x² = −1 has no real solution, because no real number squares to a negative.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A new number is defined to fill the gap.
The imaginary unit is i, the number whose square is −1.
Complex numbers
a + bi has a real part a and an imaginary part b. Every real number is complex with b = 0.
Why bother
With i, every polynomial of degree n has exactly n roots. Nothing is missing any more.
What happens when the discriminant is negative
A negative discriminant means no real root exists. Rather than stopping, mathematicians defined i with i² = −1, which supplies the missing solutions. The number system was extended because a problem demanded it, not for decoration.
The standard form
A complex number is a + bi, with a the real part and b the imaginary part. Real numbers are the case where b is zero, so the reals sit inside the complex numbers rather than beside them.
The name is unfortunate
"Imaginary" was coined dismissively and stuck. Complex numbers are no less real than negative numbers, which were also once rejected. They describe alternating current, quantum states and signal processing entirely concretely.
They live on a plane
Plotting a + bi at the point (a, b) gives the complex plane. Multiplication by i is a 90° rotation, which makes the algebra geometric and explains why complex numbers are the natural language for anything involving rotation.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Ravi Solve x² + 4 = 0
Ravi solves an equation with no real solution.
- Step 1
He rearranges to x² = −4.
Step 4: Your Turn
Practice makes it stick.
The Square
Problem 1 of 2
What is i²?
Adding
Problem 2 of 2
(3 + 2i) + (5 + 4i). What is the real part?
Complex Arithmetic
1 of 4
(2 + 3i) + (4 + 5i). Imaginary part?
2 of 4
What is i⁴?
3 of 4
(6 + 7i) − (2 + 3i). Real part?
4 of 4
x² = −9. What is the coefficient of i in the positive root?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x² = −25. What is the coefficient of i in the positive root?
Question 2 of 2
What is i defined to be?
What You Learned
- i is defined by i² = −1.
- A complex number is a + bi.
- With complex numbers, every polynomial has its full set of roots.