Imaginary Numbers

Imaginary numbers are numbers that, when squared, give negative numbers. In other words, they are the square roots of negative numbers. We get an imaginary number by multiplying a non-zero real number by the imaginary unit “i” (iota), where the value of i is √(-1) or i2 = -1.

For example: 
\( \sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3i \)

\( \sqrt{-7} = i \sqrt{7} \)


The square root of a negative number, for any positive real number x:

√(-x) = i√x 

For example:\( \sqrt{-36} = i \sqrt{36} = 6i \)


A complex number is expressed as the sum of a real number and an imaginary number.  It has the form a + ib, where ‘a’ and ‘b’ are real numbers i is the imaginary unit.

A complex number \( (a + bi) \) can be represented as a point (a, b) on the Argand plane. For example, we represent the complex number \( 2 - 4i \) using the point (2, - 4) on the plane.

Since an imaginary number bi can also be expressed as \( 0 + bi \), it is represented as (0, b). For a better understanding, let’s look at an example:

The letter “i” in a complex number \( (a + ib) \) is a unique imaginary unit, which is also known as “iota”. Since 'i' is not a real number, it is represented on the imaginary axis, perpendicular to the real number line. On the Argand plane, the point (0,1) is used to represent i.

\( (a + ib) \) is a complex number where the real part “a” moves along the real axis, and the imaginary part “ib” moves “b” units along the imaginary axis. This implies that a + ib represents the point (a, b) in the plane. Let’s look at the representation of i in math:

• The value of i can be mathematically represented as:\( i^2 = -1 \), which means \( i = \sqrt{-1} \)

• Imaginary numbers are called 'imaginary' not because they don't exist, but because they aren't real numbers.

To calculate the imaginary numbers, we follow the same methods as real numbers. We will now look at the arithmetic operations performed on imaginary numbers:

Addition and Subtraction of Imaginary Numbers

We can add or subtract imaginary numbers in the same way as combining like terms in algebra. For example:

\( 5i + 4i = 9i \)

\( 5i - 4i = i \)

Multiplication of Imaginary Numbers

We multiply imaginary numbers using algebraic multiplication rules and the property i2 = -1. 

For example: 
 

\( 5i \times 4i = 20i^2 = 20(-1) = -20 \)
 

\( 4i^2 \times (-3i^3) = -12i^5 = -12i^4 \cdot i = -12 \cdot 1 \cdot i = -12i \)

While multiplying complex numbers, apply the following rules to simplify the powers of i:

\( \begin{aligned} i^{4k} &= 1, \\ i^{4k + 1} &= i, \\ i^{4k + 2} &= -1, \\ i^{4k + 3} &= -i. \end{aligned} \)

 

Here, k is a whole number. This indicates that we can reduce any power of i to one of these four values. Example:
 

\( \begin{aligned} i^4 &= i^0 = 1, \\ i^{17} &= i^1 = i. \end{aligned} \)

Division of Imaginary Numbers

To divide imaginary numbers, we apply exponent rules:

\( \frac{a^m}{a^n} = a^{m - n} \)

To rationalize the denominator, we eliminate i by multiplying the numerator and denominator by i; we use the identity \( \frac{1}{i} = -i \) which can be expanded as:

\( \frac{1}{i} = \frac{1}{i} \times \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i \)

For example:
\( \frac{6i}{3i} = 2 \)

Imaginary numbers are essential in advanced mathematics, physics, and engineering, helping solve equations with no real solutions. These tips and tricks help students grasp the concept of imaginary numbers more easily.

 

Simplify step by step: Always break expressions into real and imaginary parts for clarity.

Use conjugates wisely: Multiply by the conjugate to simplify fractions involving imaginary numbers.

Practice arithmetic operations: Addition, subtraction, multiplication, and division of complex numbers become easier with regular practice.

Visualize on the complex plane: Plotting numbers helps understand their magnitude, direction, and operations geometrically.

Understand i and its powers: memorize that \( i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1 \) and repeat cyclically.

Although abstract, imaginary numbers have several real-life applications. Let’s see how they are applied in different fields:

• Imaginary numbers are used in designing circuits for electronic devices such as mobile phones and computers to improve their signal processing performance.

• These numbers are used in healthcare to enhance the clarity of MRI scans.

• Computers use the imaginary unit (i) in the Fourier transforms, which filter noise and enhance audio signals.

• They are used to analyze the strength of tall buildings or bridge foundations against natural disasters.

• Imaginary numbers are used to improve the quality of communication by reducing errors.