Complex Numbers – Definition, Properties & Examples

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Complex numbers are helpful in finding solutions involving the square root of negative numbers such as quadratic equations when the discriminant $b^{2} – 4ac \lt 0$.  A complex number is written as $a + ib$ where $a$ and $b$ are real numbers and $i$ is an imaginary unit.

Complex numbers have applications such as in scientific research, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and vibration analysis. Let’s understand what is a complex number.

What Are Complex Numbers?

A complex number consists of two parts – a real part and an imaginary part and is expressed as a sum of these two parts. A complex number is generally denoted by the letter $z$.

$z = a + ib$, where $a$ and $b$ are real numbers and $i$ is an imaginary unit $\left( = \sqrt{-1} \right)$.

Note 

• $a$ and $b$ are real numbers, $i$ is an imaginary unit
• $a$ is called the real part and $ib$ is called the imaginary part of a complex number
• The real part of a complex number $z$ is denoted by $Re\left(z \right)$ and the imaginary part is denoted by $Im\left(z \right)$

Graphing Complex Numbers – Argand Plane

A complex number $z = a + ib$ consists of a real part $a = \left( Re\left( z\right) \right)$ and an imaginary part $b = \left( Im\left( z\right) \right)$, which can be considered as an ordered pair $\left(Re \left(z \right), Im\left(z \right) \right)$ and can be represented as coordinates points in the Euclidean plane. 

The euclidean plane with reference to complex numbers is called the complex plane or the Argand Plane, named after Jean-Robert Argand. The complex number $z = a + ib$ is represented with the real part – $a$, with reference to the $x$-axis, and the imaginary part- $ib$, with reference to the $y$-axis.

In this argand plane, two terms are associated – modulus of a complex number and argument of a complex number. Let us try to understand these two important terms.

Modulus of Complex Number

The distance of the complex number represented as a point in the argand plane $\left(a, ib \right)$ is called the modulus of the complex number. This distance is a linear distance from the origin $\left(0, 0 \right)$ to the point $\left(a, ib \right)$, and is measured as $r = | \sqrt{a^2 + b^2}|$. 

This expression is derived from the Pythagoras theorem, where the modulus represents the hypotenuse, the real part is the base, and the imaginary part is the altitude of the right-angled triangle.

Example

Let’s consider a complex number $z = 1 + i$, to understand how the modulus of a complex number is calculated.

Here, $a = 1$ and $b = 1$.

Therefore, the modulus $r = \sqrt {a^{2} + b^{2}} = \sqrt {1^{2} + 1^{2}} = \sqrt {1 + 1} = \sqrt {2}$.

Argument of Complex Number

The angle made by the line joining the geometric representation of the complex number and the origin, with the positive $x$ – axis, in the anticlockwise direction is called the argument $\left( \theta \right)$ of the complex number. The argument of the complex number is the inverse of the $tan$ of the imaginary part $b$ divided by the real part $a$ of the complex number. $Argz \left(\theta \right) = \tan^{-1} \left(\frac {b}{a} \right)$.

Example

Let’s consider the above complex number $z = 1 + i$, to understand how the argument of a complex number is calculated.

Here, $a = 1$ and $b = 1$.

In the triangle in the figure, $\tan \theta = \frac {b}{a} = \frac {1}{1} = 1$

Therefore, $\theta = \tan^{-1}\left ( 1\right) = 45^{\circ} \text{ or }  \frac {\pi}{4}$.

Polar Form of Complex Number

In a polar form, any point is represented in terms of two values.

• The distance of a point from the origin 
• The angle made by the line joining the point and the origin with the positive direction of $x$ – axis.

A point $P\left(x, y \right)$ in polar coordinates is represented as $P\left(r, \theta \right)$, where

$r = \sqrt{x^{2} + y^{2}}$ and $\tan \theta = \frac {y}{x} => \theta = \tan^{-1}\left( \frac {y}{x}\right)$. 

