While solving equations such as \(x^2 + 1 = 0\), we observe that no real number satisfies the given condition. This limitation of the real number system motivates the introduction of a broader number system, known as complex numbers. Complex numbers extend the real numbers in such a way that every quadratic equation has a solution.

Definition of a Complex Number

A complex number is an ordered pair of real numbers written in the form \(z = a + ib\), where \(a\) and \(b\) are real numbers and \(i\) is an imaginary unit defined by the property \(i^2 = -1\). Here, \(a\) is called the real part of \(z\), denoted by \(\Re(z)\), and \(b\) is called the imaginary part of \(z\), denoted by \(\Im(z)\).

Thus, every real number can be regarded as a complex number with zero imaginary part, and every purely imaginary number is a complex number whose real part is zero.

Derivation of the Imaginary Unit

Consider the quadratic equation

\(x^2 + 1 = 0\)

Rearranging the equation, we obtain

\(x^2 = -1\)

Since no real number has a square equal to \(-1\), we define a new number \(i\) such that

\(i^2 = -1\)

This definition enables us to express solutions of equations that were previously unsolvable within the real number system.

Addition of two complex numbers

Let \(z_1 = a + ib\) and \(z_2 = c + id\) be any two complex numbers. Then, the sum \(z_1 + z_2\) is defined as follows:

The addition of complex numbers satisfy the following properties:

• The closure law:
  The sum of two complex numbers is a complex number, i.e., \(z_1 + z_2\) is a complex number for all complex numbers \(z_1\text{ and }z_2\).
• The commutative law:
  For any two complex numbers \(z_1\) and \(z_2\) , \[z_1 + z_2 = z_2 + z_1\]
• The associative law:
  For any three complex numbers \(z_1\, z_2,\,z_3\), \[(z_1 + z_2 ) + z_3 = z_1 + (z_2 + z_3)\]
• The existence of additive identity:
  There exists the complex number \((0 + i 0)\) (denoted as 0), called the additive identity or the zero complex number, such that, for every complex number \(z\), \[z + 0 = z\]
• The existence of additive inverse:
  To every complex number \(z = a + ib\), we have the complex number \(– a + i(– b)\) (denoted as \(– z\)), called the additive inverse or negative of \(z\). We observe that \[z + (–z) = 0\] (the additive identity).

Given any two complex numbers \(z_1\) and \(z_2\), the difference \(z_1 – z_2\) is defined as follows: \[z_1 – z_2 = z_1 + (– z_2 )\]

Let \(z_1 = a + ib\) and \(z_2 = c + id\) be any two complex numbers. Then, the product \(z_1 z_2\) is defined as follows: \[z_1 z_2 = (ac – bd) + i(ad + bc)\]

For example
\[\begin{aligned}(3 + i5) (2 + i6) &= (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) \\&= – 24 + i28\end{aligned}\]

The multiplication of complex numbers possesses the following properties, which we state without proofs.

• The closure law:
  The product of two complex numbers is a complex number, the product \(z_1 z_2\) is a complex number for all complex numbers \(z_1\) and \(z_2\).
• The commutative law:
  For any two complex numbers \(z_1\) and \(z_2\), \[z_1 z_2 = z_2 z_1\]
• The associative law:
  For any three complex numbers \(z_1,\, z_2,\, z_3\), \[(z_1 z_2 ) z_3 = z_1 (z_2 z_3 )\]
• The existence of multiplicative identity:
  There exists the complex number \(1 + i 0\) (denoted as 1), called the multiplicative identity such that \[z.1 = z\] for every complex number \(z\).
• The existence of multiplicative inverse:
  For every non-zero complex number \(z = a + ib\) or \(a + bi\;(a \ne 0,\; b \ne 0)\), we have the complex number \[ \frac{a}{a^2+b^2} + i\,\frac{-b}{a^2+b^2} \] (denoted by \(\frac{1}{z}\) or \(z^{-1}\)), called the multiplicative inverse of \(z\) such that \[ z \cdot \frac{1}{z} = 1 \quad \text{(the multiplicative identity)}. \]
• The distributive law:
  For any three complex numbers \(z_1,\, z_2,\, z_3\), \[\begin{aligned}&(a)\quad z_1 (z_2 + z_3 ) = z_1 z_2 + z_1 z_3 \\\\&(b)\quad (z_1 + z_2 ) z_3 = z_1 z_3 + z2 z3\end{aligned}\]

Let \(z_1 = a + ib\) and \(z_2 = c + id\) be two complex numbers, where \(c + id \neq 0\). The division of \(z_1\) by \(z_2\) is defined as the complex number obtained by multiplying the numerator and the denominator by the conjugate of the denominator.

