XSL FO Sample Copyright © 2002-2008 Antenna House, Inc. All rights reserved. - - Sample of MathML 1 Japanese High School Text "Mathematics B" N -th power root In general, a non-zero complex number, $a=r\left(\mathrm{cos}\theta +i\mathrm{sin}\theta \right)$]]>, has the following n complex numbers as n-th power roots. ${z}_{n}=\sqrt[n]{r}\left\{\mathrm{cos}\left(\frac{\theta }{n}+\frac{{360}^{\circ }}{n}×k\right)+i\mathrm{sin}\left(\frac{\theta }{n}+\frac{{360}^{\circ }}{n}×k\right)\right\}$]]> $\left(k=0,1,2,\cdots ,n-1\right),$]]> where $\sqrt[n]{r}$]]> is a positive n-th power root of a positive number r. An angle made by two vectors Suppose two vectors $\stackrel{\to }{a}=\left({a}_{1},{a}_{2}\right)$]]> and $\stackrel{\to }{b}=\left({b}_{1},{b}_{2}\right)$]]> are non-zero vectors, $\theta$]]> is the angle made by these two vectors, and ${0}^{\circ }\leqq \theta \leqq 18{0}^{\circ }$]]>. Since $\stackrel{\to }{a}\cdot \stackrel{\to }{b}=|\stackrel{\to }{a}||\stackrel{\to }{b}|\mathrm{cos}\theta$]]>, $\mathrm{cos}\theta =\frac{\stackrel{\to }{a}\cdot \stackrel{\to }{b}}{|\stackrel{\to }{a}||\stackrel{\to }{b}|}=\frac{{a}_{1}{b}_{1}+{a}_{2}{b}_{2}}{\sqrt{{{a}_{1}}^{2}+{{a}_{2}}^{2}}\sqrt{{{b}_{1}}^{2}+{{b}_{2}}^{2}}}$]]> A point that divides a segment into m:n Suppose two points, $\text{A}\left(\stackrel{\to }{a}\right)$]]> and $\text{B}\left(\stackrel{\to }{b}\right)$]]>, are not identical, $m+n\ne 0$]]>, and a point, $\text{P}\left(\stackrel{\to }{p}\right)$]]>, divides a segment AB into $m:n$]]>. Then, $\stackrel{\to }{p}=\frac{n\stackrel{\to }{a}+m\stackrel{\to }{b}}{n+m}$]]>Particularly, when the midpoint of a segment AB is $\text{M}\left(\stackrel{\to }{m}\right)$]]>, $\stackrel{\to }{m}=\frac{\stackrel{\to }{a}+\stackrel{\to }{b}}{2}$]]>Probability distribution Suppose a random variable X can take the following n values ${x}_{1}$]]>, ${x}_{2}$]]>, ......, ${x}_{n}$]]>, and the probability of an event $X={x}_{i}$]]> is ${p}_{i}$]]>. Then, Mean $m=E\left(X\right)=\sum _{i=1}^{n}{x}_{i}{p}_{i}$]]> Variance $V\left(X\right)=E\left({\left(X-m\right)}^{2}\right)=\sum _{i=1}^{n}{\left({x}_{i}-m\right)}^{2}{p}_{i}$]]> $V\left(X\right)=E\left({X}^{2}\right)-{m}^{2}=\sum _{i=1}^{n}{x}_{i}^{2}{p}_{i}-{m}^{2}$]]> Standard deviation $\sigma \left(X\right)=\sqrt{V\left(X\right)}$]]> Matrix Presentation $A=\begin{array}{cc}& \begin{array}{ccc}m& & n\\ ⎴& & ⎴\end{array}\\ \begin{array}{c}r\left[\\ s\left[\end{array}& \left(\begin{array}{ccc}{A}_{11}& ⋮& {A}_{12}\\ \cdots & ⋮& \cdots \\ {A}_{21}& ⋮& {A}_{22}\end{array}\right)\end{array}$]]>