the sum of a real number and an imaginary number. An imaginary number is a multiple of i, where i is the square root of -1. Complex numbers can be expressed in the form a + bi, where a and b are real numbers. They have the algebraic structure of a field in mathematics. In engineering and physics, complex numbers are used extensively to describe electric circuits and electromagnetic waves.
For a complex number a + bi, a is called the real part and b is called the imaginary part. Thus, the complex number -2 + 3i has the real part -2 and the imaginary part 3. Addition of complex numbers is performed by adding the real and imaginary parts separately. To add 1 + 4i and 2 - 2i, for example, add the real parts 1 and 2 and then the imaginary parts 4 and -2 to obtain the complex number 3 + 2i. The general rule for addition is
(a+ bi) + (c+di) = (a+ c) + (b+d)i
Multiplication of complex numbers is based on the premise that i×i=-1 and the assumption that multiplication distributes over addition. This gives the rule
(a+bi) × (c+di) = ( ac-bd) + (ad+bc)i
for example, (1 + 4i) × (2 - 2i) = 10 + 6i If z=a+bi is any complex number, then, by definition, the complex conjugate of z is and the absolute value, or modulus, of z is For example, the complex conjugate of 1 + 4i is 1 - 4i, and the modulus of 1 + 4i is
