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# Operating with Complex Numbers

Some equations have no real solutions. For instance, the quadratic equation

 x2 + 1 = 0 Equation with no real solution

has no real solution because there is no real number x that can be squared to produce -1. To overcome this deficiency, mathematicians created an expanded system of numbers using the imaginary unit i, defined as Imaginary unit

where i2 = -1. By adding real numbers to real multiples of this imaginary unit, we obtain the set of complex numbers. Each complex number can be written in the standard form, a + bi.

Definition of a Complex Number

For real numbers a and b, the number a + bi is a complex number. If b ≠ 0, a + bi is called an imaginary number, and bi is called a pure imaginary number.

To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately.

Addition and Subtraction of Complex Numbers

If a + bi and c + di are two complex numbers written in standard form, their sum and difference are defined as follows.

Sum: (a + bi) + (c + di) = (a + c) + (b + d)i

Difference: (a + bi) - (c + di) = (a - c) + (b - d)i

The additive identity in the complex number system is zero (the same as in the real number system). Furthermore, the additive inverse of the complex number a + bi is

 -(a + bi) = -a - bi Additive inverse

Thus, you have (a + bi) + (-a - bi) = 0 + 0i = 0.

Example 1

Adding and Subtracting Complex Numbers

 a. (3 - i) + (2 + 3i) = 3 - i + 2 + 3i Remove parentheses. = 3 + 2 - i + 3i Group like terms. = (3 + 2) + (- 1 + 3)i = 5 + 2i Standard form
 b. 2i + (-4 - 2i) = 2i - 4 -2i Remove parentheses. = -4 + 2i - 2i Group like terms. = -4 Standard form
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