In case of a complex number $a + ib$, we get 

• $r = \sqrt{a^{2} + b^{2}}$ and $\tan \theta = \frac {b}{a} => \theta = \tan^{-1}\left( \frac {b}{a}\right)$. 
• And, $\cos \theta = \frac {a}{r} => a = r \cos \theta$ and $\sin \theta = \frac {b}{r} => b = r \sin \theta$.

Therefore, in polar coordinates a complex number $z = a + ib$ can be written as $z = r \cos \theta + i r \sin \theta = r\left(\cos \theta + i \sin \theta \right)$.

Properties of Complex Numbers

Like other numbers such as natural numbers, whole numbers, integers, rational numbers, irrational numbers, or real numbers, complex numbers also exhibit certain properties. The properties of complex numbers are as follows.

Equality of Complex Numbers

For any two complex numbers $z_{1} = a_{1} + i b_{1}$ and $z_{2} = a_{2} + i b_{2}$, $z_{1} = z_{2}$, if $a_{1} = a_{2}$ and $b_{1} = b_{2}$.

Examples

Ex 1: If complex numbers $z_{1} = 3 – 2i$ and $z_{2} = x + yi$ are equal, then find $x$ and $y$.

For any two complex numbers $z_{1} = a_{1} + i b_{1}$ and $z_{2} = a_{2} + i b_{2}$, $z_{1} = z_{2}$, if $a_{1} = a_{2}$ and $b_{1} = b_{2}$.

Therefore, $x = 3$ and $y = -2$.

Ex 2: If complex numbers $z_{1} = 5 + i$ and $z_{2} = \left(a + b \right) + \left(a – b \right)i$ are equal, then find $a$ and $b$.

For any two complex numbers $z_{1} = a_{1} + i b_{1}$ and $z_{2} = a_{2} + i b_{2}$, $z_{1} = z_{2}$, if $a_{1} = a_{2}$ and $b_{1} = b_{2}$.

Therefore, 

$a + b = 5$ ——————– (1)

$a – b = 1$ ——————– (2)

Solving equations (1) & (2), we get $a = 3$ and $b = 2$.

Existence of Additive Identity

For every complex number $z = a + ib$, there exists a complex number $z_{0} = 0 + i0$, such that $z + z_{0} = z_{0} + z = z$. The complex number $z_{0} = 0 + i0$ is called the additive identity of complex numbers.

Note: In $z_{0} = 0 + i0$, $r = \sqrt{0^{2} + 0^{2}} = 0$ and $\theta = \tan^{-1} \left( \frac {0}{0}\right) = \frac {\pi}{2}$.

Existence of Multiplicative Identity

For every complex number $z = a + ib$, there exists a complex number $z_{1} = 1 + i0$, such that $z \times z_{1} = z_{1} \times z = z$. The complex number $z_{1} = 1 + i0$ is called multiplicative identity of complex numbers.

Note: In $z_{1} = 1 + i0$, $r = \sqrt{1^{2} + 0^{2}} = 1$ and $\theta = \tan^{-1} \left( \frac {0}{1}\right) = 0$.

Existence of Additive Inverse

For any complex number $z = a + ib$, there exists a complex number $-z = -\left(a + ib \right) = -a – ib$, such that $z + \left(-z \right) = -z + z = 0$. The complex number $-z$ is called the additive inverse of the complex number $z$.

Examples

Ex 1: Find additive inverse of $z = 3 – 2i$.

Additive inverse of $3 – 2i$ is $- \left(3 – 2i \right) = -3 + 2i$.

Ex 2: Find additive inverse of $z = -6 – 5i$.

Additive inverse of $z = -6 – 5i$ is $-\left(-6 – 5i \right) = 6 + 5i$.

Existence of Multiplicative Inverse

For any complex number $z = a + ib$, there exists a complex number $z^{-1} = \frac {1}{z} = \frac {1}{a + ib}$, such that $z \times z^{-1} = z^{-1} \times z = 1$. 

