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Complex numbers are those numbers which are expressed in the form of “a+ib” where, a&b stands for real numbers and ‘i’ stands for an imaginary number called “iota”. An imaginary number is mainly represented as ‘i’ or ‘j’. The value of i considered to be (√1).
For example, 2+3i is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im).
What are real numbers?
– A real number includes all the numbers existing in mathematics. Be it a negative integer or a positive one, a zero or a non zero number, a fraction, a decimal, a rational number or an irrational number. However, complex numbers are not just real numbers. They are both a real number as well as an imaginary number. For examples 0, 1.75, 13/17, 2, 4 etc.
What are Imaginary numbers?
– All those numbers which are not considered has real numbers are imaginary numbers. When an imaginary number is squared, it will give a negative result. For example: √2, √7, √11.
The main why these numbers are used is to represent periodic motions such as water waves, light waves, alternating current etc. These mention things rely on sine or cosine waves, etc. We can perform the following functions in a complex number:
– Addition: The formula for addition of complex number is “(a + ib) + (c + id) = (a + c) + i(b + d)”*
– Subtraction: The formula for Subtraction of complex number is “(a + ib) – (c + id) = (a – c) + i(b – d)”*
– Multiplication: The formula for Multiplication of complex number is”(a + ib). (c + id) = (ac – bd) + i(ad + bc)”*. Multiplication is always similar to Multiplication of two binomials.
– Division: The formula for Division of complex number is”(a + ib) / (c + id) = (ac+bd)/ (c2 + d2) + i(bc – ad) / (c2 + d2)”.*
For example:
Simplify “6i + 10i(2i)”.*
Solution: 6i + 10i(2i)
= 6i + 10i(2) + 10i (i)
= 6i +20i – 10 i2
= 26 i – 10 (1)
= 26i + 10
* – The examples and formula is taken from the below mentioned link. (https://www.toppr.com/guides/mathsformulas/complexnumberformula/)
 This reply was modified 3 years, 1 month ago by Nehal Rathi.