Thus, we consider

\[ \frac{z_1}{z_2} = \frac{a + ib}{c + id} \]

Multiplying the numerator and the denominator by the conjugate of \(c + id\), namely \(c - id\), we obtain

\[ \frac{a + ib}{c + id} \times \frac{c - id}{c - id} \]

Simplifying the expression, we get

\[ \begin{aligned} \frac{a + ib}{c + id} &= \frac{(a + ib)(c - id)}{(c + id)(c - id)} \\ &= \frac{ac - aid + ibc - i^2 bd}{c^2 + d^2} \\ &= \frac{(ac + bd) + i(bc - ad)}{c^2 + d^2} \end{aligned} \]

Hence, the quotient of two complex numbers is again a complex number, provided the denominator is not zero.

Important Observation

The denominator \(c^2 + d^2\) is a non-negative real number and is equal to \(z_2 \overline{z_2}\). Since \(z_2 \neq 0\), we have \(c^2 + d^2 \neq 0\), ensuring that the division is well-defined.

Conclusion

Therefore, the division of two complex numbers can always be expressed in the standard form \(x + iy\), where \(x\) and \(y\) are real numbers. This result shows that the set of complex numbers is closed under division, except division by zero.

In the study of complex numbers, the imaginary unit \(i\) is defined by the fundamental relation \(i^2 = -1\). Using this definition, higher powers of \(i\) can be obtained systematically.

Let us evaluate the successive powers of \(i\):

\[ \begin{aligned} i^1 &= i \\ i^2 &= -1 \\ i^3 &= i^2 \cdot i = (-1)i = -i \\ i^4 &= i^2 \cdot i^2 = (-1)(-1) = 1 \end{aligned} \]

From the above results, we observe that the powers of \(i\) repeat after every four terms. Hence, the powers of \(i\) are said to be periodic with a period of 4.

Generalisation

For any positive integer \(n\), we write

\[ n = 4q + r, \quad \text{where } r = 0,1,2,3 \]

The value of \(i^n\) depends only on the remainder \(r\) when \(n\) is divided by 4, as shown below:

\[ \begin{aligned} r = 0 &\Rightarrow i^n = i^{4q} = 1 \\ r = 1 &\Rightarrow i^n = i \\ r = 2 &\Rightarrow i^n = -1 \\ r = 3 &\Rightarrow i^n = -i \end{aligned} \]

Illustrative Result

Thus, any power of \(i\), however large, can be easily simplified by reducing the exponent modulo 4. This method is frequently used in simplifying expressions involving complex numbers.

In the real number system, the square of any real number is always non-negative. Therefore, the square root of a negative real number is not defined in the set of real numbers. This limitation leads to the introduction of the imaginary unit.

The imaginary unit \(i\) is defined by the relation

\[ i^2 = -1 \]

Using this definition, we can express the square root of a negative real number in terms of \(i\). Let \(a\) be a positive real number. Then,

\[ \begin{aligned} \sqrt{-a} &= \sqrt{a \times (-1)} \\ &= \sqrt{a}\,\sqrt{-1} \\ &= \sqrt{a}\, i \end{aligned} \]

Thus, the square root of a negative real number is an imaginary number.

Illustration

Consider the square root of \(-9\). We write

\[ \begin{aligned} \sqrt{-9} &= \sqrt{9 \times (-1)} \\ &= 3\sqrt{-1} \\ &= 3i \end{aligned} \]

Hence, \(\sqrt{-9} = 3i\).

Important Note

Every negative real number has two square roots in the complex number system. For example, the square roots of \(-a\) are \(i\sqrt{a}\) and \(-i\sqrt{a}\), since

\[ (i\sqrt{a})^2 = (-i\sqrt{a})^2 = -a \]

This extension of the number system ensures that square roots of all real numbers, whether positive or negative, are defined within the complex numbers.

• \[\large(z_1 + z_2)^2 = z_1^2 + 2z_1 z_2 + z_2^2\]
• \[\large(z_1 - z_2)^2 = z_1^2 - 2z_1 z_2 + z_2^2\]
• \[\large(z_1 + z_2)^3 = z_1^3 + 3z_1^2 z_2 + 3z_1 z_2^2 + z_2^3\]
• \[\large(z_1 - z_2)^3 = z_1^3 - 3z_1^2 z_2 + 3z_1 z_2^2 - z_2^3\]
• \[\large z_1^2 - z_2^2 = (z_1 + z_2)(z_1 - z_2)\]

Let \(z = a + ib\) be a complex number, where \(a\) and \(b\) are real numbers. Two important quantities associated with a complex number are its modulus and its conjugate. These play a key role in simplifying expressions and in division of complex numbers.

