Imaginary Numbers

Imaginary numbers are numbers as the name suggests are the number that is not real numbers. All the numbers real and imaginary come under the categories of complex numbers. Imaginary numbers are very useful in solving quadratic equations and other equations whose solutions can not easily be found using general rules.

For example, the solution of x² + x + 1 = 0 can easily be calculated using imaginary numbers. Let's learn about Imaginary numbers and their properties in detail in this article.

Imaginary Numbers Definition

The number whose square results in negative results is called an Imaginary number.

In simple words, the square root of negative numbers is called an imaginary number. They are called imaginary numbers as we cannot associate them with any real-life examples.

They are represented by "i" and are pronounced as iota at its value is,

i = √-1

Examples of Imaginary Numbers

Some examples of imaginary numbers are:

. . . -3, -2i, -i, i, 2i, 3i . . .

Note:

• i is the imaginary unit, defined as i = −1​.
• ki is the positive multiple of imaginary unit, where k > 0.
• -ki is the negative multiple of imaginary units, where k > 0.

History of Imaginary Numbers

Imaginary numbers were first encountered in the 16th century as solutions to seemingly unsolvable equations. They were first encountered by Italian mathematician Gerolamo Cardano while solving cubic equations. Later, in the 18th century, the term "imaginary" was coined for these numbers by Swiss mathematician Leonhard Euler.

Initially met with skepticism, they were eventually accepted as crucial tools in solving various mathematical problems, especially in areas like electrical engineering and quantum mechanics. Imaginary numbers are represented as multiples of the imaginary unit, "i," where i² equals -1.

What is Iota or "i"?

The term "iota" or "i" refers to the imaginary unit in mathematics. It is defined as the square root of -1, denoted by the symbol "i."

Value of i

In mathematical notation,

i ²= −1 or i = √(-1)

The use of "i" allows mathematicians to extend the number system beyond real numbers, enabling solutions to equations that would otherwise be impossible to solve.

Powers of i

We know that i² = -1 and using the power rules other powers of I can also be easily calculated.

• i = √-1
• i² = -1
• i³ = i²×i = -i
• i⁴ = (i²)² = (-1)² = 1
• i⁵ = i⁴×i = i

On observing clearly we can see a pattern, there is a cycle of  -i, 1, i... after power 2 thus,

• i^4n-1= -i
• i⁴ⁿ = 1
• i⁴ⁿ⁺¹= i

The image given below explains the multiple of "i" with various numbers.

Rules of Imaginary Number

Some of the important rules of Imaginary numbers are,

• Imaginary numbers always exist in conjugate pairs i.e. for example if the complex number a + ib exists then its conjugate pair a - ib also exists.
• Associating imaginary numbers with real values is impossible.
• The square of the imaginary numbers results in a negative number which is the polar opposite of the real numbers, i.e. i² = -1.

Geometrical Interpretation of Imaginary Numbers

We usually represent a complex number a+bi by a point (a, b) in the Argand plane. For example, a complex number 5-6i is represented by the point (5, -6) on the Argand plane. 

Imaginary numbers in the form of bi (written as 0 + bi) are represented by the point (0, b) on the plane, and hence it is a point on the vertical axis (imaginary axis). Thus, the imaginary numbers always lie on the vertical axis of an Argand plane. 

Here are a few examples of the imaginary numbers shown in the image given below.

Operations on Imaginary Numbers

We can perform basic arithmetic operations such as

• Addition
• Subtraction
• Multiplication
• Division

on the imaginary numbers so now let us discuss these operations on imaginary numbers in detail below,

Now take two complex numbers as a + bi and c + di then,

Addition of Imaginary Numbers

The addition of imaginary numbers can easily be achieved by using the basic rule of addition, i.e.

For two numbers, a+bi, and c+di when the addition is performed, then the real parts and the imaginary parts are added separately and then simplified.

Example: Add (3 + 11i) and (4 - 5i)

Solution:

(3 + 11i) + (4 - 5i) = (3 + 4) + (11i - 5i)

                             = 7 + 6i

Subtraction of Imaginary Numbers

The subtraction of imaginary numbers can easily be achieved by using the basic rule of subtraction, i.e.

For two numbers, a+bi, and c+di when the subtraction is performed, then the real parts and the imaginary parts are subtracted separately and then simplified.

Example: Subtract (3 + 11i) and (4 - 5i)

Solution:

(3 + 11i) - (4 - 5i) = (3 - 4) + (11i + 5i)

                             = -1 + 16i

Multiplication of Imaginary Numbers

The multiplication of imaginary numbers can easily be achieved by using the basic rule of multiplication and the distributive property of multiplication, i.e.

For two numbers, a+bi, and c+di the multiplication is explained with the help of the following example.

Example: Multiplying (3 + 11i) and (4 - 5i)

Solution:

(3 + 11i) × (4 - 5i) = (3 × 4) - (3 × 5i) + (11i × 4) - (11i × 5i) 

⇒ (3 + 11i) × (4 - 5i) = 12 - 15i + 44i - 55i²

⇒ (3 + 11i) × (4 - 5i) = 12 + 29i -55(-1)   {as i² = -1} 

⇒ (3 + 11i) × (4 - 5i) = 12 + 55 + 29i

⇒ (3 + 11i) × (4 - 5i) = 67 + 29i

Division of Imaginary Numbers

The division of imaginary numbers can easily be achieved by using the basic rule of division and finding the conjugate of the imaginary numbers.

For two numbers, a+bi, and c+di the division is explained with the help of the following example.

Example: Divide (3 + 11i) and (4 - 5i)

Solution:

(3 + 11i) / (4 - 5i) 

Multiplying the conjugate of the denominator (4 + 5i) to both numerator and denominator we get,

= (3 + 11i) / (4 - 5i) × (4 + 5i) / (4 + 5i)

= {(3 + 11i) × (4 + 5i)} / {(4 - 5i) × (4 + 5i)}

= (12 + 15i + 44i + 55i²) / (16 + 20i - 20i - 25i²)

= (12 - 55 + 59i) / (16 + 25)

= (- 43 + 59i) / 41

Imaginary and Real Numbers

The common differences between Imaginary and Real Numbers are:

Facts about Imaginary Numbers

Some lesser known facts related to imaginary numbers are:

• "i" is first used by Leonard Euler for √(-1).
• Square root of i has both real and imaginary parts i.e., \sqrt{i} = \pm \sqrt{1/2}(1+i)
• Euler's identity, e^iπ + 1 = 0, links five fundamental mathematical constants: e, i, π, 1, and 0.

Also, Check

• Complex Numbers
• Cube of an Imaginary Number
• Additive Identity and Inverse of Complex Numbers

Solved Examples on Imaginary Numbers

Example 1: Evaluate the square root of -121.

Solution:

√(-121) = √(-1 × 121)

             = √(-1) × √(121)      {we know that √(-1) = 1}

             = i × (±11)

             = ±11i

Example 2: Evaluate the square root of -(1/9).

Solution:

√(-1/9) = √(-1 × 1/9)

⇒ √(-1/9) = √(-1) × √(1/9)      {we know that √(-1) = 1}

⇒ √(-1/9) = i × (±1/3)

⇒ √(-1/9) = ±1/3i

Example 3: Solve the imaginary number i⁵

Solution:

Given Imaginary Number i⁵

i⁵ = i² × i² × i

⇒ i⁵ = -1 × -1 × i      (i² = -1)

⇒ i⁵ = 1 × i

⇒ i⁵ = – i

Therefore, i⁵ is i.