Algebra

Numbers

1. Natural numbers $N=\{1,2,3,4,...\}$
2. Whole numbers $W=\{0,1,2,3,...\}$
3. Integers $I=\{0,\pm 1,\pm 2,...\}$
4. Ration numbers $Q=\{x|x=\frac{a}{b},wherea,b\in I,b\ne 0\}$
5. Irrational numbers $Q^{\prime }=\sqrt[p]{x}$ such that $p^{th}$ root can't find correctly.
6. Read numbers = $Q\cup Q^{\prime }$

Ratio

If $\frac{a}{b}=\frac{c}{d}$ then

1. $\frac{b}{a}=\frac{d}{c}$
2. $\frac{a}{c}=\frac{b}{d}$
3. $\frac{a+b}{b}=\frac{c+d}{d}$
4. $\frac{a-b}{b}=\frac{c-d}{d}$
5. $\frac{a+b}{a-b}=\frac{c+d}{c-d}$
6. $\frac{a}{b}=\frac{c}{d}=\frac{al+mc}{bl+md}$
7. $\frac{a+c}{b+d}=\frac{\sqrt{a^2+c^2}}{\sqrt{b^2+d^2}}$

General Formulas

• $(a+b{)}^2=a^2+2ab+b^2$
• $(a-b{)}^2=a^2-2ab+b^2$
• $(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$
• $(a+b{)}^3=a^3-3a^{2}b+3a{b}^2+b^3$
• $(a+b{)}^n=a^n+{}^n{C}_1{a}^{n-1}b+{}^n{C}_2{a}^{n-2}{b}^2+...+b^n$
• $(a-b{)}^n=a^n-{}^n{C}_1{a}^{n-1}b+{}^n{C}_2{a}^{n-2}{b}^2-...+(-1{)}^n{b}^n$
• $a^2-b^2=(a+b)(a-b)$
• $a^3-b^3=(a-b)(a^2+ab+b^2)=(a-b{)}^3+3ab(a-b)$
• $a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b{)}^3-3ab(a+b)$