spacerun: yes;"> </span>I had, and continue to have, no other agenda other than helping someone make an informed choice.<span style="mso-spacerun: yes;"> </span>If you happen to be in a similar position to that which I was in or are using MMS and want to know more about it, you might find the information here similarly useful.<span style="mso-spacerun: yes;"> </span>Let me know if you do.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">I’m really not at all interested in sparking another flame war.<span style="mso-spacerun: yes;"> </span>Any trolls and flamers out there: that is not an invitation – I will flatly ignore you.<span style="mso-spacerun: yes;"> </span>Genuine and <i style="mso-bidi-font-style: normal;">relevant</i> (this post is about MMS and nothing more…) comments and questions are welcome.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="font-family: Calibri;">Many thanks and much credit due to Ben Goldacre for “Bad Science” – which I’d highly recommend.<span style="mso-spacerun: yes;"> </span>A number of the facts, figures and ideas here are sourced from his work.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><a href="http://www.amazon.co.uk/Bad-Science-Ben-Goldacre/dp/0007240198"><span style="font-family: Calibri;">http://www.amazon.co.uk/Bad-Science-Ben-Goldacre/dp/0007240198</span></a></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><a href="http://www.badscience.net/"><span style="font-family: Calibri;">http://www.badscience.net/</span></a></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="font-family: Calibri;">@bengoldacre</span></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br /></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="font-family: Calibri;">Whether you like any of this or not just be sure that you avoid being too “uncritically deferential”.<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="margin: 0cm 0cm 0pt;"><br /></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">Here’s the original email:</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">Basically, MMS looks to me to be inadvisable.<span style="mso-spacerun: yes;"> </span>It may not be too harmful in most circumstances if done properly.<span style="mso-spacerun: yes;"> </span>It’s highly unlikely to do you any good - I’ve found no convincing or sufficient evidence for that - and that’s not me being biased, that’s just me applying certain criteria for “convincing” and “sufficient”.<span style="mso-spacerun: yes;"> </span>It’s probably largely harmless at very low doses, will probably make you a bit poorly at slightly higher doses, quite poorly at higher doses than that - but really only even gets close to being potentially toxic at the very top end of the suggested doseage levels (i.e. those for treating cancer – 70 a day).</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">If I were to make a judgement on that I’d say that it looks like Jim Humble has been pretty canny and is exploiting the uncertain grey area that is there between what’s probably safe and what may be toxic, given currently available evidence – and is using that grey area to imply that there are previously unexplored benefits (over and above the likely side-effects) within it.<span style="mso-spacerun: yes;"> </span>To be truly fair - he could be (though I think is highly unlikely to be) right, but he really needs to demonstrate properly that that’s likely to be the case.<span style="mso-spacerun: yes;"> </span>Just because it might not do too much harm doesn’t mean that it might do any good.<span style="mso-spacerun: yes;"> </span>It would be great (and not impossible – nothing’s impossible if no-one’s collected and properly analysed and presented the data...) if there were a sudden “spike” in health benefits over damaging effects somewhere between probably safe and likely harmful levels as they’re currently understood – it’s just that on the face of it looks unlikely and I think it’s safer and more reasonable to assume, for the time being, that sodium chlorite and chlorine dioxide become more harmful to you the more you take.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">One slightly concerning aspect, is that the main toxic effect (of drinking chlorine dioxide – which is what you end up doing with MMS) is likely to be on child development.<span style="mso-spacerun: yes;"> </span>A range of developmental effects are seen in rat pups, though at relatively high doses (for rats) compared to MMS (no-one’s tested relatively high doses of chlorine dioxide on pregnant women, babies or children obviously) - </span><a href="http://www.atsdr.cdc.gov/toxprofiles/tp160.pdf" target="_blank"><span style="font-family: Calibri;">http://www.atsdr.cdc.gov/toxprofiles/tp160.pdf</span></a><span style="font-family: Calibri;">.<span style="mso-spacerun: yes;"> </span>There is a study on chlorine dioxide in the drinking water supply that claims to have found a difference in rates of neonatal jaundice, cranial circumference and body length at birth between babies born to mothers who had been drinking water treated with chlorine dioxide and babies born to mothers who hadn’t – but things like any other chemicals in the water, dietary habits etc. weren’t taken into account so it’s meaningless, because it’s impossible to attribute these differences to chlorine dioxide with any confidence.