Introduction

Have you ever tried finding the square root of negative numbers?

Before we go any further, let me first give you some brief notes on REAL NUMBERS.

Most of you may already be familiar with ‘real numbers’. Every real number is either a rational or an irrational number.

Rational Numbers

Integers, Fractions, Mixed Numbers, Repeating and Terminating decimals are all rational numbers, because, they can be expressed as fractions (in the form of a/b, where b is not zero.)

1/3, 5 6/7, –89, 0, 92.64, 5.11111… are some examples of rational numbers.

Irrational Numbers

Irrational numbers cannot be written as a simple fraction. They are non-terminating, non-repeating decimals.

Pi, square root of 2, ‘e’ are examples of irrational numbers.

Pi = 3.141592653589793238462643383279……

Square root of 2 = 1.414213562373095048801688724209……

e = 2.718281828459045235360287471352……

Let’s go back to our discussion….

Have you ever tried finding the square root of negative numbers?

The square root of a negative number CANNOT be a real number…

What does that mean?

Well...this means that you will NEVER get a real number for an answer when you square root a negative number.....your answer will NOT look like 5, –8, 9/10, 1.34, square root of 3 etc.

Is that reason clear? I’ll try to say enough to give you some idea of what I'm talking about.

NO real number, when multiplied by itself, will produce a negative number...that's why the square root of a negative number CANNOT be a real number...

HENCE...."imaginary numbers" were invented. Imaginary numbers are the square roots of negative numbers.

The imaginary number "i" is used to express the square roots of negative numbers.

Well…what is "i"?

"i" is the square root of negative 1.

That is:

i = sqrt (–1)

I^2 = i × i = sqrt (–1) × sqrt (–1) = –1

In fact, there are TWO numbers that are the square root of ‘-1’ and they are i and –i just like there are two numbers that are the square root of 9, 3 and –3.

All imaginary numbers are just multiples of "i".

We can make an imaginary number simply by multiplying a real number by "i".

For example,

Multiply the real number 7 by "i" and that gives us an imaginary number "7i".

Here are some more examples:

28.5i

2.75937683082138376058940586i

sqrt(2) × i

As I mentioned above, the imaginary number "i" is used to express the square roots of negative numbers. Let’s look at some examples.

Sqrt (–9) = +3i, –3i

Let’s check:

(+3i) × (+3i) = (3 × 3) × (i × i) = 9 × (i^2) = 9 × (–1) = –9

And

(–3i) × (–3i) = (–3 × –3) × (i × i) = 9 × (i^2) = 9 × (–1) = –9

Similarly,

Sqrt (–16) = +4i, –4i

Sqrt (–25) = +5i, –5i

Sqrt (–36) = +6i, –6i

Now that we have learned about imaginary numbers, I find it relevant to talk a bit about complex numbers too.

Complex numbers

The word "complex" means "Composed of two or more parts".

Complex numbers are composed of a REAL part and an IMAGINARY part.

They are a real number plus an imaginary number.

Here are some examples.

8 + 5i

13 + (–6) i = 13 – 6i

57 – i

In general, a complex number can be written in the form a + bi where ‘a’ and ‘b’ are real numbers (including ‘0’) and ‘i’ is the imaginary number.

Who invented imaginary numbers?

There was no one person who invented imaginary numbers. A number of mathematicians contributed to the invention and development of imaginary numbers. Gerolamo Cardano, Rafael Bombelli, Rene Descartes, Leonhard Euler, Carl Friedrich Gauss, William Rowan Hamilton are the most important mathematicians who have contributed significantly to its unfolding – directly or otherwise.

Imaginary numbers in Real Life

Imaginary numbers or complex numbers have essential concrete applications in electrical engineering and related areas like signal processing, radar... They are indispensable in a field of physics called quantum mechanics.