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Imaginary and Complex Numbers

Author: Sophia
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what's covered
In this lesson, you will learn how to determine the value of a power of i. Specifically, this lesson will cover:

Table of Contents

1. Complex Numbers

A complex number is a number that contains a real part and an imaginary part, such as 6 minus 5 i. In mathematics, we deal with real numbers all the time. These are numbers that we can place on the number line, such as integers, decimals and fractions, and rational and irrational numbers. An imaginary number is not real, and contains the imaginary unit i, which is square root of negative 1 end root. This is non-real because every real number squared is non-negative. So when a negative number is underneath a square root, there is no real number that it evaluates to.

In general, we write complex numbers in the form a plus b i. a is the real number component to the complex number, and b i is the imaginary number component to the complex number. Complex numbers represent a larger set of numbers than real numbers do, because the complex number system includes real numbers, and also includes the set of all imaginary numbers as well.

term to know
Complex Number
A number in the form a plus b i, containing a real part, a, and an imaginary part, bi, where i is the imaginary unit, square root of short dash 1 end root.

2. The Complex Plane

We can represent complex numbers on what is called the complex plane. It is similar to the coordinate plane we use for graphing, but instead of x- and y-axes, we have a real axis and an imaginary axis. This is shown below:

A graph (complex plane) representing complex numbers, where the complex plane uses terms such as ‘real axis’ and ‘imaginary axis’. Here, the x-axis represents the ‘real axis’, and the y-axis represents the ‘imaginary axis’.

We can plot complex numbers on the complex plane following a very similar process for plotting coordinate points (x, y) on the coordinate plane.

EXAMPLE

Plot the following points on the complex plane:

  • 2 plus 5 i
  • short dash 3 minus 4 i
  • short dash 2 i
  • 3
A complex plane, where the x-axis represents the ‘real axis’, the y-axis represents the ‘imaginary axis’, and both range from −7 to 7. The following four points are marked on the graph: −3 − 4i in the third quadrant, 3 units to the left and 4 units down form the origin; 2 + 5i in the first quadrant, 2 units to the right and 5 units up from the origin; −2i on the negative y-axis, two units below the origin; 3 on the positive x-axis, three units to the right of the origin.

Notice that there are positive and negative sides to the complex plane, just as there are positive and negative sides to the coordinate plane. Also, note the numbers negative 2 i and 3. These numbers lie on one of the axes of the plane because they are either purely imaginary (having no real component) or purely real (having no imaginary component).

3. Powers of i

There is an interesting pattern with the powers of the imaginary unit, i. The pattern is cyclical, which means that it repeats in cycles. Let's examine some of the powers of i:

Power of i Calculation Result
i to the power of 1 i to the power of 1 equals i i
i squared i squared equals open parentheses square root of short dash 1 end root close parentheses squared equals short dash 1 short dash 1
i cubed i cubed equals i squared times i equals short dash 1 times i equals short dash i short dash i
i to the power of 4 i to the power of 4 equals i squared times i squared equals open parentheses short dash 1 close parentheses open parentheses short dash 1 close parentheses equals 1 1
i to the power of 5 i to the power of 5 equals i to the power of 4 times i equals 1 times i equals i i
i to the power of 6 i to the power of 6 equals i to the power of 5 times i equals i times i equals i squared equals short dash 1 short dash 1
i to the power of 7 i to the power of 7 equals i to the power of 6 times i equals short dash 1 times i equals short dash i short dash i
i to the power of 8 i to the power of 8 equals i to the power of 4 times i to the power of 4 equals open parentheses 1 close parentheses open parentheses 1 close parentheses equals 1 1

At this point, you can see that the powers of i follow the pattern: i comma space short dash 1 comma space short dash i comma space 1 comma space i comma space minus 1 comma space...

Notice that if the power is a multiple of 4, then the result is 1.

Since the pattern repeats, it is important to know the first four powers of i:

formula to know
Imaginary Number
table attributes columnalign left end attributes row cell i equals square root of short dash 1 end root end cell row cell i squared equals short dash 1 end cell row cell i cubed equals short dash i end cell row cell i to the power of 4 equals 1 end cell end table

EXAMPLE

Simplify i to the power of 32.

32 equals 4 times 8 Rewrite 32 in factored form so that 4 is one if its factors.
i to the power of 32 equals i to the power of 4 times 8 end exponent equals open parentheses i to the power of 4 close parentheses to the power of 8 Use exponent properties to rewrite.
equals 1 to the power of 8 i to the power of 4 equals 1.
equals 1 Simplify.

As you can see, this process can be used if the power is any multiple of 4.

How can we find any power of i? The idea is that we look for the closest multiple of 4.

EXAMPLE

Simplify i to the power of 23.

Since 23 is not a multiple of 4, we look for the closest multiple of 4 that is smaller than 23, which is 20. Then, we split up the expression as follows:

i to the power of 23 equals i to the power of 20 plus 3 end exponent equals i to the power of 20 times i cubed Split the powers into one that’s a multiple of 4.
equals 1 times i cubed We know i to the power of 20 equals 1 since 20 is a multiple of 4.
equals 1 times open parentheses short dash i close parentheses i cubed equals short dash i.
equals short dash i Simplify.

Therefore, i to the power of 23 equals short dash i.

