What is a Complex Number?

A number which can be represented in the form of x+iy, where ‘i’ is an imaginary number, is called a complex number. By the expression, we can conclude that the complex number is a combination of a real number and an imaginary number. For example, 3+5i is a complex number, where 3 is the real number and 5i is imaginary.

Modulus and Conjugate:

If z=x+iy is a complex number, then the modulus of z is represented by;

\(\begin{array}{l}|z|=\sqrt{x^{2}+y^{2}}\end{array} \)

And conjugate of complex number, z is denoted by;

\(\begin{array}{l}\bar{z}=x-iy\end{array} \)

The value of i is equal to the root of minus one, i.e., i = √-1. This imaginary number is also called ‘iota’. The complex number is majorly used for representing periodic motions such as sound waves, water waves, alternating current, light waves, etc. Let us learn here related formulas and how to solve them.

Complex Number Formulas

The formulas are based on four major algebraic operations on complex numbers, such as:

  • Addition
  • Subtraction
  • Multiplication
  • Division

Suppose (p+iq) and (r+it) are two complex numbers, then;

  • (p + iq) + (r + is) = (p + r) + i(q + s) [Addition formula]
  • (p + iq) – (r + is) = (p – r) + i(q – s) [Subtraction formula]
  • (p + iq) . (r + is) = (pr – qs) + i(ps + qr) [Multiplication formula]
  • (p + iq) / (r + is) = (pr+qs)/ (r2 + s2) + i(qr – ps) / (r2 + s2) [Division formula]

Also, read:

Solving Complex Number

Before we solve complex number equations, first we should learn more about the imaginary number ‘i’ and its values.

As we know, if ‘i’ is an imaginary number, then;

i2=-1 or i = √-1

Based on the above expression, we can write the values of different power of i.

i √-1
i2 -1
i3 i.i2 = -i
i4 I2.i2 = (-1).(-1) = 1

By the above table, we can now derive the below expressions:

  • i4k+1 = i
  • i4k+2 = -1
  • i4k+3 = -i
  • i4k = 1

Let us solve complex number with the help of examples:

Example 1: Solve x2 = -15

Solution: Given,

x2 = -15

Applying square root to both the sides, we get;

x= √(-15)

x= √[(-1)(15)]

x= √-1. √15

As we know, √-1 = i


x=i √15

Complex Number Multiplication

By the complex number formula of multiplication, we know that;

(p + iq) . (r + is) = (pr – qs) + i(ps + qr)

Based on this formula, we can solve many multiplication problems.

Example 2: Multiply (2+i3) and (4+i5).

Solution: By the formula of multiplication of complex numbers, we have:

(p + iq) . (r + is) = (pr – qs) + i(ps + qr)

If we multiply (2+i3) and (4+i5), then we get;

(2+i3) . (4+i5) = (2.4-3.5)+i(2.5+3.4)

= (8-15)+i(10+12)

= -7+i22

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