<span style="mso-spacerun: yes;"> </span>The minimum toxic level of chlorine dioxide for rats (if you scale up by relative bodyweight from rats to people) is, I reckon, equivalent to around about 100 or more doses of MMS a day for a small person, so loads - but no-one can really say for sure how results from rats will necessarily translate to people so I’d just say it’s advisable to make sure to stay well clear of anywhere near those sorts of levels.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">For what it’s worth I reckon one dose of properly prepared MMS contains about 0.9mg of chlorine dioxide, which in a 150ml glass of water is a concentration of 6mg/l.<span style="mso-spacerun: yes;"> </span>The U.S. recommended upper limit on chlorine dioxide in the drinking water is 0.8mg/l and the “minimal risk level” is 0.1mg per kg (of your bodyweight) per day.<span style="mso-spacerun: yes;"> </span>Toxic effects are seen in rats beginning at around 5mg per kg (of their bodyweight) per day.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">MMS could be horrible, in various ways, if you do it wrong – neat MMS is, frankly, just dangerous to have around the house.<span style="mso-spacerun: yes;"> </span>Also, putting the drops of MMS and citric acid straight into a glass of water - i.e. not mixing the drops first and then adding water (as you should if you follow the instructions) - will multiply the amount of chlorine dioxide in the dose taken by a factor of about 8 (this should be born in mind with respect to toxic levels) so it is important to make sure that it’s always mixed according to instructions – especially if more than a few drops are being used.<span style="mso-spacerun: yes;"> </span>Because of its known chemical properties as an oxidant and a free-radical there are reasons to suspect that chlorine dioxide could be dangerous if you used it even at low-ish doses over the long term but no-one really knows because sufficient tests haven’t been done for that either.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">Whether MMS is dangerous in itself or not is only one thing though.<span style="mso-spacerun: yes;"> </span>Even therapies that are completely harmless <i style="mso-bidi-font-style: normal;">in themselves</i> are obviously dangerous if they are used to replace more effective treatments.<span style="mso-spacerun: yes;"> </span>This is what makes many people particularly bothered about this kind of thing, as it gives false hope and exploits (sometimes very) ill and vulnerable people (especially children), who are effectively persuaded to deny themselves (or are denied by others) the opportunity for more evidence based, demonstrably effective and even sometimes potentially life-saving, treatment.<span style="mso-spacerun: yes;"> </span>I don’t think anyone likes being (or should be) hoodwinked in this way – especially where hope and health are concerned (though it’s hardly uncommon).<span style="mso-spacerun: yes;"> </span>But, of course, it’s up to each individual to decide whether they trust this stuff (or Jim Humble over the FDA and the like, or me for that matter) or not.</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">You might remember, for example, that it took ages for the South African government to provide effective treatment for HIV/AIDS (one of the many things MMS is supposedly good for).<span style="mso-spacerun: yes;"> </span>This is covered in some detail in “Bad Science”.<span style="mso-spacerun: yes;"> </span>As I understand it the reasons were partly cultural (i.e. an understandable mistrust of “evil” white western “colonial” drugs companies), partly due to the “evil” drugs companies’ own price demands but were also due to some people successfully promoting their own cheap treatment for HIV/AIDS (vitamin supplements) right at the top level of government (which was sympathetic because it believed that “natural” was best).<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">So national policy on HIV/AIDS treatment basically became eat your greens and top up with vitamin supplements.<span style="mso-spacerun: yes;"> </span>Not bad advice in general and likely to improve general health, but maybe not so good for the treatment of a serious illness.<span style="mso-spacerun: yes;"> </span>The Western Cape province defied national policy and provided anti-retroviral drugs.<span style="mso-spacerun: yes;"> </span>One study, which compared what ultimately happened in the Western Cape with the rest of South Africa estimated that, had anti-retroviral drugs been provided nationwide in South Africa between 1999 and 2007, this would have prevented 343,000 deaths.<span style="mso-spacerun: yes;"> </span>Another estimated that 330,000 deaths would have been prevented between 2000 and 2005.<span style="mso-spacerun: yes;"> </span>This kind of thing is in part why what I see as the abuse of people, language and science, as I think I put it to you previously, tends to raise my ire... </span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt;"><span style="font-family: Calibri;">Anyway, I hope this is all helpful! </span></div><ul style="margin-top: 0cm;" type="disc"><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Mixing MMS with citric acid creates chlorine dioxide </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Chlorine dioxide is an oxidant, a free radical and a very effective industrial biocide (i.e. a powerful disinfectant) </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Sodium chlorite is used to make chlorine dioxide for use in (amongst other things) some water and food treatment in the U.S.