Imaginary Numbers

What Are Imaginary Numbers?

If you are wondering what are imaginary numbers? And think that it is about the imagination of numbers and that there must be an imaginary meaning of an imaginary number, then no, you’re wrong. We don’t have an imaginary meaning of an imaginary number but we have the real imaginary numbers definition that actually exists and is used by many electricians in the application of electricity, specifically alternating current (AC). Imaginary numbers cannot be quantified on a number line, it is because of this reason that it is called an imaginary number and not real numbers.  

Imaginary Numbers Definition

Imaginary numbers are the numbers that give a negative number when squared. In other words, we can say that an imaginary number is basically the square root of a negative number which does not have a tangible value. The imaginary number unlike real numbers cannot be represented on a number line but are real in the sense that it is used in Mathematics. Imaginary numbers are also known as complex numbers.  Imaginary numbers also show up in equations of quadratic planes where the imaginary numbers don’t touch the x-axis. Imaginary numbers are also very useful in advanced calculus. imaginary numbers are denoted as “i”

What is Complex Number?

In Mathematics, Complex numbers do not mean complicated numbers; it means that the two types of numbers combine together to form a complex. Complex numbers are made of two types of numbers, i.e., real numbers and imaginary numbers. They are the building blocks of more obscure math, such as algebra. Complex numbers are applied to many aspects of real life, for example, in electronics and electromagnetism.

Complex numbers are represented as a + bi, where the real number is at the first and the imaginary number is at the last. 

Real And Imaginary Numbers

If we do a “real vs imaginary numbers”, the first thing we would notice is that a real number, when squared, does not give a negative number whereas imaginary numbers, when squared, gives negative numbers. Real numbers are denoted as R and imaginary numbers are denoted by “i”.   A set of real numbers forms a complete and ordered field but a set of imaginary numbers has neither ordered nor complete field. 

A real number can be algebraic as well as transcendental depending on whether it is a root of a polynomial equation with an integer coefficient or not. Also, it can be either rational or irrational depending on whether it can be expressed as a ratio of two integers or not.

Imaginary Numbers Chart

A very interesting property of “i” is that when we multiply it, it circles through four very different values. Here is an example, 

i x i = -1, -1 x i = -i, -i x i = 1, 1 x i = i. 

We can also call this cycle as imaginary numbers chart as the cycle continues through the exponents. This knowledge of the exponential qualities of imaginary numbers. 

Addition Of Numbers Having Imaginary Numbers

When we add two numbers, for example, a+bi, and c+di, we have to separately add and simplify the real parts first followed by adding and simplifying the imaginary parts. 

Here is an answer: (a+c) + i(b+d).

Subtraction Of Numbers Having Imaginary Numbers

When we subtract c+di from a+bi, we will find the answer just like in addition. In other words, we group all the real terms separately and imaginary terms separately before doing the simplification. Here is an example: (a+bi)-(c+di) = (a-c) +i(b-d).

Multiplication Of Numbers Having Imaginary Numbers

We take this (a+bi)(c+di) and multiply it.

(a+bi)(c+di) = (a+bi)c + (a+bi)di

                   = ac+bci+adi+bdi2

                   = (ac-bd)+i(bc+ad)

Division Of Numbers Having Imaginary Numbers

The division of one imaginary number by another is done by multiplying both the numerator and denominator by its conjugate pair and then make it real. 

For example: multiplication of: (a+bi) / ( c+di) is done in this way:

(a+bi) / ( c+di) = (a+bi) (c-di) / ( c+di) (c-di) = [(ac+bd)+ i(bc-ad)] / c2 +d2.

Fun Fact

  • Imaginary numbers have made their appearance in pop culture. The protagonist Robert Langdon in Dan Brown’s "The Da Vinci Code," referred to Sophie Neveu’s belief in the imaginary number.

  • The short story  “The Imaginary,” by Isaac Asimov has also referred to the idea of imaginary numbers where imaginary numbers along with equations explain the behavior of a species of squid.

Solved Imaginary Numbers Examples

Question 1) Simplify and add 2i+3i

Solution 1) Simplifying 2i+3i as (2+3)i

                        Adding (2+3) = 5

                                       = 5i

Question 2) Simplify and multiply (3i)(4i)

Solution 2) Simplifying (3i)(4i) as (3 x 4)(i x i)

                                                 = (12)(i2)

                                                 = (12)(-1)

                                                 = -12

FAQ (Frequently Asked Questions)

1. Why Are Imaginary Numbers Useful? 

Imaginary numbers are extremely essential in various mathematical proofs, such as the proof of the impossibility of the quadrature of a circle with a compass and a straightedge only.

Imaginary numbers are often used to represent waves. We multiply a measure of the strength of the waves by the imaginary number i. The advantage of this is that multiplying by an imaginary number is seen as rotating something 90º. So if one is at 90º to another, it will be useful to represent both mathematically by making one of them an imaginary number. The other can be a non-imaginary number and together the two will be a complex number for example 3+4i. 

2. How Will You Explain Imaginary Numbers To A Layperson? 

All numbers are mostly abstract. The most simple abstractions are the countable numbers: 1, 2, 3, 4, and so on. But what if someone is asked to explain negative numbers! What does "minus two" mean? How would we interpret that number? How would we assign meaning to that number? In this sense, imaginary numbers are no different from the negative numbers. They too are completely abstract concepts, which are created entirely by humans.

Imaginary numbers don't exist, but so do negative numbers. The best way to explain imaginary numbers would be to draw a coordinate system and place the pen on the origin and then draw a line of length 3. The question anyone would ask will be  "where to" or "which direction". If you tell them to go right, they reach the point (3, 0). This direction will correspond to the positive numbers. Now if you tell them to go left instead, they will reach the point (-3, 0). This "left" direction will correspond exactly to the negative numbers. Lastly, if you tell them to go straight up, they will reach the point 

(0, 3). The "up" direction will correspond exactly to the imaginary numbers.

In this sense, imaginary numbers are basically "perpendicular" to a preferred direction. In other sense, imaginary numbers are just the y-coordinates in a plane.