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Algebra 2 Complex numbers

Introduction to complex numbers (i, a + bi form)

20 practice questions 0 video lessons Theory + worked examples

Introduction to Complex Numbers

California Algebra 2 • Standard N-CN.1 • Complex Numbers

Introduction to Complex Numbers is the opening topic of Complex Numbers in the California Common Core State Standards. It is aligned to Standard N-CN.1, which requires students to know there is a complex number i with i squared equal to negative one, and that every complex number has the form a + b i.

Complex numbers extend the reals with the imaginary unit \(i=\sqrt{-1}\); a complex number \(a+bi\) has a real part \(a\) and imaginary part \(b\).

California Algebra 2 › Complex Numbers › Introduction to Complex Numbers  —  Standard N-CN.1

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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 \(i\): \(\sqrt{-n}=i\sqrt{n}\).
A complex number on the plane The complex number 3 + 2i is plotted with real part 3 and imaginary part 2. Re Im 3 + 2i
\(3+2i\) has real part \(3\), imaginary part \(2\).
Imaginary unit i Imaginary unit i Imaginary unit i i = √(-1), so i² = -1 i³ = -i, i⁴ = 1 (cycle of 4) a + bi: real part a, imaginary part b
The imaginary unit and its powers.

Key facts:

\[i=\sqrt{-1},\quad i^2=-1,\quad \sqrt{-n}=i\sqrt{n}\]
i is the square root of negative one, and i squared is negative one
Reduce \(i^n\) by dividing the exponent by 4 and using the remainder.

How to work with i

  1. Rewrite \(\sqrt{-n}\) as \(i\sqrt{n}\).
  2. Replace \(i^2\) with \(-1\).
  3. For \(i^n\), use the remainder of \(n\div4\).
  4. Keep real and imaginary parts separate.
Example 1 — Square root of a negative
Simplify \(\sqrt{-16}\).
Solution

Factor out \(i=\sqrt{-1}\).

\(\sqrt{-16}\)\(=\)\(\sqrt{16}\cdot\sqrt{-1}\)
\(=\)\(4i\)
the square root of negative 16 is 4 i
Example 2 — Powers of i
Simplify \(i^{23}\).
Solution

Powers of \(i\) cycle every 4; \(23=4(5)+3\).

\(i^{23}\)\(=\)\(i^{3}\)
\(=\)\(-i\)
i to the 23 is negative i
Example 3 — Simplify a radical
Write \(\sqrt{-49}\) in terms of \(i\).
Solution

Split off \(\sqrt{-1}\).

\(\sqrt{-49}\)\(=\)\(7i\)
the square root of negative 49 is 7 i
Example 4 — Real and imaginary parts
State the real and imaginary parts of \(3-5i\).
Solution

Compare with \(a+bi\).

\(\text{real part}\)\(=\)\(3\)
\(\text{imaginary part}\)\(=\)\(-5\)
real part 3, imaginary part negative 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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