Sample of MathML 1 ｜ Sample of MathML 1 Japanese High School Text "Mathematics B" N -th power root In general, a non-zero complex number, $a=r(\cos \theta +i\sin \theta )$]]>, has the following n complex numbers as n-th power roots. $$z_n=\sqrt[n]{r}\left\{\cos \left(\frac{\theta }{n}+\frac{{360}^{\circ }}{n}\times k\right)+i\sin \left(\frac{\theta }{n}+\frac{{360}^{\circ }}{n}\times k\right)\right\}$$]]> $$(k=0,1,2,\cdots ,n-1),$$]]> 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 $\overrightarrow{a}=(a_1,a_2)$]]> and $\overrightarrow{b}=(b_1,b_2)$]]> are non-zero vectors, $\theta$]]> is the angle made by these two vectors, and $0^{\circ }\leqq \theta \leqq 180^{\circ }$]]>. Since $\overrightarrow{a}\cdot \overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta$]]>, $$\cos \theta =\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|}=\frac{a_1{b}_1+a_2{b}_2}{\sqrt{{a_1}^2+{a_2}^2}\sqrt{{b_1}^2+{b_2}^2}}$$]]> A point that divides a segment into m:n Suppose two points, $\text{A}(\overrightarrow{a})$]]> and $\text{B}(\overrightarrow{b})$]]>, are not identical, $m+n\ne 0$]]>, and a point, $\text{P}(\overrightarrow{p})$]]>, divides a segment AB into $m:n$]]>. Then, $$\overrightarrow{p}=\frac{n\overrightarrow{a}+m\overrightarrow{b}}{n+m}$$]]>Particularly, when the midpoint of a segment AB is $\text{M}(\overrightarrow{m})$]]>, $$\overrightarrow{m}=\frac{\overrightarrow{a}+\overrightarrow{b}}{2}$$]]>Probability distribution Suppose a random variable X can take the following n values $x_1$]]>, $x_2$]]>, ......, $x_{\mathrm{n}}$]]>, and the probability of an event $X=x_i$]]> is $p_i$]]>. Then, Mean $$m=E(X)={\displaystyle \sum _{i=1}^n{x}_i{p}_i}$$]]> Variance $$V(X)=E({(X-m)}^2)={\displaystyle \sum _{i=1}^n{(x_i-m)}^2{p}_i}$$]]> $$V(X)=E(X^2)-m^2={\displaystyle \sum _{i=1}^n{x}_i^2{p}_i}-m^2$$]]> Standard deviation $$\sigma (X)=\sqrt{V(X)}$$]]> Matrix Presentation $A=\begin{array}{ccccccc}& & & m& & n& \\ & & & ⎴& & ⎴& \\ r& [& (& A_{11}& ⋮& A_{12}& )\\ & & \cdots & ⋮& \cdots \\ s& [& A_{21}& ⋮& A_{22}\end{array}$]]>