Algebra of Complex Numbers

Complex Numbers

A. Definition

Complex numbers are expressions of the form a + ib where a, b ∈ ℝ. It is denoted by z, i.e. z = a + ib. The number a is called the real part of z and is written Re z. The number b is called the imaginary part of z and is written Im z.

Every complex number can be regarded as:

• Purely real - if b = 0
• Purely imaginary - if a = 0
• Imaginary - if b ≠ 0

Remark :

• (a) The set ℝ of real numbers is a proper subset of the complex numbers. Hence the complete number system is
  
• (b) Zero is both purely real as well as purely imaginary but not (strictly) imaginary.
• (c)
  
  is called the imaginary unit. Also i² = -1; i³ = -i; i⁴ = 1, and so on.
• (d)
  
  only if at least one of a or b is non-zero.

B. Algebraic Operations

The algebraic operations on complex numbers are carried out exactly like on polynomials in i, using the relation i² = -1. Inequalities such as "greater than" or "less than" are not defined for complex numbers; expressions like z > 0 or 4 + 2i < 2 + 4i are meaningless.

Note that in real numbers, if a² + b² = 0 then a = 0 = b. This kind of inference does not translate in simple form to complex numbers: for complex numbers z₁² + z₂² = 0 it need not follow that z₁ = z₂ = 0.

Equality of Complex Numbers

Two complex numbers

C. Conjugate of a Complex Number

If z = a + ib then its complex conjugate is obtained by changing the sign of its imaginary part and is denoted by

Remarks and useful properties:

• (i)
  
• (ii)
  
• (iii)
  
  which is a real number.
• (iv) If z lies in the first quadrant then
  
  lies in the fourth quadrant and -
  
  lies in the second quadrant.

D. Modulus (Absolute Value)

The modulus of a complex number z = a + ib is defined as

|z| = √(a² + b²).

Properties of modulus:

• |z| ≥ 0, and |z| = 0 iff z = 0.
• |z₁z₂| = |z₁||z₂| for all complex numbers z₁, z₂.
• |z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality).
• |z̄| = |z| where z̄ denotes the conjugate of z.

E. Argument and Argand Plane (Geometric Representation)

A complex number z = a + ib can be represented as the point (a, b) in the plane called the Argand plane. The distance of this point from the origin is |z|. The angle θ made with the positive real axis is the argument of z, denoted arg z, satisfying

cos θ = a / |z| and sin θ = b / |z|.

F. Polar and Exponential Form

Using modulus and argument, a non-zero complex number can be written in polar form as

z = r(cos θ + i sin θ), where r = |z| and θ = arg z.

By Euler's formula this is equivalent to the exponential form:

z = r e^{iθ}.

G. Division and Reciprocal

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator to obtain a real denominator. In particular, for non-zero z,

1/z = z̄ / |z|².

H. Square Roots of a Complex Number

Every non-zero complex number has two distinct square roots. To find square roots of a + ib one can set

a + ib = (x + iy)² with real x, y and solve for x, y using real and imaginary parts and the relation x² + y² = √(a² + b²).

Examples

Ex.1 Express (1 + 2i)²/(2 + i)² in the form x + iy.

Sol.

Compute the numerator and denominator separately.

(1 + 2i)² = 1 + 4i + 4i² = 1 + 4i - 4 = -3 + 4i.

(2 + i)² = 4 + 4i + i² = 4 + 4i - 1 = 3 + 4i.

Now divide (-3 + 4i) by (3 + 4i) by multiplying numerator and denominator by conjugate of denominator.

(-3 + 4i)(3 - 4i) = (-3)(3) + (-3)(-4i) + (4i)(3) + (4i)(-4i)

= -9 + 12i + 12i -16i²

= -9 + 24i + 16 = 7 + 24i.

(3 + 4i)(3 - 4i) = 3² + 4² = 9 + 16 = 25.

Therefore the quotient is

(7/25) + i(24/25).

Ex.2 Show that a real value of x will satisfy the equation

Sol.

Write the given equation in the form (complex expression) = (another complex expression) and use componendo and dividendo if appropriate.

By componendo and dividendo we obtain

Therefore, x will be real if

Ex.3 Find the square root of a + ib

Sol.

Let

Squaring both sides gives

Equating real and imaginary parts gives

Now, (x² + y²)² = (x² - y²)² + 4x²y² = a² + b², hence

x² + y² = √(a² + b²) ...(iii)

[

From (i) and (iii) we obtain

If b is positive, both x and y have the same sign; if b is negative, x and y have opposite signs (by the equation for the imaginary part).

Summary

This chapter introduced complex numbers as extensions of real numbers, defined algebraic operations, equality, conjugation, modulus, argument and polar/exponential forms. Techniques for addition, multiplication, division (using conjugates), and finding square roots were explained with worked examples. The geometric interpretation in the Argand plane and basic properties such as the triangle inequality and relations between a number and its conjugate and modulus were presented for a clear, exam-oriented understanding.