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What are imaginary numbers?

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  • Last Updated : 30 Jun, 2021

If you are wondering what are imaginary numbers and thinking that there must be a meaning for imaginary number, Then lets get into the article to learn about what exactly are imaginary numbers.

Imaginary numbers :
The imaginary numbers are the numbers which, when squared, give a negative number. Imaginary numbers are the square roots of negative numbers where they do not have any definite value.
The imaginary numbers are represented as the product of a real number and the imaginary value i.

Example –

3i, 5i, 25i are some examples of imaginary numbers.

The value of i2 is given as -1.
So the value of (5i)2 is -25, and that implies \sqrt{-1}   is i.

Complex number :
Complex numbers are a combination of real numbers and imaginary numbers. 
The complex number is of the standard form – a + bi.
Where a and b are real numbers. i is an imaginary unit.
Now let’s get a quick understanding by using some examples:

  1. Real numbers – 
    -1,2, 10, 10000.
  2. Imaginary numbers – 
    4i, -5i, 2400i.
  3. Complex numbers – 
    2+3i, -5-4i.

Conjugate pair of an imaginary number :

  • a+bi is a complex number and the conjugate pair of a +bi  is a-bi.
  • When an imaginary number is multiplied by its conjugate pair, then the result will be a real number.

Rules of imaginary numbers :

  1. i = \sqrt{-1}
  2. i^2 = -1
  3. i^3 = -i
  4. i^4 = 1
  5. i^{4n} = 1

Arithmetic Operations on Imaginary Numbers :
1. Addition –

  • When two imaginary numbers are added, then the real part is added into one , and then the imaginary part is added into one.

Examples –

1. (2 + 2i) + (3 + 4i)
  = (2 + 3) + (2 + 4)i
 = (5 + 6i)

2. (3 + 4i) + (5 + 3i)
= (8) + (7)i
=  8 + 7i

3. (5 + 3i) + (4 + 2i) 
= (5 + 4) + (3 + 2)i
= 9 + 5i.

2. Subtraction –

  • When two imaginary numbers are subtracted, then the real part is subtracted, and then the imaginary part is subtracted.

Examples –

1. (2 + 2i) – (3 + 4i)
= (2 – 3) + (2 – 4)i
= (-1 – 2i)

2. (3 + 4i) – (5 + 3i)
= (-2) + (1)i
=  -2 + i

3. (5 + 3i) – (4 + 2i) 
 = (5 – 4) + (3 – 2)i
= 1 + i.

3. Multiplication

  • When two imaginary numbers are multiplied, then the result will be as follows –

Examples –

1. (a + bi) (c + di)
= (a + bi)c + (a + bi)di
= ac + bci + adi+ bdi^2
= (ac – bd)+i(bc + ad)

 2. (3 + 4i)*(3 – 4i)
= (9 + 12i -12i – 16i^2 )
= 25

 3. (2 + 2i) * (3 – 4i) 
= (6 + 6i -8i – 8i^2 )
= (14 – 2i)

4) Division
The numerator and denominator will be multiplied by its conjugate  pair of denominators.

Examples –

1. (2 + 2i) / (3 + 4i)
Multiplying the numerator and denominator with the conjugate pair of denominators.
=(2 + 2i) * (3 – 4i) / (3 + 4i)*(3 – 4i)
=(6 +6i -8i – 8i^2 ) / (9 + 12i -12i – 16i^2 )
=(14 – 2i) / 25

2. (3 + 4i) / (2 + 2i)
Multiplying the numerator and denominator with the conjugate pair of denominators.
= ((3 + 4i) * (2 – 2i)) / ((2 + 2i) * (2 – 2i))
=(6 + 2i – 8i^2 ) / (4-4i^2 )
=(14 + 2i) / (8)
=(7 + i) / 4

3. (2 + 2i) / (2 + 2i)
Multiplying the numerator and denominator with the conjugate pair of denominators.
= ((2 + 2i) * (2 – 2i)) / ((2 + 2i) * (2 – 2i))
= (4 – 4i^2 ) / (4 – 4i^2 )
= 8 / 8
= 1

Where do we use imaginary numbers :

  • Imaginary numbers are very useful in various mathematical proofs.
  • 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.
  • Combining AC currents is very difficult as they may not match properly on the waves. 
  • Using imaginary currents helps in making the calculations easy.
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