Another method to simplify higher powers of i is to use this algorithm:

  • Divide the power by 4 using long division, then record the remainder.
  • Replace the power with the remainder.
  • If the reminder is 0, then the power was a multiple of 4, which means the result is 1.

EXAMPLE

Simplify i to the power of 58.

When 4 is divided into 58, the quotient is 14 and the remainder is 2.

Then, i to the power of 58 equals i squared equals short dash 1.

We can use our knowledge of powers of i to simplify more complex expressions.

EXAMPLE

Simplify open parentheses 3 i close parentheses squared.

If you think you should simplify this like an ordinary algebraic expression, you are right!

open parentheses 3 i close parentheses squared equals 3 squared i squared Use properties of exponents.
equals 9 open parentheses short dash 1 close parentheses 3 squared equals 9 comma i squared equals short dash 1
equals short dash 9 Simplify.

Therefore, open parentheses 3 i close parentheses squared equals short dash 9.

Looking at the previous example, we saw that open parentheses 3 i close parentheses squared equals short dash 9. This also means that square root of short dash 9 end root equals 3 i!

With the use of imaginary numbers, we can rewrite square roots of negative numbers. To do so, we use the property square root of a b end root equals square root of a square root of b.

EXAMPLE

Suppose we want to rewrite square root of short dash 9 end root. Since i equals square root of short dash 1 end root comma we have the following:

square root of short dash 9 end root equals square root of 9 open parentheses short dash 1 close parentheses end root Write -9 in factored form with -1 as one factor.
equals square root of 9 square root of short dash 1 end root Use properties of radicals to write as a product of radicals.
equals 3 i square root of 9 equals 3 and square root of short dash 1 end root equals i.

Let’s look at an example where the radicand is not a perfect square.

EXAMPLE

Simplify square root of short dash 50 end root.

Since the radicand is negative, we know the final answer will include i.

square root of short dash 50 end root equals square root of 50 open parentheses short dash 1 close parentheses end root Write -50 in factored form with -1 as one factor.
equals square root of 50 square root of short dash 1 end root Use properties of radicals to write as a product of radicals.
equals square root of 50 times i square root of short dash 1 end root equals i.

Note that the “times” sign is left in the expression so that the i is not "too close" to the radical.

Now let's simplify square root of 50 times i.

square root of 50 times i equals square root of 25 times 2 end root times i Rewrite 50 equals 25 times 2 since 25 is a perfect square.
equals square root of 25 square root of 2 times i Separate the square roots.
equals 5 square root of 2 times i square root of 25 equals 5

While this answer is simplified, it is preferable to write the answer as 5 i square root of 2. Basically, the “nonradical” terms are written together before the radical term is. The expression 5 square root of 2 times i could also be written 5 square root of 2 i comma but this can be confusing since it’s difficult to tell whether the i is not under the radical (it’s not).

Recall that the solutions to the quadratic equation a x squared plus b x plus c equals 0 are x equals fraction numerator short dash b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction.

When the discriminant b squared minus 4 a c is negative, the equation is no real solution. With our knowledge of imaginary numbers, we can express the nonreal solutions in terms of i.

EXAMPLE

Find the solutions of the equation x squared plus 6 x plus 13 equals 0.

We’ll use the quadratic formula, then simplify as usual.

a equals 1 comma space b equals 6 comma space c equals 13 Identify a comma b, and c.
x equals fraction numerator short dash 6 plus-or-minus square root of 6 squared minus 4 open parentheses 1 close parentheses open parentheses 13 close parentheses end root over denominator 2 open parentheses 1 close parentheses end fraction Replace all variables with their values.
x equals fraction numerator short dash 6 plus-or-minus square root of short dash 16 end root over denominator 2 end fraction Simplify the radicand: 6 squared minus 4 open parentheses 1 close parentheses open parentheses 13 close parentheses equals short dash 16
Simplify the denominator: 2 open parentheses 1 close parentheses equals 2
square root of short dash 16 end root equals square root of 16 square root of short dash 1 end root equals 4 i Simplify the square root.
x equals fraction numerator short dash 6 plus-or-minus 4 i over denominator 2 end fraction Replace square root of short dash 16 end root with 4 i.
x equals short dash 3 plus-or-minus 2 i Perform the division by 2 across all terms in the numerator.

Therefore, the solutions to the equation are x equals short dash 3 plus 2 i and x equals short dash 3 minus 2 i.

Finding complex solutions to a quadratic equation is very useful in the field of electrical engineering.

summary
Complex numbers consist of a real part and an imaginary part. Looking at the complex plane, complex numbers encompass a larger set of numbers than real numbers, because they include imaginary numbers. The square root of negative 1 is imaginary, because no real number squared results in negative number. When looking at increasing powers of i, the solutions follow a pattern of i, negative 1, negative i, and 1, continuously.

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE. ACCESS FOR FREE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
Complex Number

A number in the form a plus b i, containing a real part, a, and an imaginary part, bi, where i is the imaginary unit, square root of short dash 1 end root.

Formulas to Know
Imaginary Number

table attributes columnalign left end attributes row cell i equals square root of short dash 1 end root end cell row cell i squared equals short dash 1 end cell row cell i cubed equals short dash i end cell row cell i to the power of 4 equals 1 end cell end table

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