Algebra

Complex Numbers

Basic Algebraic Operations

Commutative Property:

$\begin{aligned} a + b & = b + a \\ a \times b & = b \times a \\ \end{aligned}$

Associative Property:

$\begin{aligned} (a + b) + c & = a + (b + c) \\ (a \times b) \times c & = a \times (b \times c) \\ \end{aligned}$

Distributive Property:

$\begin{aligned} a \cdot (b + c) & = ab + ac \\ (a + b)(c + d) & = ac + ad + bc + bd \\ \end{aligned}$

Exponents and Powers

$\begin{aligned} a^m \times a^n & = a^{m+n} \\ \frac{a^m}{a^n} & = a^{m-n} \\ (a^m)^n & = a^{m \times n} \\ a^0 & = 1 \\ a^1 & = a \\ a^{-n} & = \frac{1}{a^n} \\ a^{\frac{m}{n}} & = \sqrt[n]{a^m} \\ \end{aligned}$

Polynomials

Quadratic:

$\begin{aligned} (a + b)^2 & = a^2 + 2ab + b^2 \\ & = (a - b)^2 + 4ab \\ (a - b)^2 & = a^2 - 2ab + b^2 \\ & = (a + b)^2 - 4ab \\ a^2 + b^2 & = (a + b)^2 - 2ab \\ & = (a - b)^2 + 2ab \\ a^2 - b^2 & = (a - b)(a + b) \\ (a + b)^2 + (a - b)^2 & = 2(a^2 + b^2) \\ (a + b)^2 - (a - b)^2 & = 4ab \\ (x + a)(x + b) & = x^2 + (a + b)x + ab \\ (a + b + c)^2 & = a^2 + b^2 + c^2 + 2(ab + ac + bc) \\ \end{aligned}$

Cubic:

$\begin{aligned} (a + b)^3 & = a^3 + b^3 + 3ab(a + b) \\ & = a^3 + 3a^2b + 3ab^2 + b^3 \\ (a - b)^3 & = a^3 - b^3 - 3ab(a - b) \\ & = a^3 - 3a^2b + 3ab^2 - b^3 \\ a^3 + b^3 & = (a + b)(a^2 - ab + b^2) \\ a^3 - b^3 & = (a - b)(a^2 + ab + b^2) \\ (a + b + c)^3 & = a^3 + b^3 + c^3 − 3(a + b)(b + c)(c + a) \\ \end{aligned}$

Special:

$\begin{aligned} a^2 + b^2 + c^2 - ab - bc - ca & = \frac{1}{2}\left[ (a - b)^2 + (b - c)^2 + (c - a)^2 \right] \\ a^3 + b^3 + c^3 - 3abc & = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \\ & = (a + b + c)^3 − 3(a + b + c)(ab + bc + ca) \\ \end{aligned}$

Higher Order Polynomials:

$\begin{aligned} (a \pm b)^4 & = a^4 \pm 4a^3b + 6a^2b^2 \mp 4ab^3 + b^4 \\ a^4 - b^4 & = (a - b)(a + b)(a^2 + b^2) \\ (a \pm b)^5 & = a^5 \pm 5a^4b + 10a^3b^2 \mp 10a^2b^3 + 5ab^4 \pm b^5 \\ a^5 - b^5 & = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) \\ \end{aligned}$

Quadratic Equations

For a quadratic equation of the form $ ax^2 + bx + c = 0 $,
the solution is given by the quadratic formula.

$ x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

Discriminant: $ \Delta \text{ (or) } D = b^2 - 4ac $

Nature of roots:

  • Distinct real roots: $ \Delta > 0 $
  • Equal real roots: $ \Delta = 0 $
  • Distinct complex roots: $ \Delta < 0 $
Completing the Square
  • $ ax^2 + bx + c = \left(x + \dfrac{b}{2a}\right)^2 - \dfrac{b^2-4ac}{4a^2} $
  • If $a = 1$ then,
    $ x^2 + bx + c = \left(x + \dfrac{b}{2}\right)^2 - \dfrac{b^2}{4} + c $

Logarithms

$ a^x = b \implies x = \log_a b $

$\begin{aligned} \ln a & = \log_e a \\ \log_a 1 & = 0 \\ \log_a a & = 1 \\ \log_a x^n & = n \log_a x \\ \log_{a^n} x & = \dfrac{1}{n} \log_a x \\ \log_a \sqrt[n]{x} & = \dfrac{1}{n} \log_a x \\ \log_a(x \cdot y) & = \log_a x + \log_a y \\ \log_a\left(\dfrac{x}{y}\right) & = \log_a x - \log_a y \\ a^{\log_a x} & = x \\ \log_a b & = \dfrac{1}{\log_b a} \\ \log_a b & = \dfrac{\log_c b}{\log_c a} \\ \end{aligned}$

Subsections of Algebra

Complex Numbers

Conjugate

If $z = x + iy$ is a complex number, $\bar{z} = x - iy$

Properties

For any complex number $z, z_1, z_2$, we have:
$ \begin{aligned} \text{(i)} \quad & \overline{\overline{z}} = z \\ \text{(ii)} \quad & z + \overline{z} = 2 \operatorname{Re}(z), \quad z + \overline{z} = 0 \Leftrightarrow z \text{ is purely imaginary.} \\ \text{(iii)} \quad & z - \overline{z} = 2i \operatorname{Im}(z), \quad z - \overline{z} = 0 \Leftrightarrow z \text{ is purely real.} \\ \text{(iv)} \quad & \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \\ \text{(v)} \quad & \overline{z_1 - z_2} = \overline{z_1} - \overline{z_2} \\ \text{(vi)} \quad & \overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2} \\ \text{(vii)} \quad & \overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}, \quad z_2 \neq 0 \\ \text{(viii)} \quad & z_1 \overline{z_2} + \overline{z_1} z_2 = 2 \operatorname{Re}(\overline{z_1} z_2) = 2 \operatorname{Re}(z_1 \overline{z_2}) \\ \text{(ix)} \quad & \overline{z^n} = (\overline{z})^n \\ \text{(x)} \quad & \text{If } z = f(z_1), \text{ then } \overline{z} = f(\overline{z_1}) \\ \text{(xi)} \quad & \text{If } z = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}, \text{ then } \overline{z} = \begin{vmatrix} \overline{a_1} & \overline{a_2} & \overline{a_3} \\ \overline{b_1} & \overline{b_2} & \overline{b_3} \\ \overline{c_1} & \overline{c_2} & \overline{c_3} \end{vmatrix} \\ \text{(xii)} \quad & z \cdot \overline{z} = \{\operatorname{Re}(z)\}^2 + \{\operatorname{Im}(z)\}^2 \end{aligned} $