Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple.
In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it.
We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary numberis defined as the square root of
So, using properties of radicals,
We can write the square root of any negative number as a multiple ofConsider the square root of
We useand notbecause the principal root ofis the positive root.
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when writtenwhereis the real part andis the imaginary part. For example,is a complex number. So, too, is
Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.
A complex number is a number of the formwhere
Ifthenis a real number. Ifandis not equal to 0, the complex number is called a pure imaginary number. An imaginary number is an even root of a negative number.
Given an imaginary number, express it in the standard form of a complex number.
Expressin standard form.
In standard form, this is
Expressin standard form.
We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number, we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairswhererepresents the coordinate for the horizontal axis andrepresents the coordinate for the vertical axis.
Let’s consider the numberThe real part of the complex number isand the imaginary part is 3. We plot the ordered pairto represent the complex numberas shown in [link].
In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis, as shown in [link].
Given a complex number, represent its components on the complex plane.
Plot the complex numberon the complex plane.
The real part of the complex number isand the imaginary part is –4. We plot the ordered pairas shown in [link].
Plot the complex numberon the complex plane.
Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts.
Adding complex numbers:
Subtracting complex numbers:
Given two complex numbers, find the sum or difference.
Add or subtract as indicated.
We add the real parts and add the imaginary parts.
Subtractfrom
Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.
Lets begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. Consider, for example,:
Given a complex number and a real number, multiply to find the product.
Find the product
Distribute the 4.
Find the product:
Now, lets multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term,it equals
Given two complex numbers, multiply to find the product.
Multiply
Multiply:
Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard formWe need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate ofisFor example, the product ofandis
The result is a real number.
Note that complex conjugates have an opposite relationship: The complex conjugate ofisand the complex conjugate ofisFurther, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.
Suppose we want to dividebywhere neithernorequals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.
Multiply the numerator and denominator by the complex conjugate of the denominator.
Apply the distributive property.
Simplify, remembering that
The complex conjugate of a complex numberisIt is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.
Find the complex conjugate of each number.
Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by
Find the complex conjugate of
Given two complex numbers, divide one by the other.
Divide:by
We begin by writing the problem as a fraction.
Then we multiply the numerator and denominator by the complex conjugate of the denominator.
To multiply two complex numbers, we expand the product as we would with polynomials (using FOIL).
Note that this expresses the quotient in standard form.
The powers ofare cyclic. Let’s look at what happens when we raiseto increasing powers.
We can see that when we get to the fifth power ofit is equal to the first power. As we continue to multiply by increasing powers, we will see a cycle of four. Let’s examine the next four powers of
The cycle is repeated continuously:every four powers.
Evaluate:
Sincewe can simplify the problem by factoring out as many factors ofas possible. To do so, first determine how many times 4 goes into 35:
Evaluate:
Can we writein other helpful ways?
As we saw in [link], we reducedtoby dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization ofmay be more useful. [link] shows some other possible factorizations.
Factorization of | ||||
Reduced form | ||||
Simplified form |
Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.
Access these online resources for additional instruction and practice with complex numbers.
Explain how to add complex numbers.
Add the real parts together and the imaginary parts together.
What is the basic principle in multiplication of complex numbers?
Give an example to show that the product of two imaginary numbers is not always imaginary.
Possible answer:timesequals 1, which is not imaginary.
What is a characteristic of the plot of a real number in the complex plane?
For the following exercises, evaluate the algebraic expressions.
Ifevaluategiven
Ifevaluategiven
Ifevaluategiven
Ifevaluategiven
Ifevaluategiven
Ifevaluategiven
For the following exercises, plot the complex numbers on the complex plane.
For the following exercises, perform the indicated operation and express the result as a simplified complex number.
25
For the following exercises, use a calculator to help answer the questions.
EvaluateforPredict the value if
EvaluateforPredict the value if
128i
EvaluateforPredict the value for
Show that a solution ofis
Show that a solution ofis
For the following exercises, evaluate the expressions, writing the result as a simplified complex number.
0