Imaginary and Complex Numbers

Imaginary Unit

A solution of the equation

x^2 = -1

is defined as the imaginary unit, written as

i

A real number is a number in the set of all rational and irrational numbers. The equation

x^2 = -1

does not have a solution in the set of real numbers because the square of any real number is positive. To solve certain types of problems, mathematicians defined a number to be the solution of this equation. The imaginary unit is the square root of –1, written as

i

i=\sqrt{-1}

Even though square roots of negative numbers are referred to as imaginary, they have uses in many areas of mathematics and science. For example, electrical engineers work with square roots of negative numbers when analyzing electric circuits.

Pure Imaginary Numbers

Multiples of

i

are called pure imaginary numbers, written in the form

bi

, where

b

is a real number.

An imaginary number, also known as a pure imaginary number, is a number of the form $bi$ , where $b$ is a real number and $i$ is the imaginary unit. Some examples are $12i$ and $i\sqrt{19}$ .

The square root of any negative number can be rewritten as a pure imaginary number. For example, if

x

is a positive number, then

-x

is a negative number, and the square root of

-x

can be written as a pure imaginary number.

\sqrt{-x}=i\sqrt{x}

Simplifying the Square Root of a Negative Number

Rewrite

\sqrt{-5}

as a pure imaginary number.

Rewrite the radicand, factoring out –1.

\sqrt{-5}=\sqrt{-1\cdot5}

To factor out the imaginary unit, rewrite the square root of the product as the product of square roots.

\sqrt{-1\cdot5}=\sqrt{-1}\cdot\sqrt{5}

Rewrite the product in terms of

i

\begin{aligned}\sqrt{-5}&=\sqrt{-1}\cdot\sqrt{5}\\&=i\cdot\sqrt{5}\\&=i\sqrt{5}\end{aligned}

Defining Complex Numbers

A complex number is the sum of a real number and a pure imaginary number. Complex numbers can be written in the form

a + bi

, where

a

and

b

are real numbers.

Pure imaginary numbers can be combined with real numbers to form a different type of number. A complex number is the sum of a real number and a pure imaginary number. The standard form of a complex number is

a+bi

for any real numbers

a

and

b

. The real part of the complex number is

a

, and the imaginary part is

b

Complex Number	Real Part	Imaginary Part
$6+i\sqrt3$	$6$	$\sqrt3$
$\sqrt7 -5i$	$\sqrt7$	$-5$
$12i$	$0$	$12$
$\sqrt{19}$	$\sqrt{19}$	$0$

All complex numbers have a real part and an imaginary part, although one or both of these parts may be equal to zero. For example, the standard form of the complex number $12i$ is $0+12i$ , which shows that its real part is zero. The standard form of the complex number $\sqrt{19}$ is $\sqrt{19}+0i$ , which shows that its imaginary part is zero.

Real and imaginary numbers are both subsets of complex numbers:

When $a=0$ , $a+bi=bi$ , which is a pure imaginary number.
When $b=0$ , $a+bi=a$ , which is a real number.

All real numbers and all pure imaginary numbers are also complex numbers. However, some complex numbers, such as

8+7.4i

and

3.9-5i

, are neither real numbers nor pure imaginary numbers.

Simplifying a Complex Number

Write the standard form of the complex number:

\sqrt{-81}+3

Rewrite any square roots of negative numbers as pure imaginary numbers.

\begin{aligned}\sqrt{-81}&=\sqrt{-1 \cdot 81}\\&=\sqrt{-1} \cdot \sqrt{81}\\&=9i\end{aligned}

Substitute the pure imaginary number into the original expression.

\begin{gathered}\sqrt{-81}+3\\(9i)+3\end{gathered}

Rearrange the number in the form

a+bi

3+9i

Complex Plane

Complex numbers can be graphed in a coordinate plane with a real axis and an imaginary axis.

A coordinate plane is used to locate points in terms of distance from the $x$ - and $y$ -axes. The complex plane is used to locate points that represent complex numbers in terms of distance from the real axis and the imaginary axis. The real axis is the horizontal axis in the complex plane and represents the set of real numbers. The imaginary axis is the vertical axis in the complex plane and represents the set of pure imaginary numbers.

The number $a+bi$ is located at the point $(a,b)$ .
If $a=0$ , then the number $bi$ is located on the imaginary axis.
If $b=0$ , then the number $a$ is located on the real axis.

To graph the complex number

a+bi

, plot the point

(a,b)

in the complex plane.

Graphing Numbers in the Complex Plane

Graph each number in the complex plane:

1) $9i-3$

2) $\sqrt{-4}$

3) $-\sqrt{-64}+5$

4) $\sqrt{3}+\sqrt{-2}$

5) –3

Write each number in the standard form of a complex number:

a+bi

For

9i-3

, rearrange the number in standard form:

9i - 3=-3+9i

For

\sqrt{-4}

, rewrite the square root of the negative number as a pure imaginary number, and then write in standard form:

\begin{aligned}\sqrt{-4}&=\sqrt{-1\cdot4}\\&=2i\\&=0+2i\end{aligned}

For

-\sqrt{-64}+5

, rewrite the square root of the negative number as a pure imaginary number, and then write in standard form:

\begin{aligned}-\sqrt{-64}+5&=-\sqrt{-1\cdot64}+5\\&=-8i+5\\&=5-8i\end{aligned}

For

\sqrt{3}+\sqrt{-2}

, rewrite the square root of the negative number as a pure imaginary number:

\begin{aligned}\sqrt3+\sqrt{-2}&=\sqrt{3}+\sqrt{-1 \cdot 2}\\&=\sqrt3+i\sqrt{2}\end{aligned}

For -3, write the number in standard form:

-3=-3+0i

Identify the coordinates of each point, and write them in the form $(a,b)$ .

For $-3+9i$ , the value of $a$ is –3. The value of $b$ is 9. The coordinates of the point are $(-3,9)$ .

For $0+2i$ , the value of $a$ is zero. The value of $b$ is 2. The coordinates are $(0,2)$ .

For $5-8i$ , the value of $a$ is 5. The value of $b$ is –8. The coordinates are $(5,-8)$ .

For $\sqrt{3}+i\sqrt{2}$ , the value of $a$ is $\sqrt{3}$ . The value of $b$ is $\sqrt2$ . The coordinates are $(\sqrt3,\sqrt2)$ , or about $(1.7,1.4)$ .

For $-3+0i$ , the value of $a$ is $-3$ . The value of $b$ is zero. The coordinates are $(-3,0)$ .

Plot the points.

<Vocabulary >Operations with Complex Numbers

Complex Numbers

Contents

Imaginary and Complex Numbers

Imaginary Unit

Pure Imaginary Numbers

Defining Complex Numbers

Complex Plane