<span style="mso-spacerun: yes;"> </span>As far as I can tell it was banned for these and some other uses in the E.U. in 2009 </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">If you are going to take MMS you should follow the instructions to the letter.<span style="mso-spacerun: yes;"> </span>Although, having said that, there would be no extra harm (in fact it would be advisable) in using more drops of citric acid than instructed.<span style="mso-spacerun: yes;"> </span>The greater the amount of citric acid, the greater the guarantee that there will be no sodium chlorite left over after the reaction, and sodium chlorite is more toxic than chlorine dioxide.<span style="mso-spacerun: yes;"> </span>Any less than 5 drops of citric acid per drop of MMS and there will definitely be some sodium chlorite left over </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">It’s a bit uncertain what the exact dosage of chlorine dioxide you end up taking with MMS is.<span style="mso-spacerun: yes;"> </span>The amount of chlorine dioxide in one properly prepared dose of MMS will change, depending on the size of each “drop” and the surrounding temperature.<span style="mso-spacerun: yes;"> </span>It could be higher in colder conditions.<span style="mso-spacerun: yes;"> </span>However: </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">One “dose” probably has about the same concentration of chlorine dioxide as the water in an average sized fish tank disinfected with Sanogene: </span><a href="http://www.lenntech.com/sanogene.htm" target="_blank"><span style="font-family: Calibri;">http://www.lenntech.com/sanogene.htm</span></a><span style="font-family: Calibri;"> </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">This is about 7-8 times the recommended limit (U.S.) on the concentration of chlorine dioxide in the drinking water supply </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">The approximate <i style="mso-bidi-font-style: normal;">concentration</i> of chlorine dioxide you get from one dose is higher than the (U.S.) maximum for drinking water but the <i style="mso-bidi-font-style: normal;">amount</i> of chlorine dioxide in one dose (one glass) is lower than the “minimal risk level” for most people (it depends on how little you are) </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">A number of doses gets well above both the drinking water limit and the “minimal risk level” but would nearly always be well below estimated toxic levels </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">It would be quite hard, but not impossible, to prepare and drink so much chlorine dioxide from MMS to get to levels at which there are likely to be toxic effects </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">The toxic levels have largely been established through tests on rats.<span style="mso-spacerun: yes;"> </span>Most of the known effects (of drinking chlorine dioxide solution) are on the development of rat pups </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Preparing MMS releases chlorine dioxide gas (this is why it smells when you mix a few drops of it with citric acid) and there are also toxic risks associated with that, so it’s advisable to keep the room well ventilated when making it </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">There is no selection of “good” or “bad” bacteria by chlorine dioxide (in fact, you can’t really generally define any such thing as “good” and “bad” bacteria).<span style="mso-spacerun: yes;"> </span>Chlorine dioxide kills them all and is capable of killing all types of cells (not just bacteria), including the cells that we’re all made of </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">You can expect to feel poorly when you take it because it kills all types of microbes in your gut, including those you need (and possibly the cells of your actual gut too), not because you’re “detoxing” (there’s actually no such thing as a “detox” – it’s basically a marketing invention) </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Chlorine dioxide behaves like a completely indiscriminate antibiotic in the gut.<span style="mso-spacerun: yes;"> </span>The apparent “detox” effects of nausea, vomiting, diarrhoea, stomach cramps etc. as you increase the dose of MMS are the same side effects you can get with antibiotics, which in a very crude sense is kind of what MMS is.<span style="mso-spacerun: yes;"> </span>MMS could leave you open to further infection, as they can </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">The effect on your gut is a bit like the effect of spraying your whole garden with weed killer.<span style="mso-spacerun: yes;"> </span>Weed killer doesn’t just kill weeds </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">It’s hard to imagine a drug where the side-effects can be expected to dominate over any potential benefit more than MMS </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">There’s no evidence MMS has any benefit.<span style="mso-spacerun: yes;"> </span>At the very least there is no evidence that it is both <i style="mso-bidi-font-style: normal;">safer and more effective</i> than existing treatments for anything </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">There are a wide range of things that could explain the apparent “success” of MMS – e.g. just getting better naturally anyway, the placebo effect etc.