COMPLEX NUMBERS

ARITHMETIC OPERATIONS WITH COMPLEX NUMBERS Which complex representation is the best to use? It depends on the operation we want to perform.

ADDITION When performing addition/subtraction on two complex numbers, the rectangular form is the easiest to use. Addition of two complex numbers, C 1 = R 1 + jI 1 and C 2 = R 2 + jI 2, is merely the sum of the real parts plus j times the sum of the imaginary parts.

R 1 + R 2 I 1 - I 2 C1C1 C2C2 C1+ C2C1+ C2

MULTIPLICATION OF COMPLEX NUMBERS We can use the rectangular form to multiply two complex numbers If we represent the two complex numbers in exponential form, the product takes a simpler form.

CONJUGATION OF A COMPLEX NUMBERS The complex conjugate of a complex number is obtained by merely changing the sign of the number’s imaginary part. If then, C* is expressed as

SUBTRACTION Subtraction of two complex numbers, C 1 = R 1 + jI 1 and C 2 = R 2 + jI 2, is merely the sum of the real parts plus j times the sum of the imaginary parts.

DIVISION OF COMPLEX NUMBERS The division of two complex numbers is also convenient using the exponential and magnitude and angle forms, such as or

DIVISION (continued) Although not nearly so handy, we can perform complex division in rectangular notation by multiplying the numerator and denominator by the complex conjugate of the denominator

INVERSE OF A COMPLEX NUMBER A special form of division is the inverse, or reciprocal, of a complex number. If C = Me j , its inverse is given by In rectangular form, the inverse of C = R + jI is given by