Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 complex numbers? Then this Algebra 2 lesson on Real and Complex Numbers: Working with i explains each idea step by step, with worked examples.
Quick answer: A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit with i2 = −1. Every real number is complex with b = 0. Complex numbers add, subtract, and multiply like binomials, and each i2 becomes −1.
Standards: N-RN.A.1, N-RN.A.2, N-CN.A.1, N-CN.A.2 · Level: Grade 11 Algebra 2
How do the number sets fit together?
Every number belongs to a nested family. Natural numbers are 1, 2, 3, …; integers add 0 and the negatives; rational numbers are quotients of integers such as −7/4, and their decimals stop or repeat. Irrational numbers such as √17 and π have decimals that never stop and never repeat. Rational and irrational numbers together fill the number line: they are the real numbers.
What is the imaginary unit?
Squaring a real number never gives a negative result, so a new number is defined to fill the gap.
The imaginary unit is i = √(−1), so i2 = −1. A complex number is any number a + bi with a and b real. Here a is the real part and b is the imaginary part. For n > 0, √(−n) = i√n.
How do you add, subtract, and multiply complex numbers?
| Operation | Rule | Example |
|---|---|---|
| Add | Combine real parts and imaginary parts | (2 + 5i) + (4 − 9i) = 6 − 4i |
| Subtract | Distribute the minus sign, then combine | (2 + 5i) − (4 − 9i) = −2 + 14i |
| Multiply | Use FOIL, then replace i2 with −1 | (1 + 3i)(2 + i) = 2 + i + 6i + 3i2 = −1 + 7i |
How do you find a power of i?
The powers of i repeat in a cycle of length 4: i, −1, −i, 1.
- Divide the exponent n by 4.
- Keep the remainder 0, 1, 2, or 3.
- Match it: 0 gives 1, 1 gives i, 2 gives −1, 3 gives −i.
Evaluate i62. Since 62 = 4 · 15 + 2, the remainder is 2, so i62 = i2 = −1.
The rule √a · √b = √(ab) fails for negative radicands. Write i first: √(−4) · √(−9) = 2i · 3i = −6, not 6.
Pull out i before doing any other step. Once every root is written as a multiple of i, ordinary algebra finishes the job.
- The imaginary unit satisfies i2 = −1.
- A complex number a + bi has real part a and imaginary part b.
- Real numbers are the complex numbers whose imaginary part is 0.
- Powers of i repeat every 4 steps: i, −1, −i, 1.
- Add and subtract by parts; multiply with FOIL and replace i2 by −1.
Real and Complex Numbers: Working with i: frequently asked questions
Why is the imaginary unit called imaginary?
The name dates from a time when negative square roots seemed useless. Today the numbers are as legitimate as any other and are used daily in electrical engineering and signal processing. The word is only a historical label.
Is every real number also a complex number?
Yes. A real number such as 7 can be written 7 + 0i, so it is a complex number whose imaginary part is zero. The real numbers form a subset of the complex numbers.
Why does √(−4) · √(−9) equal −6 instead of 6?
The product rule for square roots only holds when the radicands are not negative. Rewrite each root first as 2i and 3i, then multiply to get 6i2 = −6.
How do you evaluate a very large power of i?
Divide the exponent by 4 and look only at the remainder. A remainder of 0, 1, 2, or 3 gives 1, i, −1, or −i respectively.
Where do complex numbers appear outside of math class?
Alternating-current circuits use them to track voltage, current, and impedance together. Computer graphics and sound processing also use complex arithmetic because multiplying by i turns a point by 90 degrees.
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