<span style="mso-spacerun: yes;"> </span>These at least need to be eliminated as possible explanations of how it “works” before it can be accepted to be effective </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">“Confounding” factors like these are one of the reasons it took ages for people to realise that things like leeches and blood-letting etc. didn’t actually work and for conventional, evidence based medicine to begin to develop </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Pretty much any treatment can appear to “work” (in a sense) for everyday illnesses and can confuse everybody – even the people who developed the treatment.<span style="mso-spacerun: yes;"> </span>They may think they’re onto a good thing but until proper tests are done they can’t say so for sure </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Remind me to tell you about my “miracle toe-stubbing therapy” one day… basically if someone tells you that doing something will make you better and you do it and then you get better, naturally you associate the fact that you’re better with doing whatever it was they told you to do </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">You usually take medicine when you’re feeling unwell and are generally (unless it’s serious) likely to get better naturally anyway.<span style="mso-spacerun: yes;"> </span>It’s very difficult to distinguish for yourself the effect of your immune system from the effect of the medicine </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Your friend knows plenty of people who’ve been taking MMS for years and are fit and well - which is excellent to hear!<span style="mso-spacerun: yes;"> </span>The pertinent question though, is not whether they are fit and well but whether they are more, the same or less fit and well than they would have been had they not taken it – and that’s impossible to answer </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">The “evidence” for MMS working in serious illnesses is deeply flawed.<span style="mso-spacerun: yes;"> </span>It’s largely anecdotal (so impossible to substantiate) but even where trials have been done they’ve been very poorly conducted and designed and the results misrepresented somewhat </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Established, “conventional”, evidence based medicine is an enormous success story, in terms of improvements in infant mortality, life expectancy, survival and recovery rates from many illnesses etc. - it plainly works </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Death and illness are natural.<span style="mso-spacerun: yes;"> </span>Is successfully preventing death and curing illness unnatural? </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">An apple a day keeps the doctor away is fundamentally true – eating at least five fresh fruit and veg a day and avoiding processed junk is a very good idea indeed – for general health </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Your immune system is pretty good at getting you better from many everyday bugs etc. so reaching straight for the medicine cabinet (full of side-effects), whatever it contains, is not necessarily the best and most natural idea – but if you’re really ill conventional medicine is plainly a very good idea </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Which is more natural: MMS or Sanogene (see above)? </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">MMS could be very unsafe in a number of ways if you don’t follow the instructions </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Mixing MMS the right way releases quite a bit of chlorine dioxide gas.<span style="mso-spacerun: yes;"> </span>If you mix sodium chlorite and citric acid drops straight into a glass of water (instead of adding the water last) then there is more than enough water present to dissolve all of the chlorine dioxide generated – so it all stays in the water rather than being released to the atmosphere </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">You multiply the dose of chlorine dioxide about 8-fold just by mixing things together like this – i.e. in exactly the right amount but in slightly the wrong order.<span style="mso-spacerun: yes;"> </span>It wouldn’t smell so bad but it would taste worse and would be worse for you </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Drinking one (or even part of a) bottle of neat MMS (without any mixing with citric acid or diluting it) might not kill you but would be very likely to lead to kidney failure at the very least (a Chinese man attempted suicide by drinking the equivalent of about a third of a bottle of neat MMS and suffered kidney failure - but didn’t die) </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">If you mixed one bottle of neat MMS with citric acid (rather than just a few drops) in the proportions suggested you could release enough chlorine dioxide gas to kill you (I doubt anyone would do this on purpose but there’s always a risk of spills) </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">There is no reason for neat MMS to be such a dangerously high concentration of sodium chlorite (28%) - preparing MMS as instructed produces much more chlorine dioxide than can be dissolved in a few drops of water, so most if it is released as a gas </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">You’d get the same dose<i style="mso-bidi-font-style: normal;"> </i>of chlorine dioxide solution if neat MMS was only ~4% sodium chlorite (and mixing it with citric acid would release little or no chlorine dioxide gas). </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">It goes without saying that 4% sodium chlorite solution would be generally much safer to have around the house! </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">It could be unsafe even if you do follow instructions but no-one really knows.