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GEOMETRY AND ARITHMETIC OF
COMPLEX NUMBERS COMPLEX
CHAPTER 19 FUNCTIONS THE EXPONENTIAL AND
TRIGONOMETRIC FUNCTIONS
Complex

Numbers and

Functions

19.1 Geometry and Arithmetic of Complex Numbers

Complex Numbers

A complex number is a symbol x + iy,or x + yi, where x and y are real numbers and
2
i =−1. Arithmetic of complex numbers is defined by:
Equality a + ib = c + id exactly when a = c and b = d.
Addition (a + ib) + (c + id) = (a + c) + i(b + d).
Multiplication (a + ib)(c + id) = (ac − bd) + i(ad + bc).

In multiplying two complex numbers, we proceed exactly as we would with polynomials
2
a + bx and c + dx with i in place of x and i =−1. For example,
2
(6 − 4i)(8 + 13i) = (6)(8) + (−4)(13)i + i[(6)(13) + (−4)(8)]= 100 + 46i.

The number a is called the real part of a + bi, denoted Re(a + bi). We call b the imaginary
part of a + bi, denoted Im(a + bi). For example,

Re(−4 + 12i) =−4 and Im(−4 + 12i) = 12.

The real and imaginary parts of any complex numbers are themselves real.
We may think of any real number a as the complex number a + 0i. In this way, the com-
plex numbers are an extension of the real numbers. This extension has profound implications in
2
algebra and analysis. The equation x + 1 = 0 has no solutions if we restrict x to be real. In the
complex numbers, it has two solutions: i and −i.

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