5.8: Defining Complex Numbers
The coldest possible temperature, known as absolute zero is almost –460 degrees Fahrenheit. What is the square root of this number?
Watch This
First, watch this video.
Khan Academy: Introduction to i and Imaginary Numbers
Then, watch this video.
Guidance
Before this concept, all numbers have been real numbers. 2, -5, , and are all examples of real numbers. Look at #1 from the Review Queue. With what we have previously learned, we cannot find because you cannot take the square root of a negative number. There is no real number that, when multiplied by itself, equals -25. Let’s simplify .
In order to take the square root of a negative number we are going to assign a variable, . represents an imaginary number . Now, we can use to take the square root of a negative number.
All complex numbers have the form , where and are real numbers. is the real part of the complex number and is the imaginary part . If , then is left and the number is a real number. If , then the number is only and called a pure imaginary number . If and , the number will be an imaginary number.
Example A
Find .
Solution: First pull out the . Then, simplify .
Investigation: Powers of i
In addition to now being able to take the square root of a negative number, also has some interesting properties. Try to find and .
1. Write out and simplify.
2. Write out and simplify.
3. Write out and simplify.
4. Write out and simplify.
5. Write out and simplify.
6. Do you see a pattern? Describe it and try to find .
You should see that the powers of repeat every 4 powers. So, all the powers that are divisible by 4 will be equal to 1. To find , divide 19 by 4 and determine the remainder. That will tell you what power it is the same as.
Example B
Find:
a)
b)
c)
Solution:
a) 32 is divisible by 4, so .
b) , with a remainder of 2. Therefore, .
c) , with a remainder of 3. Therefore,
Example C
Simplify the complex expressions.
a)
b)
Solution: To add or subtract complex numbers, you need to combine like terms. Be careful with negatives and properly distributing them. Your answer should always be in standard form , which is .
a)
b)
Intro Problem Revisit We're looking for .
First we need to pull out the . Then, we need to simplify .
Guided Practice
Simplify.
1.
2.
3.
4.
Answers
1. Rewrite in terms of and simplify the radical.
2. Rewrite in terms of and simplify the radical.
3. , with a remainder of 2. Therefore, .
4. Distribute the negative and combine like terms.
Vocabulary
- Imaginary Numbers
- Any number with an associated with it. Imaginary numbers have the form or .
- Complex Numbers
- All real and imaginary numbers. Complex numbers have the standard form , where or can be zero. is the real part and is the imaginary part .
- Pure Imaginary Numbers
- An imaginary number without a real part, only .
Practice
Simplify each expression and write in standard form.
Absolute Value
The absolute value of a number is the distance the number is from zero. The absolute value of a complex number is the distance from the complex number on the complex plane to the origin.Complex Conjugate
Complex conjugates are pairs of complex binomials. The complex conjugate of is . When complex conjugates are multiplied, the result is a single real number.i
is an imaginary number. .Imaginary Number
An imaginary number is a number that can be written as the product of a real number and .imaginary part
The imaginary part of a complex number is .Pure Imaginary Numbers
The pure imaginary numbers are the subset of complex numbers without real parts, only .Real Number
A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.real part
The real part of a complex number is .rectangular coordinates
A point is written using rectangular coordinates if it is written in terms of and and can be graphed on the Cartesian plane.rectangular form
The rectangular form of a point or a curve is given in terms of and and is graphed on the Cartesian plane.standard form
The standard form of a complex number is where and are real numbers.Image Attributions
Description
Learning Objectives
Here you'll define, discover the “powers of ,” and add and subtract complex and imaginary numbers.