<span style="mso-spacerun: yes;"> </span>No-one knows the long term effects of the estimated MMS dosage levels on people but there are reasons to expect that they could be potentially damaging simply because of the oxidant/free-radical nature of chlorine dioxide </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Maybe big drug companies are evil – but that doesn’t lead to the conclusion that any alternatives are necessarily not</span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Drug companies spend a lot of time and effort trying to get new drugs round the FDA – they certainly don’t see the FDA as friends </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">That big drug companies are evil doesn’t mean established conventional medicine is evil, just that there can be (sometimes very serious) issues, especially with new, insufficiently tested and heavily promoted treatments (which is basically what MMS is too - and with MMS there’s no reason it might even be expected to work) </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">The position the FDA are in (with criticism from both Jim Humble and from supporters of MMS) is very similar to the position any regulator can find itself in when a big drug company wants to get early approval for a new drug (i.e. under pressure from the drug company and from patients’ groups).<span style="mso-spacerun: yes;"> </span>The regulator will often refuse approval because trials are not complete and there is insufficient evidence that the drug is safe and effective.<span style="mso-spacerun: yes;"> </span>There’s no difference between Jim Humble and big drug companies in this context<span style="mso-spacerun: yes;"> </span></span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">It may not always be perfect (I don’t know) but I think the FDA are just the fall guy (and probably the good guy) in this instance.<span style="mso-spacerun: yes;"> </span>It’s just applying the law (as it should) and the law (with respect to claims made in drug marketing) has demonstrably been broken </span><a href="http://www.fda.gov/ICECI/EnforcementActions/WarningLetters/ucm152422.htm" target="_blank"><span style="font-family: Calibri;">http://www.fda.gov/ICECI/EnforcementActions/WarningLetters/ucm152422.htm</span></a><span style="font-family: Calibri;"> </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Claims for the benefits of MMS are unfounded and people are, in effect, being used as lab rats in a (very poorly designed) drug trial on a potentially dangerous substance </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Until proper (“proper” includes ethical as well as scientific considerations) trials have been done anyone using MMS is doing so simply on the basis of trust in Jim Humble and not on the basis of evidence.<span style="mso-spacerun: yes;"> </span>I don’t trust Jim Humble much myself, if at all - but that’s just a personal opinion and it’s up to each individual to decide for themselves on that </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Not having enough money to be able to disprove the hypothesis that MMS is dangerous (which is a very valid hypothesis given the oxidant, free radical properties of chlorine dioxide), or that it is not effective, is not a valid reason for encouraging people to take it, no matter what anyone thinks the benefits could be </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">Much of the blurb, the explanation of “how it works”, is pseudo-science – i.e. it (mis)uses lots of scientific terms and concepts to make apparent sense but misses out key facts and details, so completely misrepresents the action of chlorine dioxide on cells in the body </span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">MMS has all the signs of a scam, particularly in the way the “science” is presented and distorted</span></li><li class="MsoNormal" style="margin: 0cm 0cm 10pt; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;"><span style="font-family: Calibri;">If you really don’t trust the FDA and want to look for a conspiracy you could argue (though I think this is a fairly weak way to argue) that: </span></li></ul><div class="MsoNormal" style="margin: 0cm 0cm 10pt 35.45pt;"><span style="font-family: Calibri;">a) it could be in the FDA’s interest (as chlorine dioxide is already present in U.S. water supplies and food) to promote (not challenge) the safety of chlorine dioxide</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt 35.45pt;"><span style="font-family: Calibri;">b) the FDA’s attitude to MMS could be based on knowledge that the current limits (in water etc.) are not as safe as is stated by government agencies in public (and therefore that the higher levels in most doses of MMS are more dangerous than is stated)</span></div><div class="MsoNormal" style="margin: 0cm 0cm 10pt 35.45pt;"><span style="font-family: Calibri;">c) liberal laws covering the claims that can be made about a drug or treatment in any marketing would be of benefit to big drug companies.<span style="mso-spacerun: yes;"> </span>If the FDA is in the hands of these companies then regulations are probably too loose rather than too tight.<span style="mso-spacerun: yes;"> </span>Even then, MMS has contravened them</span></div>Our Frankhttp://www.blogger.com/profile/02765024367933720371noreply@blogger.com1