Numbers and Operations

3.2 Complex Numbers


The complex numbers are formed by “attaching” imaginary numbers to real numbers using a plus sign (+). The standard form of a complex number is a + bi , where a and b are real. The number a is called the real part of the complex number, and the b number is called the imaginary part. If b = 0, then the complex number is just a real number. If b  0, the complex number is called imaginary. If a = 0, bi is called a pure imaginary number. Examples of imaginary numbers are 2 + 3i, –  + 4i, 6i, 0.11 + (–0.45)i, and  – i.

When the imaginary part of a complex number is a radical, write the i to the left in order to avoid ambiguity about whether i is under the radical.

Finding sums, differences, products, quotients, and reciprocals of complex numbers can be accomplished directly on your calculator. The imaginary unit i is 2nd decimal point. If you enter an expression with i in it, the calculator will do imaginary arithmetic in REAL mode. If the expression entered does not include i but the output is imaginary, the calculator gives you the error message NONREAL ANS. For example, if you tried to calculate  in REAL mode, you would get this error message. In a + bi mode,  would calculate as 1.732···i . You should use a + bimode exclusively for the Level 2 test. Although complex number arithmetic per se is not likely to be on a Level 2 test, an understanding of how it is done may be. A review of the main features of complex number arithmetic is provided in the next several paragraphs.

To add or subtract complex numbers, add or subtract their real and imaginary parts. For example,

(5 – 7i ) + (2 + 4i ) = 7 – 3i .

To multiply complex numbers, multiply like you would any two binomial expressions, using FOIL. Thus

(a + bi )(c + di ) = ac + adi + bci + bdi 2 = (ac – bd ) + (ad + bc )i .

The difference of the first and last terms makes the real part, and the sum of the outer and inner terms makes the imaginary part.

To find the quotient of two complex numbers, multiply the denominator and numerator by the conjugate of the denominator. Then simplify. For example,


1. Write the product of (2 + 3i )(4 – 5i ) in standard form.

      (A)  –7 – 23i

      (B)  –7 + 2i

      (C)  23 – 7i

      (D)  23 + 2i

      (E)  23 – 2i

2. Write  in standard form.




      (D)  –1 + 2i

      (E)  –1–2i

3. If z = 8 – 2i , 2 =

      (A)  60 − 32i

      (B)  64 + 4i

      (C)  64–4i

      (D)  60

      (E)  68