What is an imaginary number and how are complex numbers associated with it?

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. In other words, imaginary numbers are those numbers, when squared give a negative value. To find out the value of i, we have to consider complex numbers and the role of i in a new equation formed.  

Simplifying ‘i’  

x2 = 0 – 1

x2 = -1

x = √-1

x = i

Calculate powers of a number represented by i

The value of iota, denoted as i, is √-1. The value of the imaginary unit number  i exists, when there is a negative number inside the square root, such that a unit imaginary number is equal to the root of -1. Therefore, the square of the imaginary unit is equal to -1 and its cube is equal to the value, – i. In the same way, we can find the value of iota by solving it for different exponents.

The imaginary unit, i, is the square root of -1. Hence, the square of the imaginary unit is -1. This is how it is if we change the powers of i.

i0 = 1

i1 = i

i2 = -1 

i3 = i2 X i         = -1 x i =  -i

i4 = i2 X i2      = -1 x -1 = 1

i5 = i4 X i         = 1 x i = i

i6 = i4 X i2      = 1 x -1 = -1

To understand imaginary numbers, it is necessary to understand the powers of the imaginary unit. It can be seen that there is a pattern for the powers of the imaginary unit.

Understanding the powers of the imaginary unit is essential in understanding imaginary numbers. Following the examples above, it always simplifies to -1, -i, 1, or i. A simple shortcut to simplify an imaginary unit raised to a power is to divide the power by 4 and then raise the imaginary unit to the power of the reminder.

For example: to simplify i23, first divide 23 by 4.

23/4 = 5 remainder 3. So i23 = i3 = -i …… as already shown above.

The square of an imaginary number, ib, is (ib)2 = – b2 ,where b is a real number. An imaginary number can be added to a real number to form another complex number. For example, a + ib is a complex number with ‘a’ as the real part of the complex number and ‘b’ as the imaginary part of the complex number.

What is a Complex Number?

A complex number is an element of a number system that contains a real number that is multiplied by an imaginary unit, i, where i is the square root of -1. 

Every complex number can be expressed in the form a + bi, where a and b are real numbers.

No real number satisfies the above equation, i2 = -1, and complex numbers allow solutions to all polynomial equations, and even to those which have no solutions in real numbers. That is why the complex number is crucial to imaginary number i.        

Operations That Can Work on Imaginary Numbers

The basic arithmetic operations are addition, subtraction, multiplication, and division. Let us discuss these operations on imaginary numbers.

We also know that imaginary numbers are a part of complex numbers.

Imaginary number operations & calculating powers of i

• Addition or subtraction: The sum or difference of two complex numbers is a complex number whose real part and the imaginary part are obtained by just adding or subtracting the corresponding real parts and imaginary parts, respectively. i.e.,

(a+bi)±(x+yi)=(a±x)+i(b±y)

• Multiplication: The product of two complex numbers is found by multiplying them considering them as binomials:

(a+bi)(x+yi)=ax+iay+ibx+i2by

=ax + i(ay+bx)-by [i2= -1]

=(ax-by)+i(ay+bx)

• Division: We rationalise the denominator by multiplying and dividing by the conjugate of the complex number.

a+bix+yi=a+bix+yi*x-yix-yi

=(a+bi)(x-yi)x2-y2i2

=(ax+by)+i(bx-ay)x2+y2 

Some interesting tidbits about imaginary numbers

• The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians: See Niccolò Fontana Tartaglia, Gerolamo Cardano.
• Imaginary numbers are not “imaginary”, they really exist and have many uses.
• Imaginary numbers are used to represent waves.
• Imaginary numbers show up in equations that don’t touch the x-axis.
• Imaginary numbers are very useful in advanced calculus.
• Complex numbers have a role in quantum mechanics, a theory that describes the behavior of nature at the scale of atoms and subatomic particles.
• Imaginary numbers can also be applied to signal processing, which is useful in cellular technology and wireless technologies, as well as radar and even biology (brain waves).
• Usually denoted by the symbol i, imaginary numbers are denoted by the symbol j in electronics (because i already denotes “current”).