Complex Numbers

Complex Numbers and their Identities

What are Complex Numbers?

Complex numbers are those that are expressed as \(x+iy\), where \(x\) and \(y\) are real numbers and \(i\) is an imaginary number known as “iota”. ( \(\sqrt{-1}\) ) is the value of \(i\). For example, \(2+3i\) is a complex number, where \(2\) is a real number (Re) and \(3i\) is an imaginary number (Im).

What are Real Numbers?

Real numbers are any numbers that exist in a number system, such as positive, negative, zero, integer, rational, irrational, fractions, and so on. It is represented by the symbol Re(). 12, -45, 0, 1/7, 2.8, 5, and so on are all real numbers.

What are Imaginary Numbers?

Numbers that are not real are called imaginary numbers. When we square an imaginary number, the result is negative. It is written as Im(). For example, the numbers \(\sqrt{-2}, \sqrt{-7}, \sqrt{-11}\) are all imaginary.

To solve the equation \(x^2+1 = 0\) , complex numbers were introduced. The equation’s roots are of the form \(x = ±\sqrt{-1}\), and there are no real roots. As a result of the introduction of complex numbers, we have imaginary roots.

• Some examples of complex number :

$$ 1 + j $$ $$ -13 – 3i$$ $$ 0.89 + 1.2 i$$ $$ √5 + √2i$$

• An imaginary number is usually represented by the letters \(i\) or \(j\) which are both equal to -1. As a result, the square of an imaginary number is negative.

Notation

An equation of the form \(z = x+iy\), where \(x\) and \(y\) are real numbers, is defined to be a complex number. The real part is denoted by Re(z) = \(x\) and the imaginary part is denoted by Im(z) = \(iy\).

$$ z = x + i y $$

• Through Euler’s formula, a complex number \(z = x+iy\) may be written in “phasor” form as

$$ z = |z|(cos\theta+isin\theta)=|z|e^{i/\theta} $$

• Here, \(|z|\) is known as the complex modulus (or sometimes the complex norm) and \(\theta\) is known as the complex argument or phase.
• The geometric representation of a complex number as simply a point in the plane was historically significant because it made the concept of a complex number more acceptable. “Imaginary” numbers, in particular, gained acceptance in part due to their visualization.
• Because complex numbers, unlike real numbers, do not have a natural ordering, there is no analogue of complex-valued inequalities. This property is not surprising when viewed as elements in the complex plane, because points in a plane also lack natural ordering.

Considering \(z = x + iy\),

• Modulus of \(z\) is given by

$$ |z|=\sqrt{x^2 + y^2} $$

• The absolute square of z is defined by

$$ |z|^2= z . \bar{z} $$

• The complex conjugate, and the argument may be computed respectively as

$$ \bar{z} = x - iy $$ $$ arg(z) = \theta = tan^{-1}(\dfrac{y}{x}) $$

• The real Re(z) and imaginary parts Im(z) are given by

$$ Re(z) = (\dfrac{1}{2})(z+\bar{z}) $$ $$ Im(z) = (\dfrac{1}{2i})(z-\bar{z}) $$

• de Moivre’s identity relates powers of complex numbers for real n by

$$ z^n = |z|^n[cos(n\theta)+isin(n\theta)] $$

• A power of complex number \(z\) to a positive integer exponent \(n\) can be written in closed form as

$$ z^n = [x^n-(\dfrac{n}{2})x^{(n-2)}y^2+(\dfrac{n}{4})x^{(n-4)}y^4-...] + i[(\dfrac{n}{1})x^{(n-1)}y-(\dfrac{n}{3})x^{(n-3)}y^3+...] $$

• The first few are explicitly

$$ z^2 = (x^2-y^2)+i(2xy) $$ $$ z^3 = (x^3-3xy^2)+i(3x^2y-y^3) $$ $$ z^4 = (x^4-6x^2y^2+y^4)+i(4x^3y-4xy^3) $$ $$ z^5 = (x^5-10x^3y^2+5xy^4)+i(5x^4y-10x^2y^3+y^5) $$

Some useful identities

$$ (z1 + z2)^2 = (z1)^2 + (z2)^2 + 2 (z1 × z2) $$ $$ (z1 – z2)^2 = (z1)^2 + (z2)^2 – 2 (z1 × z2) $$ $$ (z1)^2 – (z2)^2 = (z1 + z2)(z1 – z2) $$ $$ (z1 + z2)^3 = (z1)^3 + 3(z1)^2 z2 +3(z2)^2 z1 + (z2)^3 $$ $$ (z1 – z2)^3 = (z1)^3 – 3(z1)^2 z2 +3(z2)^2 z1 – (z2)^3 $$

Properties

• The addition of two complex conjugate numbers will result in a real number.
• The multiplication of two complex conjugate numbers will also result in a real number.
• If the sum of two complex numbers is real, and the product of two complex numbers is also real, then these complex numbers are conjugate to each other.
• If \(x\) and \(y\) are real numbers and \(x + iy = 0\), then \(x = 0\) and \(y = 0\).
• If \(p, q, r\), and \(s\) are real numbers and \(p + iq = r + is\), then \(p = r\), and \(q = s\).
• Complex numbers obey the commutative laws of addition and multiplication :

$$ z1 + z2 = z2 + z1 $$ $$ z1 . z2 = z2 . z1 $$

• Complex numbers obey the associative laws of addition and multiplication :

$$ (z1 + z2) + z3 = z1 + (z2 + z3) $$ $$ (z1 . z2) . z3 = z1 . (z2 . z3) $$

• Complex numbers obey the distributive law :

$$ z1 . (z2 + z3) = z1.z2 + z1.z3 $$

• For any two complex numbers, say \(z1\) and \(z2\),

$$ |z1 + z2| ≤ |z1|+|z2| $$

• Following tool can be used to find the conjugate, square root, magnitude and argument of any complex number

• Following tool can be used on two complex numbers to find their multiplication, addition, subtraction and division.