$z^{-1} = \frac {1}{z} = \frac {1}{a + ib}$

Multiplying the numerator and denominator by $a – ib$, we get

$z^{-1} = \frac {1}{a + ib} \times \frac {a – ib}{a – ib} = \frac {1 \times \left(a – ib \right)}{\left(a + ib \right) \left(a – ib \right)} = \frac {a – ib}{a^{2} – \left(ib \right)^{2}} = \frac {a – ib}{a^{2} – i^{2}b^{2}} = \frac {a – ib}{a^{2} + b^{2}}$.

Therefore, multiplicative inverse of any complex number $z = a + ib$ is $z^{-1} = \frac {a}{a^{2} + b^{2}} – i \frac {b}{a^{2} + b^{2}}$.

Examples

Ex 1: Find multiplicative inverse of $z = 1 + i$.

In $z = 1 + i$, $a = 1$ and $b = 1$, therefore, multiplicative inverse of $z = 1 + i$ is $z^{-1} = \frac {a}{a^{2} + b^{2}} – i \frac {b}{a^{2} + b^{2}} = \frac {1}{1^{2} + 1^{2}} – i \frac {1}{1^{2} + 1^{2}} = \frac {1}{2} – i \frac {1}{2}$

Ex 2: Find the multiplicative inverse of $z = 2 + 3i$.

In $z = 2 + 3i$, $a = 2$ and $b = 3$, therefore, multiplicative inverse of $z = 2 + 3i$ is $z^{-1} = \frac {a}{a^{2} + b^{2}} – i \frac {b}{a^{2} + b^{2}} = \frac {2}{2^{2} + 3^{2}} – i \frac {2}{2^{2} + 3^{2}} = \frac {2}{4 + 9} – i \frac {2}{4 + 9} = \frac {2}{13} – i \frac {2}{13}$.

Closure Property of Complex Numbers

The closure property of a complex number states that for any two complex numbers $z_{1}$ and $z_{2}$ the result of addition, subtraction, multiplication, and division is also a complex number, i.e., the closure property holds for all the four arithmetic operations.

Closure Property of Addition of Complex Numbers

For any two complex numbers $z_{1}$ and $z_{2}$, the sum $z_{1} + z_{2}$ is also a complex number.

For example, for two complex numbers $2 – 4i$ and $6 + 7i$, the sum $ \left(2 – 4i \right) + \left(6 + 7i \right) = 8 + 3i$ is also a complex number.

Closure Property of Subtraction of Complex Numbers

For any two complex numbers $z_{1}$ and $z_{2}$, the difference $z_{1} – z_{2}$ is also a complex number.

For example, for two complex numbers $9 + 5i$ and $-3 + 4i$, the difference $ \left(9 + 5i \right) – \left(-3 + 4i \right) = 12 + i$ is also a complex number. Also $ \left(-3 + 4i \right) – \left(9 + 5i \right) = -12 – i$ is also a complex number.

Closure Property of Multiplication of Complex Numbers

For any two complex numbers $z_{1}$ and $z_{2}$, the product $z_{1} \times z_{2}$ is also a complex number.

For example, for two complex numbers $1 + 3i$ and $3 + 2i$, the product $\left(1 + 3i \right) \times \left(3 + 2i \right) = -3 + 11i$ is also a complex number.

Closure Property of Division of Complex Numbers

For any two complex numbers $z_{1}$ and $z_{2}$, the quotient $z_{1} \div z_{2}$ is also a complex number.

For example, for two complex numbers $2 + i$ and $1 – i$, the quotient $\left(2 + i \right) \div \left(1 – i \right) = \frac {1}{2} – \frac {3}{2}i$ is also a complex number.

Commutative Property of Complex Numbers

The commutative property states that the result of the operation remains the same, even if the order of the numbers is changed. The commutative property holds for the addition and multiplication of complex numbers. 

