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.