Algebra 2
Complex numbers
Introduction to complex numbers (i, a + bi form)
20 practice questions
0 video lessons
Theory + worked examples
Theory
The imaginary unit is \(i=\sqrt{-1}\), so \(i^2=-1\). A complex number is written
\[a+bi,\quad a=\text{real part},\ b=\text{imaginary part}.\]
Powers of \(i\) cycle with period 4: \(i,\ -1,\ -i,\ 1,\ \dots\)
Every negative square root can be written with \(i\): \(\sqrt{-n}=i\sqrt{n}\).
\(3+2i\) has real part \(3\), imaginary part \(2\).
The imaginary unit and its powers.
Key facts:
\[i=\sqrt{-1},\quad i^2=-1,\quad \sqrt{-n}=i\sqrt{n}\]
Reduce \(i^n\) by dividing the exponent by 4 and using the remainder.
How to work with i
- Rewrite \(\sqrt{-n}\) as \(i\sqrt{n}\).
- Replace \(i^2\) with \(-1\).
- For \(i^n\), use the remainder of \(n\div4\).
- Keep real and imaginary parts separate.
Example 1 — Square root of a negative
Simplify \(\sqrt{-16}\).
Solution
Factor out \(i=\sqrt{-1}\).
| \(\sqrt{-16}\) | \(=\) | \(\sqrt{16}\cdot\sqrt{-1}\) |
| \(=\) | \(4i\) |
Example 2 — Powers of i
Simplify \(i^{23}\).
Solution
Powers of \(i\) cycle every 4; \(23=4(5)+3\).
| \(i^{23}\) | \(=\) | \(i^{3}\) |
| \(=\) | \(-i\) |
Example 3 — Simplify a radical
Write \(\sqrt{-49}\) in terms of \(i\).
Solution
Split off \(\sqrt{-1}\).
| \(\sqrt{-49}\) | \(=\) | \(7i\) |
Example 4 — Real and imaginary parts
State the real and imaginary parts of \(3-5i\).
Solution
Compare with \(a+bi\).
| \(\text{real part}\) | \(=\) | \(3\) |
| \(\text{imaginary part}\) | \(=\) | \(-5\) |
Common pitfalls
\(i^2=-1\), not \(1\).
\(\sqrt{-a}\,\sqrt{-b}\neq\sqrt{ab}\): convert to \(i\) first.
Keep the parts separate — \(a\) and \(bi\) don't combine.
Frequently asked questions
What is the imaginary unit?
\(i=\sqrt{-1}\), so \(i^2=-1\).
What is a complex number?
A number \(a+bi\) with a real part \(a\) and imaginary part \(b\).
What is \(i^3\)?
\(i^3=-i\).
How do you simplify \(\sqrt{-25}\)?
\(\sqrt{-25}=5i\).
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