Commutative Property of Addition of Complex Numbers

The commutative property of addition states that for any two complex numbers $z_{1}$ and $z_{2}$, $z_{1} + z_{2} = z_{2} + z_{1}$.

Examples

Ex 1: Verify the commutative property of addition of complex numbers for $z_{1} = 5 + 3i$ and $z_{2} = -3 + 2i$.

$z_{1} + z_{2} = \left(5 + 3i \right) + \left(-3 + 2i \right) = (5 – 3) + (3 + 2)i = 2 + 5i$

And $z_{2} + z_{1} = \left(-3 + 2i \right) + \left(5 + 3i \right) = (-3 + 5) + (2 + 3)i = 2 + 5i$

Therefore, $z_{1} + z_{2} = $z_{2} + z_{1}$.

Commutative Property of Multiplication of Complex Numbers

The commutative property of multiplication states that for any two complex numbers $z_{1}$ and $z_{2}$, $z_{1} \times z_{2} = z_{2} \times z_{1}$.

Examples

Ex 1: Verify the commutative property of multiplication of complex numbers for $z_{1} = 1 + 2i$ and $z_{2} = 1 + i$.

$z_{1} \times z_{2} = \left(1 + 2i \right) \times \left(1 + i \right) = -1 + 3i$

And, $z_{2} \times z_{1} = \left(1 + i \right) \times \left(1 + 2i \right) = -1 + 3i$

Therefore, $z_{1} \times z_{2} = z_{2} \times z_{1}$.

Note: The commutative property does not hold for subtraction and division for complex numbers.

Associative Property of Complex Numbers

The associative property states that the result of the operation remains the same, even if the grouping of the numbers in the operation is changed. The associative property holds for the addition and multiplication of complex numbers.

Associative Property of Addition of Complex Numbers

The associative property of addition states that for any three complex numbers $z_{1}$, $z_{2}$ and $z_{3}$, $\left(z_{1} + z_{2} \right) + z_3 = z_{1} + \left(z_{2} + z_{3} \right)$.

Examples

Ex 1: Verify the associative property of addition of complex numbers for $z_{1} = 2 – 3i$,  $z_{2} = 3 + 5i$, and $z_{3} = 4 + 6 i$.

$\left(z_{1} + z_{2} \right) + z_3 = $\left( \left(2 – 3i \right) + \left(3 + 5i$ \right) \right) + \left(4 + 6 i \right) = \left(5 + 2i \right) + \left(4 + 6i \right) = 9 + 8i$

And, $z_{1} + \left(z_{2} + z_{3} \right) = \left(2 – 3i \right) + \left( \left(3 + 5i \right) + \left(4 + 6 i \right) \right) = \left(2 – 3i \right) + \left( 7 + 11i\right) = 9 + 8i$

Therefore, $\left(z_{1} + z_{2} \right) + z_3 = z_{1} + \left(z_{2} + z_{3} \right)$

Associative Property of Multiplication of Complex Numbers

The associative property of multiplication states that for any three complex numbers $z_{1}$, $z_{2}$ and $z_{3}$, $\left(z_{1} \times z_{2} \right) \times z_3 = z_{1} \times \left(z_{2} \times z_{3} \right)$.

Examples

Ex 1: Verify the associative property of multiplication of complex numbers for $z_{1} = 1 – 2i$,  $z_{2} = 2 + 3i$, and $z_{3} = 3 – 4 i$.

$\left(z_{1} \times z_{2} \right) \times z_3 = \left(\left(1 – 2i \right) \times \left(2 + 3i \right) \right) \times \left(3 – 4 i \right) = \left(8 – i \right) \times \left(3 – 4i \right) = 20 – 35i$

And, $z_{1} \times \left(z_{2} \times z_{3} \right) = \left(1 – 2i \right) \times \left( \left(2 + 3i \right) \times \left(3 – 4 i \right)\right)  = \left(1 – 2i \right) \times \left(18 + i \right) = 20 – 35i$

Therefore, $\left(z_{1} \times z_{2} \right) \times z_3 = z_{1} \times \left(z_{2} \times z_{3} \right)$

Note: The associative property does not hold for subtraction and division for complex numbers.

