Complex Numbers
INTRODUCTION
A complex number is a number that can be represented by an expression of the form: a + bi
where a and b are real numbers, and i is a symbol with the property
i2 = −1
We call a the real part of the complex number and we call b the imaginary part.
ADDING AND SUBTRACTING COMPLEX NUMBERS
To add or subtract complex numbers, simply add or subtract their real parts and their imaginary parts separately.
\( (a+bi)\pm (c+di) = (a\pm c)+(b\pm d)i \)
Example: If \( x=3+2i\) and \(y=-2-5i\), find \(x+y\).
Solution:
\( \begin{align} x+y &= (3+2i)+(-2-5i) \\ &= (3-2)+(2-5)i \\ &= (1)+(-3)i \\ &= 1-3i \end{align} \)
MULTIPLYING COMPLEX NUMBERS
Multiplication is defined so that the usual laws hold: i.e., do FOIL as usual, but simplify at the end using the fact that \(i^2 = -1\).
\( \displaylines{(a+bi)x(c+di) &=& ac+adi+bci+bdi^2 \\ &=& (ac-bd)+(ad+bc)i} \)
Example: If \(x=3 +2i\) and \(y=-2-5i\), find \(xy\)
