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Complex Numbers

Abraham de Moivre was a talented mathematician. He wanted to become a professor of mathematics in England, but as a Frenchman, he faced discrimination. His first paper came from studying Newton’s Principia. In 1697, he became a Fellow of the Royal Society. In 1710, he helped settle the Newton–Leibniz calculus dispute. He contributed to analytic geometry and probability theory. He introduced the approximation of the binomial distribution by the normal distribution for a large number of trials. He also studied mortality statistics and annuities. De Moivre’s formula connected trigonometry with complex numbers. He even predicted his own death using his sleep pattern.

Introduction

For real numbers, . To handle negative squares, we define such that . Example: . A complex number is written as where and are real numbers and . Real part: Re(z) = a, Imaginary part: Im(z) = b. Purely imaginary: a = 0, purely real: b = 0, 0 + i0 is both purely real and purely imaginary. Equality: . Powers of i: i = i, i^2 = -1, i^3 = -i, i^4 = 1, i^{4n} = 1, i^{4n+1} = i, i^{4n+2} = -1, i^{4n+3} = -i.

Geometrical Representation

A complex number is a point P(x,y) on the Argand plane, where the x-axis is real and the y-axis is imaginary.

Modulus and Argument

Modulus: |z| = √(x^2 + y^2). Argument: θ = tan^(-1)(y/x). Principal value: -π < θ ≤ π.

Polar and Euler Form

If r = |z|, then z = r(cosθ + i sinθ) (Polar form) and z = r e^(iθ) (Euler form).

Uni-modular Complex Numbers

If |z| = 1, z lies on the unit circle. 1/z = cosθ - i sinθ.

Algebra of Complex Numbers

For z1 = a + ib, z2 = c + id: Addition: z1 + z2 = (a+c) + i(b+d), Subtraction: z1 - z2 = (a-c) + i(b-d), Multiplication: z1 z2 = (ac - bd) + i(ad + bc), Division: z1 / z2 = (ac + bd)/(c^2 + d^2) + i(bc - ad)/(c^2 + d^2).

Geometrical Meaning

Addition → z1 + z2 is the diagonal of a parallelogram with sides z1 and z2. Product → multiply moduli, add arguments: z1 z2 = r1 r2 (cos(θ1 + θ2) + i sin(θ1 + θ2)). Division → divide moduli, subtract arguments: z1 / z2 = (r1 / r2)(cos(θ1 - θ2) + i sin(θ1 - θ2)).

De Moivre’s Theorem

(cosθ + i sinθ)^n = cos(nθ) + i sin(nθ) for any real θ and integer n.

Illustrations

Illustration 1: Find x^4 + 9x^3 + 35x^2 - x + 4 if x = -5 + 4i.

Solution: x + 5 = 4i. Squaring: x^2 + 10x + 25 = -16 → x^2 + 10x + 41 = 0. Then x^4 + 9x^3 + 35x^2 - x + 4 = (x^2 + 10x + 41)(x^2 - x + 4) - 160 = 0 - 160 = -160.

Illustration 2: Express 1/(cosθ + i sinθ) in a + ib form.

Solution: Multiply numerator and denominator by (cosθ - i sinθ):

1/(cosθ + i sinθ) = (cosθ - i sinθ)/(cos^2θ + sin^2θ) = cosθ - i sinθ = cosθ - i sinθ.

Illustration 3: Represent sinθ - i cosθ in polar form (θ acute).

Real part > 0, Imaginary part < 0 → Argument is negative acute angle. Modulus: √(sin^2θ + cos^2θ) = 1. Polar form: z = 1(cos(-α) + i sin(-α)) = 1 e^(-iα).

Illustration 4: Represent (1/2)(cos(π/3) + i sin(π/3)) in polar form.

r = 1/2, θ = π/3 → Polar form: z = (1/2)(cos(π/3) + i sin(π/3)) = (1/2) e^(i π/3).

Finding Principal Argument:

Step 1: tanθ = y/x. Step 2: Determine quadrant based on signs of x and y. Step 3: Argument = θ, -π+θ, π+θ, or -θ depending on quadrant.

Summary of Key Formulas:

Modulus: |z| = √(x^2 + y^2)

Argument: θ = tan^(-1)(y/x)

Polar form: z = r(cosθ + i sinθ)

Euler form: z = r e^(iθ)

Addition: z1 + z2 = (a+c) + i(b+d)

Subtraction: z1 - z2 = (a-c) + i(b-d)

Multiplication: z1 z2 = (ac-bd) + i(ad+bc)

Division: z1/z2 = (ac+bd)/(c^2+d^2) + i(bc-ad)/(c^2+d^2)

De Moivre: (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ)