Complex Numbers. Introduction.

Assumption is the mother of all screw-ups, Anonymous.

Numbers, Numbers, and More Numbers

A natural number is a mathematical object. It is a number that occurs commonly in nature. As such, it is a whole, non-negative number,that is, a member of the sequence 0, 1, 2, 3, …, obtained by starting at 0 and repeatedly adding 1. The set of natural numbers is denoted by ℕ. Some texts start ℕ at 1; here we include 0. In set‐builder notation: ℕ = {0, 1, 2, 3, …}.

An integer number is a whole number that can be positive or negative, but does not have fractions or decimal part. Integers are the collection of whole numbers and negative numbers. The set of integer numbers is denoted by ℤ. Hence, ℤ = { ···, -3, -2, -1, 0, 1, 2, 3, ···}.

A rational number is a number that is of the form p/q where p and q are integers and q is not equal to 0. The set or rational numbers is denoted by the doublestruck capital letter ℚ. It is the set of all fractions with integer numerators and non-zero integer denominators. ℚ = {$\frac{p}{q}|~ p, q \in ℤ, q \ne 0$} where p and q are integers (with q not equal to zero). Rational numbers can have either a terminating (e.g., $\frac{1}{2} = 0.5$) or repeating decimal (e.g., $\frac{1}{3} = 0.333...$) representation.

Rational numbers fill in “gaps” between integers but still form a countable set. In calculus, ℚ is dense in ℝ: between any two reals there’s a rational.

An irrational number is any number that is not a rational number, i.e., it cannot be expressed as a fraction ratio of two integers. The decimal representation of an irrational number goes on forever without repeating -non-terminating and non-repeating. The following are examples of irrational numbers: $\sqrt{2} = 1.41421356..., \pi = 3.14159265..., \sqrt{3}$, e (Euler’s number), and φ (the golden ration).

Together, rationals and irrationals form the real numbers ℝ, a complete ordered field It is denoted by ℝ. It includes the positive and negative integers, the fractions made from those integers (or rational numbers) and also the irrational numbers.

Complex Numbers

Complex numbers are fundamental in various fields of mathematics, physics, and engineering.

A complex number is specified by an ordered pair of real numbers (a, b) ∈ ℝ² and expressed or written in the form z = a + bi, where a and b are real numbers, and i is the imaginary unit, defined by the property i² = −1 ⇔ i = $\sqrt{-1}$, e.g., 2 + 5i, $7\pi + i\sqrt{2}.$ ℂ= { a + bi ∣a, b ∈ ℝ}.

ℂ is algebraically closed — every polynomial equation has a root in ℂ

Equality, Real & Imaginary Parts

Two complex numbers a + ib and c + id are equal if and only if a = c and b = d. a + bi = c + di ⟺ a = c, b = d.

In the complex number z = a + bi: a is called the real part of the complex number, denoted as ℜ(z) = a. b is called the imaginary part, denoted as ℑ(z) = b. Example: z = 3 +4i, the real part is ℜ(z) = 3 and the imaginary part is ℑ(z) = 4.

Geometric Interpretation

We often use symbols z, w, ζ to denote complex numbers. They are the points on the plane, expressed as ordered pairs (a, b), where the real part “a” represents the coordinate for the horizontal axis and the imaginary part “b” represents the coordinate for the vertical axis. The set of complex numbers is denoted or expressed as ℂ = ℝ² = {(a, b) or a +bi: a, b ∈ ℝ}.

a + i·0, 0 + i·b, 0 + i·1, 0 + i·(-1), a + i·(-b) are often abbreviated as a, ib or bi, i, -i, and -ib respectively. We identify a real number a with the complex number a + i·0. In this sense, the real numbers are contained within the complex numbers. Numbers of the form 0 + ib = ib are called purely imaginary numbers.

Arithmetic of Complex Numbers

• Addition: (a +bi) + (c +di) = (a +c) + (b +d)i, e.g., (1 + 3i) + (3 + 4i) = 4 + 7i.
• Subtraction: (a +bi) − (c +di) = (a −c) + (b −d)i, e.g., (1 + 3i) - (3 + 4i) = -2 -i.
• Multiplication. We can multiply complex numbers by applying the distributive property and using i² = −1: (a + bi)(c + di) = (ac −bd) + (ad + bc)i, e.g., (1 + 3i)·(4 + 2i) = 4 + 6i² + 12i + 2i =[i²=-1] 4 -6 + 14i = -2 + 14i.

Complex conjugates & Modulus

The complex conjugate of a complex number z = a + bi is denoted by $\bar z$ and is defined as $\bar z = a -bi$, e.g, $\overline{3 + i} = 3 -i$, $\overline{\sqrt{2}-\frac{\pi}{3}i } = \sqrt{2} +\frac{\pi}{3}i$. If you plot the complex number on the complex plane, the conjugate is a reflection across the real axis (x-axis), changing the sign of the imaginary component while the real part remains unchanged.

Properties:

• Multiplying a complex number by its conjugate yields a real number: $z·\bar z = (a + bi)(a -bi) = a^2 + b^2, ∣z∣ = \sqrt{a^2 + b^2} = \sqrt{z·\bar z}$, e.g., z = 3 + 4i, $z·\bar z = (3 + 4i)(3 - 4i) = 3^2 + 4^2 = 9+16 = 25.$

The modulus of a complex number is the square root of the product of the number itself and its conjugate. (vi)

• Suppose z = $\bar z ⇒ b = 0$, z is a real number. A complex number is real if and only if its imaginary part is zero, making it equal to its conjugate.
• Similarly, if z = $-\bar z ⇒ a = 0$, meaning z is a pure imaginary number. If a complex number equals its negative conjugate, then it is purely imaginary.
• $\bar {\bar z} = z$
• $Re(z) = \frac{z+ \bar z}{2}$
• $Im(z) = \frac{z - \bar z}{2i}$
• The conjugate of the sum of two complex numbers is the sum of their conjugates: $\overline {z_1 + z_2} = \overline {(a + bi) + (c+di)} = \overline{(a + c) + (b+d)i} = (a + c) - (b+d)i = (a -bi) + (c-di) = \overline {z_1}+\overline {z_2}$ (iv).
• Similarly, $\overline {z_1 · z_2} = \overline {z_1}· \overline {z_2}$ (v). The modulus of the product of two complex numbers is equal to the product of their moduli.

$\overline {z}\overline {w} =\text{[vi]}~ \sqrt{zw · \overline {zw}} =\text{[v]}~ \sqrt{z·w·\overline{z}·\overline {w}} = \sqrt{z·\overline{z}·w·\overline{w}} =\text{[vi]}~ |z|·|w|$.

• |z| ≥ 0
• |z| = 0 ↔ z = 0
• Triangle inequality: |z + w| ≼ |z| + |w|

Division of Complex Numbers

To divide complex numbers, z₂≠0 (x₂ and y₂ are not simultaneously zero), we eliminate the imaginary unit from the denominator by multiplying both the numerator and the denominator by the complex conjugate of the denominator. $\frac{z_1}{z_2} = \frac{x_1+iy_1}{x_2+iy_2} = \frac{x_1+iy_1}{x_2+iy_2}\frac{x_2-iy_2}{x_2-iy_2}=\frac{x_1x_2 + y_1y_2 + i(x_2y_1-x_1y_2)}{x_2²-y_2²} = \frac{x_1x_2 + y_1y_2}{x_2²-y_2²} + i(\frac{x_2y_1-x_1y_2}{x_2²-y_2²})$

Example: Calculate $\frac{2+i}{1-3i}$

$\frac{2+i}{1-3i}$ =[To simplify the expression, multiply both numerator and denominator by the conjugate of the denominator:] $\frac{2+i}{1-3i}·\frac{1+3i}{1+3i} = \frac{-1+7i}{10} =[\text{Thus, the simplified form is:}] \frac{-1}{10}+\frac{7}{10}i$.

Properties

• Commutativity: $z_1 + z_2 = z_2 + z_1, z_1·z_2 = z_2·z_1$
• Associativity: $z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3, z_1 · (z_2 · z_3) = (z_1 · z_2) · z_3$
• Distributivity: $z_1·(z_2 + z_3) = z_1·z_2 + z_1·z_3$
• z = a + ib, -z = -a -ib, z + (-z) = 0.
• Multiplicative inverse. z = a + ib, $z \ne 0, z^{-1} = \frac{1}{z} = \frac{a}{a²+b²}-i\frac{b}{a²+b²} = \frac{\bar z}{|z|^2}, z \dot (\frac{1}{z}) = 1$

Polar Form & Euler’s Formula

A complex number z = a +bi can also be expressed in polar form, which uses its modulus (magnitude) r and argument (angle) θ:

• The modulus r = |z| = $\sqrt{a^2+b^2}$. It represents the distance of the complex number from the origin in the complex plane.

  |z₁-z₂| is the distance between these two points in the complex plane. |Re(z)| ≤ |z|, |Im(z)| ≤ |z|, $z · \bar z = |z|², |z_1+z_2| \le |z_1| + |z_2|$ and the equality holds if and only if both complex numbers lie on the same half ray through the origin in the complex plane.

Proof:

$|z_1+z_2|² = (z_1+z_2) · \overline{z_1+z_2} = (z_1+z_2)(\overline{z_1}+\overline{z_2}) = z_1·\overline{z_1} + z_2· \overline{z_2} + z_1·\overline{z_2} + z_2· \overline{z_1} = |z_1|² + |z_2|² + z_1·\overline{z_2} + \overline{z_1·\overline{z_2}} = |z_1|² + |z_2|² + 2Re(z_1·\overline{z_2}) \le |z_1|² + |z_2|² + 2|z_1·\overline{z_2}| = |z_1|² + |z_2|² + 2|z_1||\overline{z_2}| = |z_1|² + |z_2|² + 2|z_1||z_2| = (|z_1| + |z_2|)², \text{ hence } |z_1+z_2| \le |z_1| + |z_2|$

By induction, it can be demonstrated that $|z_1 + z_2 + \cdots z_n| \le |z_1| + |z_2| \cdots |z_n|$

Furthermore, $|z_1| = |z_1 - z_2 + z_2| \le |z_1 - z_2| + |z_2| \leadsto |z_1| - |z_2| \le |z_1 - z_2| \leadsto |z_1 - z_2| \ge |z_1| - |z_2|, \text{ and mutatis mutandis } |z_1 - z_2| \ge |z_2| - |z_1| \leadsto |z_1 - z_2| \ge \big||z_2| - |z_1|\big|$

• The argument $θ = arg(z) = tan^{-1}(\frac{b}{a})$is the angle between the positive real axis and the line representing the complex number. The angle θ should be determined in the correct quadrant, considering the signs of a and b. Furthermore, the argument of a complex number z = a + bi (≠ 0) is the set {$\theta_0 + 2n\pi: 0 \le \theta_0 < 2\pi, n \in \mathbb{Z}, a = rcos(\theta_0), b = rsin(\theta_0)$}, $\theta_0$ is called the principal argument of z (≠ 0), $\theta_0$ = Arg(z).

The polar form of a complex number is: z = a + bi = rcos(θ) + rsin(θ)i = r(cos(θ) + isin(θ)) =[This can be written compactly using Euler’s formula], re^iθ, e.g., z = 1 + i = $\sqrt{2}e^{\frac{π}{4}i}$. It satisfies e^i(θ+β) = e^iθe^iβ.

Euler’s formula is a fundamental formula in complex analysis. It establishes the mathematical relationship between trigonometric functions and the complex exponential function. It states that e^iθ = cos(θ) + isin(θ). This allows us to represent complex numbers compactly in exponential form.

The multiplication of two complex numbers in polar form results in a new complex number whose modulus is the product of the moduli and whose argument is the sum of the arguments.

Proof:

Given z = |z|e^iθ, w = |w|e^iβ, zw = |z|e^iθ|w|e^iβ =[e^i(θ+β) = e^iθe^iβ] |z||w|e^i(θ+β) = |zw|e^i(θ+β) where:

• ∣zw∣=∣z∣∣w∣ (the modulus of the product)
• θ+β is the resulting argument of the product.

Given a complex number z, it can be expressed as z = |z|e^iθ₁ = |z|e^iθ₂ ⇒[From this equality, we can infer:] e^iθ₁ = e^iθ₂ ⇒[Using Euler’s formula, we can express the complex exponentials in terms of sine and cosine:] cos(θ₁) + isin(θ₁) = cos(θ₂) + isin(θ₂) ⇒[From this equality, we can separate the real and imaginary parts:] cos(θ₁) = cos(θ₂) and sin(θ₁) = sin(θ₂) ⇒ θ₂ - θ₁ ∈ 2πℤ, this means that the difference between θ₁ and θ₂ is an integer multiple of 2π, indicating that the angles are coterminal.

In complex analysis, the argument (arg) of a complex number z = a+bi is the angle θ formed with the positive real axis. However, because angles can differ by multiples of 2π radians, the argument isn’t unique.

• $\text{arg}(z)$ refers to all possible angles that a complex number z can make with the positive real axis in the complex plane. It's a multi-valued function.
• $\text{Arg}(z)$ refers to the principal argument, which is typically restricted to a specific range, usually between $\pi$ and $\pi$. Therefore, $\text{arg}(z) = \text{Arg}(z) + 2\pi k$, where k is any integer ℤ. This accounts for the periodic nature of angles in polar coordinates, e.g., z = 1 + i = $\sqrt{2}e^{\frac{π}{4}i}, \text{arg}(z) =\frac{π}{4} + 2\pi k$.
• For a complex number z = x + iy, we can calculate Arg(z) using the following formula: $Arg(z) = ε + tan^{-1}(\frac{y}{x})$ where ε = 0, ±πdepending on the quadrant of z, it adjusts the angle based on the quadrant in which the complex number lies.

1. If x > 0, y ≥ 0, then the point is in the first quadrant: ε = 0, e.g., z = 1 + i, Arg(z) = $tan^{-1}(\frac{y}{x})$ = arctan(1) = π/4, z = $\sqrt{2} e^{\frac{\pi}{4}}$
2. If x < 0 and y ≥ 0, then the point is in the second quadrant: ε = π, e.g., z = -1 + i, Arg(z) = $tan^{-1}(\frac{y}{x}) + \pi$ = arctan(-1) + π = -π/4 + π = 3π/4, z = $\sqrt{2} e^{\frac{3\pi}{4}}$
3. If x < 0 and y < 0, then the point is in the third quadrant: ε = -π, e.g., z = -1 - i, arctan(y/x) = arctan(1) = π/4. Therefore, Arg(z) = -π + π/4 = -3π/4, z = $\sqrt{2} e^{\frac{-3\pi}{4}}$
4. If x > 0 and y < 0, then the point is in the fourth quadrant: ε = 0, e.g., z = 1 - i, arctan(y/x) = arctan(-1) = -π/4. Therefore, Arg(z) = -π/4, z = $\sqrt{2} e^{\frac{-\pi}{4}}$

It satisfies $\text{arg}(z_1)+\text{arg}(z_2) = {θ_1+θ_2, θ_1 ∈ \text{arg}(z_1), θ_2 ∈ \text{arg}(z_2)} = \text{arg}(z_1·z_2)$. This property holds because the argument function $\text{arg}(z)$ considers all possible angles differing by multiples of 2π. For any complex numbers z₁ and z₂, this property ensures that the sum of their arguments corresponds correctly to the argument of their product, accounting for the periodic nature of angles.

The same does not apply to $\text{Arg}(z_1)+\text{Arg}(z_2)$ which is typically restricted to $-\pi < \text{Arg}(z) \leq \pi$. Since $\text{Arg}(z)$ is confined to a specific range, the sum $\text{Arg}(z_1) + \text{Arg}(z_2)$ can fall outside this range.

If z ≠ 0, $z\bar z = |z|² ⇒ z(\frac{\bar z}{|z|²}) = 1$ ⇒ Hence, the multiplicative inverse of z is $z^{-1} =\frac{\bar z}{|z|²}$, e.g., $(1+i)^{-1} =\frac{1-i}{|\sqrt{2}|²} = \frac{1-i}{2}$. Alternatively, $z = \sqrt{2}e^{\frac{π}{4}i}, z^{-1} =\frac{1}{z} = \frac{1}{\sqrt{2}e^{i\frac{π}{4}}} = \frac{e^{-\frac{π}{4}i}}{\sqrt{2}}$

This last step need to be demonstrated. Let $z = e^{iθ} = cos(θ) + isin(θ) ≠ 0, z^{-1}=\frac{\bar z}{|z|²} = \frac{cos(θ)-sin(θ)i}{cos(θ)²+sin(θ)²}$ =[Since cosine is even (cos(−θ) = cos(θ)) and sine is odd ( sin(−θ) = −sin(θ))] $\frac{e^{-iθ}}{1} = e^{-iθ}$. Hence, $z^{-1} = e^{-iθ}$. This aligns with the property of exponents: $(e^{-iθ})^{-1} = e^{-iθ}$

Product Property of Exponents. The product of two exponentials with the same base is equal to the base raised to the sum of their exponents, $a^m·a^n = a^{m+n}$

• Calculate z = -1 -i, z¹⁰⁰.

z = -1 -i =[The point is in the third quadrant, ε = −π] Therefore, Arg(z) = −π-π/4 = −3π/4 or equivalently 5π/4, z = $\sqrt{2} e^{\frac{-\pi}{4}}$ = $\sqrt{2}e^{\frac{5π}{4}i}$.

$z^{100} =[ \text{De Moivre's Theorem } ] \sqrt{2}^{100}e^{\frac{5π∙100}{4}i} = 2^{50}e^{125πi} = 2^{50}(e^{πi})^{125} = [e^{πi} = -1] -2^{50}$