What is the conjugate of a complex number?

The conjugate of a complex number is obtained by changing the sign of its imaginary part.

A complex number is of the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. The conjugate of a complex number a + bi is denoted by a - bi. It is obtained by changing the sign of the imaginary part of the complex number. For example, the conjugate of 3 + 4i is 3 - 4i.

The conjugate of a complex number has several important properties. Firstly, the product of a complex number and its conjugate is always a real number. This can be shown as follows:

(a + bi)(a - bi) = a^2 - abi + abi - b^2i^2
= a^2 + b^2

Since i^2 = -1, the imaginary terms cancel out and we are left with a real number.

Secondly, the conjugate of a complex number is used to find the modulus (or absolute value) of the complex number. The modulus of a complex number a + bi is denoted by |a + bi| and is defined as the square root of the sum of the squares of its real and imaginary parts. It can be shown that:

|a + bi| = √(a^2 + b^2)

Finally, the conjugate of a complex number is also used to find the quotient of two complex numbers. To divide one complex number by another, we multiply the numerator and denominator by the conjugate of the denominator. For example:

(3 + 4i)/(1 + 2i) = (3 + 4i)(1 - 2i)/(1 + 2i)(1 - 2i)
= (3 - 8i - 4)/(1 + 4)
= -1 + i