Complex Numbers

Imaginary and Complex Numbers

Imaginary Unit

A solution of the equation x2=1x^2 = -1 is defined as the imaginary unit, written as ii.
A real number is a number in the set of all rational and irrational numbers. The equation x2=1x^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 ii.
i=1i=\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 ii are called pure imaginary numbers, written in the form bibi, where bb is a real number.

An imaginary number, also known as a pure imaginary number, is a number of the form bibi, where bb is a real number and ii is the imaginary unit. Some examples are 12i12i and i19i\sqrt{19}.

The square root of any negative number can be rewritten as a pure imaginary number. For example, if xx is a positive number, then x-x is a negative number, and the square root of x-x can be written as a pure imaginary number.
x=ix\sqrt{-x}=i\sqrt{x}
Step-By-Step Example
Simplifying the Square Root of a Negative Number
Rewrite 5\sqrt{-5} as a pure imaginary number.
Step 1
Rewrite the radicand, factoring out –1.
5=15\sqrt{-5}=\sqrt{-1\cdot5}
Step 2
To factor out the imaginary unit, rewrite the square root of the product as the product of square roots.
15=15\sqrt{-1\cdot5}=\sqrt{-1}\cdot\sqrt{5}
Solution
Rewrite the product in terms of ii.
5=15=i5=i5\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+bia + bi, where aa and bb 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+bia+bi for any real numbers aa and bb. The real part of the complex number is aa, and the imaginary part is bb.
Complex Number Real Part Imaginary Part
6+i36+i\sqrt3 66 3\sqrt3
75i\sqrt7 -5i 7\sqrt7 5-5
12i12i 00 1212
19\sqrt{19} 19\sqrt{19} 00

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 12i12i is 0+12i0+12i, which shows that its real part is zero. The standard form of the complex number 19\sqrt{19} is 19+0i\sqrt{19}+0i, which shows that its imaginary part is zero.

Real and imaginary numbers are both subsets of complex numbers:

  • When a=0a=0, a+bi=bia+bi=bi, which is a pure imaginary number.
  • When b=0b=0, a+bi=aa+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+7.4i8+7.4i and 3.95i3.9-5i, are neither real numbers nor pure imaginary numbers.
Step-By-Step Example
Simplifying a Complex Number
Write the standard form of the complex number:
81+3\sqrt{-81}+3
Step 1
Rewrite any square roots of negative numbers as pure imaginary numbers.
81=181=181=9i\begin{aligned}\sqrt{-81}&=\sqrt{-1 \cdot 81}\\&=\sqrt{-1} \cdot \sqrt{81}\\&=9i\end{aligned}
Step 2
Substitute the pure imaginary number into the original expression.
81+3(9i)+3\begin{gathered}\sqrt{-81}+3\\(9i)+3\end{gathered}
Solution
Rearrange the number in the form a+bia+bi.
3+9i3+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 xx- and yy-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+bia+bi is located at the point (a,b)(a,b).
  • If a=0a=0, then the number bibi is located on the imaginary axis.
  • If b=0b=0, then the number aa is located on the real axis.
To graph the complex number a+bia+bi, plot the point (a,b)(a,b) in the complex plane.
Step-By-Step Example
Graphing Numbers in the Complex Plane

Graph each number in the complex plane:

1) 9i39i-3

2) 4\sqrt{-4}

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

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

5) –3

Step 1
Write each number in the standard form of a complex number:
a+bia+bi
For 9i39i-3, rearrange the number in standard form:
9i3=3+9i9i - 3=-3+9i
For 4\sqrt{-4}, rewrite the square root of the negative number as a pure imaginary number, and then write in standard form:
4=14=2i=0+2i\begin{aligned}\sqrt{-4}&=\sqrt{-1\cdot4}\\&=2i\\&=0+2i\end{aligned}
For 64+5-\sqrt{-64}+5, rewrite the square root of the negative number as a pure imaginary number, and then write in standard form:
64+5=164+5=8i+5=58i\begin{aligned}-\sqrt{-64}+5&=-\sqrt{-1\cdot64}+5\\&=-8i+5\\&=5-8i\end{aligned}
For 3+2\sqrt{3}+\sqrt{-2}, rewrite the square root of the negative number as a pure imaginary number:
3+2=3+12=3+i2\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-3=-3+0i
Step 2

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

For 3+9i-3+9i, the value of aa is –3. The value of bb is 9. The coordinates of the point are (3,9)(-3,9).

For 0+2i0+2i, the value of aa is zero. The value of bb is 2. The coordinates are (0,2)(0,2).

For 58i5-8i, the value of aa is 5. The value of bb is –8. The coordinates are (5,8)(5,-8).

For 3+i2\sqrt{3}+i\sqrt{2}, the value of aa is 3\sqrt{3}. The value of bb is 2\sqrt2. The coordinates are (3,2)(\sqrt3,\sqrt2), or about (1.7,1.4)(1.7,1.4).

For 3+0i-3+0i, the value of aa is 3-3. The value of bb is zero. The coordinates are (3,0)(-3,0).

Solution
Plot the points.