Complex Numbers for the CSET Math

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Excerpt from the CSET Math Study Guide

Teacher candidates studying for the CSET should know understand the complex number system.

THE SYSTEM C OF COMPLEX NUMBERS

The system C of complex numbers is the number system of ordinary algebra. It is the smallest set in which, for example, the equation x² = a can be solved when a is any element of R.

(a, b) = (c, d) if and only if a = c and b = dIn other words, a + bi = c + di says that a and c must be equal and b and d must be equal.ADDITION AND MULTIPLICATION ON CAddition and multiplication on C are defined by(i) (a, b) + (c, d) = (a + c, b + d)
(ii) (a, b) · (c, d) = (ac - bd, ad + bc)

for all (a, b), (c, d) Î C.

PROPERTIES OF COMPLEX NUMBERS

The real numbers are a proper subset of the complex numbers C. The elements (a, b) Î C in which b ¹ 0, are called imaginary numbers.

In the set of real numbers, negative numbers do not have square roots. A new kind of number, called imaginary was invented so that negative numbers would have a square root. These numbers start with the number i, which equals the square root of -1, or i² = -1.

All imaginary numbers consist of two parts, the real part, b, and the imaginary part, i.

Example

Simplify: Ö(-5)

Solution: Write -5 as a product of prime factors.
Ö(-1 · 5)

Write as separate square roots.

Continue Lesson - Pages: 1 2 3

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