Distributive Property of Multiplication of Complex Numbers

The distributive property states that the result of multiplication remains the same, even if the numbers are split using addition and subtraction. The distributive property of multiplication of complex holds for addition and subtraction.

Distributive Property of Multiplication Over Addition of Complex Numbers

The distributive property of multiplication over addition states that for any three complex numbers $z_{1}$, $z_{2}$ and $z_{3}$, $z_1 \times \left(z_{2} + z_{3}\right) = z_{1} \times z_{2} + z_{1} \times z_{3}$.

Examples

Ex 1: Verify the distributive property of multiplication over addition of complex numbers for $z_{1} = 2 + i$,  $z_{2} = 1 + i$, and $z_{3} = 4 – 3i$.

$z_1 \times \left(z_{2} + z_{3}\right) = \left(2 + i \right) \times \left(\left(1 + i \right) + \left(4 – 3i \right)\right)  = \left(2 + i \right) \times \left(5 – 2i \right) = 12 + i$.

And, $z_{1} \times z_{2} + z_{1} \times z_{3} = \left(2 + i \right) \times \left(1 + i \right) + \left(2 + i \right) \times \left(4 – 3i \right) = \left(1 + 3i \right) + \left(11 – 2i \right) = 12 + i$

Therefore, $z_1 \times \left(z_{2} + z_{3}\right) = z_{1} \times z_{2} + z_{1} \times z_{3}$

Distributive Property of Multiplication Over Subtraction of Complex Numbers

The distributive property of multiplication over subtraction states that for any three complex numbers $z_{1}$, $z_{2}$ and $z_{3}$, $z_1 \times \left(z_{2} – z_{3}\right) = z_{1} \times z_{2} – z_{1} \times z_{3}$.

Note: The distributive property does not hold for division in complex numbers.

Examples

Ex 1: Verify the distributive property of multiplication over subtraction of complex numbers for $z_{1} = 7 – 4i$,  $z_{2} = 2 + 3i$, and $z_{3} = 5 + 2i$.

$z_1 \times \left(z_{2} – z_{3}\right) = \left(7 – 4i \right) \times \left( \left(2 + 3i \right) – \left(5 + 2i \right)\right) = \left(7 – 4i \right) \times \left(-3 + i \right) = -17 + 19i$

And, $z_{1} \times z_{2} – z_{1} \times z_{3} = \left(7 – 4i \right) \times \left(2 + 3i \right) – \left(7 – 4i \right) \times \left(5 + 2i \right) = \left(26 + 13i \right) – \left(43 – 6i \right) = -17 + 19i$

Therefore, $z_1 \times \left(z_{2} – z_{3}\right) = z_{1} \times z_{2} – z_{1} \times z_{3}$

Conclusion

In mathematics, the complex number $a + ib$ is an element of a number system that contains the real numbers and a special element denoted by $i$ called an imaginary unit. Complex numbers like numbers of other categories have certain properties that are used in solving the problems based on them.

Practice Problems

1. Find the modulus and argument of the following complex numbers.
   • $1 + i$
   • $1 – i$
   • $-1 + i$
   • $-1 – i$
2. Find the value of $a$ and $b$, if $z_{1} = z_{2}$
   • $z_{1} = 2 + 5i$ and $z_{2} = a + ib$
   • $z_{1} = -1 + 2i$ and $z_{2} = a – ib$
   • $z_{1} = -3 + 7i$ and $z_{2} = -a + ib$
   • $z_{1} = 1 + i$ and $z_{2} = -a – ib$

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)

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