Conjugate of a Complex Number

The conjugate of the complex number \(z = a + ib\), denoted by \(\overline{z}\), is defined as

\[ \overline{z} = a - ib \]

Thus, the conjugate of a complex number is obtained by changing the sign of its imaginary part while keeping the real part unchanged.

Using the definition, we can verify the following:

\[ \begin{aligned} z + \overline{z} &= (a + ib) + (a - ib) = 2a \\ z - \overline{z} &= (a + ib) - (a - ib) = 2ib \end{aligned} \]

Modulus of a Complex Number

The modulus of the complex number \(z = a + ib\), denoted by \(|z|\), is defined as the non-negative real number

\[ |z| = \sqrt{a^2 + b^2} \]

Geometrically, \(|z|\) represents the distance of the point corresponding to \(z\) from the origin in the Argand plane.

Relation Between Modulus and Conjugate

The modulus and the conjugate of a complex number are closely related. This relation can be established as follows:

\[ \begin{aligned} z\,\overline{z} &= (a + ib)(a - ib) \\ &= a^2 + b^2 \\ &= |z|^2 \end{aligned} \]

Hence, the square of the modulus of a complex number is equal to the product of the number and its conjugate.

Important Observations

From the above definitions and results, we conclude that: \(|z| \geq 0\) for every complex number \(z\), and \(|z| = 0\) if and only if \(z = 0\). Also, the conjugate of the conjugate of a complex number is the number itself.

• \[\mid z_1\,z_2\mid=\mid z_1\mid\,\mid z_2\mid\]
• \[ \left| \dfrac{z_1}{z_2} \right|=\dfrac{|z_1|}{|z_2|}\quad \text{ provided } z_2\ne0 \]
• \[ \overline{z_1\,z_2}=\overline{z_1}\,\overline{z_2} \]
• \[ \overline{z_1\,\pm\,z_2}=\overline{z_1}\,\pm\,\overline{z_2} \]
• \[ \overline{\left(\dfrac{z_1}{z_2}\right)}=\dfrac{\overline{z_1}}{\,\overline{z_2}}\quad\text{provided } z_2\ne0 \]

These properties are fundamental in the study of complex numbers and are extensively used in algebraic manipulations and problem-solving.

A complex number can be represented geometrically in a plane. This geometrical interpretation helps in visualising complex numbers and understanding their properties more clearly. Two commonly used representations are the Cartesian plane representation and the polar representation.

Cartesian Plane (Argand Plane) Representation

Let \(z = a + ib\) be a complex number, where \(a\) and \(b\) are real numbers. We associate the complex number \(z\) with the ordered pair \((a, b)\) in a plane. This plane is known as the Argand plane or complex plane.

In this plane, the horizontal axis represents the real part and is called the real axis, while the vertical axis represents the imaginary part and is called the imaginary axis. The point \((a, b)\) uniquely represents the complex number \(z = a + ib\).

Thus, every complex number corresponds to a unique point in the Cartesian plane, and conversely, every point in the plane represents a unique complex number.

Polar Representation of a Complex Number

Consider a complex number \(z = a + ib\) represented by the point \(P(a, b)\) in the Argand plane. Let the distance of the point \(P\) from the origin be \(r\), and let the angle made by the line joining the origin to \(P\) with the positive real axis be \(\theta\).

The distance \(r\) is the modulus of \(z\), given by

\[ r = |z| = \sqrt{a^2 + b^2} \]

The angle \(\theta\), called the argument of \(z\), satisfies

\[ \begin{aligned} \cos \theta &= \frac{a}{r} \\ \sin \theta &= \frac{b}{r} \end{aligned} \]

Using these relations, we can express the complex number \(z\) as

\[ \begin{aligned} z &= a + ib \\ &= r(\cos \theta + i \sin \theta) \end{aligned} \]

This form of expressing a complex number is known as its polar representation.

Conclusion

The Cartesian plane representation expresses a complex number in terms of its real and imaginary parts, while the polar representation expresses it in terms of its modulus and argument. Both representations are equivalent and are used according to convenience in different problems involving complex numbers.