Cogito
Integrated Math 2 · Chapter 3 · Lesson 1
The Imaginary Unit
A square root that had to be invented.
12 problems · about 20 minutes · N-CN.A.1
What this lesson teaches
The student defines i, simplifies square roots of negatives, and evaluates powers of i.
- i is defined by i² = −1.
- Split the negative out of a root: √(−9) = 3i.
- Powers of i cycle every four.
Warm Up
Straightforward practice. Get the method working first.
5 problems√(−36) = ki. What is k?
Answer 6
Why 6.
Why was i introduced?
Answer No real number squares to a negative, so some equations had no solutions.
Why To give every quadratic a solution.
√(−25) = ki. What is k?
Answer 5
Why √25.
√(−49) = ki. What is k?
Answer 7
Why √49.
What is i⁴?
Answer 1
Why (i²)².
Build It Up
The same ideas with more to keep track of.
3 problemsWhat is i⁸?
Answer 1
Why 8 divided by 4 leaves no remainder.
For i²³, what is the remainder when 23 is divided by 4?
Answer 3
Why 23 = 20 + 3.
For 3 + 5i, what is the real part?
Answer 3
Why The term without i.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each power of i with its value.
Answer i squared → Negative one; i to the fourth → One; i cubed → Negative i
Why The cycle has length four.
√(−81) = ki. What is k?
Answer 9
Why √81.
The Root: √(−16) = ki. What is k?
Answer 4
Why 4.
The Square: What is i² ?
Answer -1
Why −1.