In your study of mathematics, you have probably worked with only real numbers, which can be represented graphically on the real number line. The system of numbers is expanded to include imaginary numbers. The real numbers and imaginary numbers compose the set of complex numbers.

Not all quadratic equations have real-number solutions. For example, x² = −3 has no real-number solutions because the square of any real number is never a negative number. To overcome this problem, mathematicians created an expanded system of numbers using the imaginary unit i, defined as i = √−1. Note that i² = −1. The imaginary unit i can be used to write the square root of any negative number.

A complex number written in standard form is a number a + bi where a and b are real numbers. The number a is the real part, and the number bi is the imaginary part.

a + bi If b ≠ 0, then a + bi is an imaginary number. If a = 0 and b ≠ 0, then a + bi is a pure imaginary number. The diagram shows how different types of complex numbers are related.

Operations with Complex numbers:

EXAMPLE:

Add or subtract. Write the answer in standard form.

1. (8 − i ) + (5 + 4i )

2. (7 − 6i) − (3 − 6i)

SOLUTION:
A. (8 − i ) + (5 + 4i )

= (8 + 5) + (−1 + 4)i

= 13 + 3i

B.  (7 − 6i ) − (3 − 6i )

= (7 − 3) + (−6 + 6)i

= 4 + 0i

= 4

EXAMPLE:

Multiply. Write the answer in standard form.

4i(−6 + i ) b. (9 − 2i )(−4 + 7i )

SOLUTION:

4i(−6 + i )

= −24i + 4i²

= −24i + 4(−1)

= −4 − 24i