x² + 1 = 0 has no real solution, because no real number squares to a negative. The number system needed extending.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The definition
i is defined so that i² = −1. Everything else follows from that single fact.
Roots of negatives
√(−9) = √9 × √(−1) = 3i. Split off the negative first, then simplify what remains.
Powers cycle
i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and then it repeats. Divide the exponent by 4 and use the remainder.
Complex numbers
A complex number is a + bi, with a the real part and b the imaginary part. Every real number is one, with b = 0.
Not imaginary in the ordinary sense
The name is historical and unfortunate. These numbers describe alternating current and quantum states, which are entirely real phenomena.
A square root that had to be invented
No real number squares to a negative, so equations with negative discriminants had no solutions. Defining i with i² = −1 supplies them. The extension was forced by a problem, not chosen for elegance.
This has happened before
Negative numbers and irrational numbers were each rejected as impossible before being accepted. Every extension of the number system began as a response to an equation with no solution.
Powers of i cycle in fours
i, −1, −i, 1, then repeat. Any power reduces by dividing the exponent by 4 and reading the remainder. The cycle reflects multiplication by i being a quarter turn in the plane.
The name is unfortunate
"Imaginary" was coined dismissively. Complex numbers describe alternating current, wave mechanics and signal processing entirely concretely, and they are no less real than negative numbers.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 4) · length 5
Step 3: Watch an Example
One step at a time.
Watch Rosa Evaluate a Power
Rosa needs i²³.
- Step 1
The powers of i repeat every 4, so she divides 23 by 4.
Step 4: Your Turn
Practice makes it stick.
The Root
Problem 1 of 2
√(−16) = ki. What is k?
The Square
Problem 2 of 2
What is i² ?
Work With i
1 of 8
√(−25) = ki. What is k?
2 of 8
√(−49) = ki. What is k?
3 of 8
What is i⁴?
4 of 8
What is i⁸?
5 of 8
For i²³, what is the remainder when 23 is divided by 4?
6 of 8
For 3 + 5i, what is the real part?
7 of 8
Match each power of i with its value.
Tap a card on the left to start.
8 of 8
√(−81) = ki. What is k?
Step 5: Quick Check
Show what you know.
Question 1 of 2
√(−36) = ki. What is k?
Question 2 of 2
Why was i introduced?
What You Learned
- i is defined by i² = −1.
- Split the negative out of a root: √(−9) = 3i.
- Powers of i cycle every four.