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Complex Numbers

In this section we define a set of numbers that has the real numbers as a subset.

Definition

The equation 2x = 1 has no solution in the set of integers, but in the set of rational numbers, 2x = 1 has a solution. The situation is similar for the equation x² = -4. It has no solution in the set of real numbers because the square of every real number is nonnegative. However, in the set of complex numbers x² = -4 has two solutions. The complex numbers were developed so that equations such as x2 4 would have solutions.

The complex numbers are based on the symbol . In the real number system this symbol has no meaning. In the set of complex numbers this symbol is given meaning. We call it i. We make the definition that

Complex Numbers

The set of complex numbers is the set of all numbers of the form a + bi, where a and b are real numbers, , and i² = -1.

In the complex number a + bi, a is called the real part and b is called the imaginary part. If b ≠ 0, the number a + bi is called an imaginary number.

In dealing with complex numbers, we treat a + bi as if it were a binomial, with i being a variable. Thus we would write 2 + (-3)i as 2 - 3i. We agree that 2 + i3, 3i + 2, and i3 + 2 are just different ways of writing 2 + 3i (the standard form). Some examples of complex numbers are

For simplicity we write only 7i for 0 + 7i. The complex number 9 + 0i is the real number 9, and 0 + 0i is the real number 0. Any complex number with b = 0 is a real number. For any real number a, a + 0i a. The set of real numbers is a subset of the set of complex numbers. Take a look at the figure below.