## SAT SUBJECT TEST MATH LEVEL 1

## MISCELLANEOUS TOPICS

## CHAPTER 17

Imaginary and Complex Numbers

### COMPLEX NUMBERS

The imaginary unit can be added to and multiplied by real numbers to form ** complex numbers**. Every complex number can be written in the form

*a*+

*bi*, where

*a*and

*b*are real numbers.

*a*is called the

**and**

*real part**bi*the

**of the complex number**

*imaginary part**a*+

*bi*. Two complex numbers are equal if, and only if, their real parts are equal and their imaginary parts are equal.

**Key Fact P3**

**If a** +

**=**

*bi***+**

*c***=**

*di*, then*a***=**

*c*and*b*

*d.***EXAMPLE 4:** If 2(3 + *yi* ) = *x* + 8*i*, what are the values of *x* and *y* ?

*x* + 8*i* = 2(3 + *yi* ) *x* + 8*i* = 6 + 2*yi*

So, *x* = 6 and 8 = 2*y* ⇒ *x* = 6 and *y* = 4

The arithmetic of complex numbers follows all the rules you are familiar with for real numbers.

**Key Fact P4**

• **To add complex numbers, add their real parts and add their imaginary parts. For example:**

**(3** + **5 i )** +

**(2**+

**3**=

*i*)**5**+

**8**

*i*• **To subtract complex numbers, subtract their real parts and subtract their imaginary parts. For example:**

**(3** + **5 i )** –

**(2**+

**3**=

*i*)**1**+

**2**

*i*• **To multiply complex numbers, “FOIL” them as if they were binomials and replace i ^{2} by** –

**1. For example:**

Complex numbers can also be divided, but you do not need to know how to do this for the Math 1 test.