# Complex Numbers

The complex number is a special type of number that can be represented as a combination of a real number and an imaginary number. It can be denoted in the format of a +ib where a is the real number part and ib is the imaginary number part. Real numbers have definite tangible values and can be plotted on a number line. Examples of real numbers are fractions, integers, or any countable number. Imaginary numbers are abstract numbers, the square of which gives a negative value. In this case, the term complex does not indicate complicated, but it signifies that this type of number is formed by joining two numbers together, like a building complex.

Therefore, a complex number is represented by an addition of two numbers, in the form of z= a + ib where a and b both are real numbers and i is an imaginary number. Here ‘i’ is termed as ‘iota’ and the value of i equals √-1. For example, 3+4i is a complex number where 3 is a real number part and 4i is an imaginary number part as real number 4 when multiplied by √-1 becomes an imaginary number.

An interesting fact is that 0 is considered a complex number because it can be denoted in the format of 0+0i. Again 0 is a real number as well. So, zero is a real number that can be included in the set of complex numbers.

Arithmetic operations like addition and subtraction involving complex numbers can be done by combining similar parts separately which means the addition or subtraction of real number with the real number, and imaginary number with the imaginary number gives the final result.

For example, (m + in) + (p + iq) = (m + p) + i(n + q) and (m + in) – (p +iq) = (m– p) + i(n – q).

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## Properties of Complex Numbers

Some of the properties of complex numbers are mentioned below:

• If a and b are real numbers and a + bi =0, then a =0 and b =0
• If two complex numbers a + ib = c + id, where a, b, c and d are real numbers then a = c, and b=d
• The complex numbers follow the commutative law of addition and multiplication.

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