Algebra — Complex Numbers, Quadratic Equations, Progressions & Binomial Theorem — Engineering Mathematics Notes

Last reviewed 16 Sept 2026 · 7 min read

Complex numbers

A complex number $z = x + i y$ , where $i = - 1$ ( $i^{2} = - 1$ , $i^{3} = - i$ , $i^{4} = 1$ ); $x = Re (z)$ , $y = Im (z)$ .

Modulus $∣ z ∣ = r = x^{2} + y^{2}$ .
Argument $θ = ar g z$ with $tan θ = y / x$ , chosen in the correct quadrant; principal argument in $(- π, π]$ .
Conjugate $\overset{z}{ˉ} = x - i y$ ; $z \overset{z}{ˉ} = ∣ z ∣^{2}$ .
Euler's formula: $e^{i θ} = cos θ + i sin θ$ ; $e^{iπ} + 1 = 0$ .

$∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣$ ; $ar g (z_{1} z_{2}) = ar g z_{1} + ar g z_{2}$ .
$\frac{z _{1}}{z _{2}} = \frac{∣ z _{1} ∣}{∣ z _{2} ∣}$ ; $ar g \frac{z _{1}}{z _{2}} = ar g z_{1} - ar g z_{2}$ .
Triangle inequality: $∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣$ .
Division: multiply numerator and denominator by the conjugate of the denominator.

(cos θ + i sin θ)^{n} = cos n θ + i sin n θ

The $n$ distinct $n$ th roots of $z = r e^{i θ}$ :

z_{k} = r^{1/ n} [cos \frac{θ + 2 k π}{n} + i sin \frac{θ + 2 k π}{n}], k = 0, 1, \dots, n - 1

They lie on a circle of radius $r^{1/ n}$ , equally spaced by $2 π / n$ .

$1, ω = \frac{- 1 + i 3}{2}, ω^{2} = \frac{- 1 - i 3}{2}$

a x^{2} + b x + c = 0, x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Discriminant $D = b^{2} - 4 a c$	Roots
$D > 0$	Real and distinct (rational if $D$ is a perfect square and coefficients rational)
$D = 0$	Real and equal ( $- b /2 a$ )
$D < 0$	Complex conjugates

Sum $α + β = - \frac{b}{a}$ ; product $α β = \frac{c}{a}$ .
$α^{2} + β^{2} = (α + β)^{2} - 2 α β$ ; $(α - β)^{2} = (α + β)^{2} - 4 α β$ ; $α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$ .
Equation with roots $α, β$ : $x^{2} - (α + β) x + α β = 0$ .
For a cubic $a x^{3} + b x^{2} + c x + d = 0$ : $\sum α = - b / a$ , $\sum α β = c / a$ , $α β γ = - d / a$ .
Quadratic $a x^{2} + b x + c$ has a minimum (if $a > 0$ ) or maximum (if $a < 0$ ) at $x = - b /2 a$ , value $- D /4 a$ .

FormulaArithmetic progression (AP)

Terms $a, a + d, a + 2 d, \dots$

T_{n} = a + (n - 1) d, S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)

Arithmetic mean of $a$ and $b$ : $A = \frac{a + b}{2}$

FormulaGeometric progression (GP)

Terms $a, a r, a r^{2}, \dots$

T_{n} = a r^{n - 1}, S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1} (r \neq = 1)

Infinite GP ( $∣ r ∣ < 1$ ): $S_{\infty} = \frac{a}{1 - r}$ Geometric mean of $a$ and $b$ (positive): $G = ab$

FormulaHarmonic progression (HP)

Reciprocals form an AP. Harmonic mean of $a$ and $b$ : $H = \frac{2 ab}{a + b}$ For positive numbers: $A \geq G \geq H$ and $G^{2} = A H$ .

Useful sums: $\sum n = \frac{n ( n + 1 )}{2}$ ; $\sum n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ ; $\sum n^{3} = [\frac{n ( n + 1 )}{2}]^{2}$ .

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