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MathChapter 13: Polynomial and Radical Functions

Notes

Complex Numbers Notes

Use i squared equals negative one to simplify negative roots, operate on complex numbers, use conjugates in division, and compare real and imaginary parts.

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Complex numbers extend the real numbers using \(i=\sqrt{-1}\), so \(i^2=-1\). Arithmetic remains algebraic: combine like parts, distribute carefully, and replace every \(i^2\) by \(-1\).

Definition and anatomy

Anatomy of a complex number

Text labels distinguish the real part, imaginary coefficient, and complete imaginary term.

\[a+bi\]
Real part
\(a\)

The term without i.

Imaginary coefficient
\(b\)

The real-number coefficient multiplying i.

Imaginary term
\(bi\)

The coefficient together with i.

Negative square roots
\[\sqrt{-a}=i\sqrt a\quad(a>0)\]

Simplify the positive radical after factoring out \(\sqrt{-1}=i\). For example, \(\sqrt{-45}=3i\sqrt5\).

Powers of i

The four-value cycle of powers of i
Exponent remainder modulo 4Value
\(0\)\(1\)
\(1\)\(i\)
\(2\)\(-1\)
\(3\)\(-i\)

Add, subtract, and multiply

Complex-number operation rules
OperationRule
Addition\((a+bi)+(c+di)=(a+c)+(b+d)i\)
Subtraction\((a+bi)-(c+di)=(a-c)+(b-d)i\)
Multiplication\((a+bi)(c+di)=(ac-bd)+(ad+bc)i\)
Equality\(a+bi=c+di\iff a=c\text{ and }b=d\)
Worked example

Multiply and simplify

Simplify \((4-3i)(2+i)\).

  1. FOIL

    \(8+4i-6i-3i^2\).

  2. Replace i squared

    Because \(i^2=-1\), \(-3i^2=3\).

  3. Combine

    \(8+3+(4-6)i=11-2i\).

\(11-2i\).

Conjugates and division

A complex number times its conjugate is realA four-stage flow expands a conjugate product and uses i squared equals negative one to produce a real sum of squares.
  1. \((a+bi)(a-bi)\)

    Change only the sign of the imaginary term.

  2. \(a^2-(bi)^2\)

    The cross terms cancel.

  3. \(a^2-b^2(-1)\)

    Because i squared is negative one, subtraction changes to addition.

  4. \(a^2+b^2\)

    No imaginary term remains.

Worked example

Divide using the denominator's conjugate

Write \(\frac{5+i}{2-i}\) in \(a+bi\) form.

  1. Multiply by the conjugate

    Use \(\frac{2+i}{2+i}\).

  2. Expand numerator

    \((5+i)(2+i)=10+7i+i^2=9+7i\).

  3. Make denominator real

    \((2-i)(2+i)=2^2+1^2=5\).

\(\frac95+\frac75i\).

Equality and parameters

Worked example

Match real and imaginary parts

If \((k+2)+(3m-1)i=7+8i\), find \(k\) and \(m\).

  1. Real parts

    \(k+2=7\), so \(k=5\).

  2. Imaginary coefficients

    \(3m-1=8\), so \(m=3\).

\(k=5\) and \(m=3\).

Common mistakes and traps

  • Using \(i^2=1\) instead of \(-1\).
  • Writing \(\sqrt{-a}=-\sqrt a\) instead of \(i\sqrt a\).
  • Combining real and imaginary terms as though they were alike.
  • Failing to distribute a subtraction sign across both parts.
  • Choosing the wrong sign for a conjugate.
  • Comparing equal complex numbers without matching the two parts independently.
Mini check

Use the i cycle

What is \(i^{27}\)?

  1. \(1\)
  2. \(i\)
  3. \(-1\)
  4. \(-i\)
Show answer and explanation

Answer: \(-i\)

Since \(27\) leaves remainder \(3\) when divided by \(4\), \(i^{27}=i^3=-i\).

Key takeaways

Key takeaways

What to remember

  • \(i=\sqrt{-1}\) and \(i^2=-1\).
  • Combine real parts with real parts and imaginary coefficients with imaginary coefficients.
  • Multiplying conjugates produces the real value \(a^2+b^2\).
  • Equal complex numbers have equal real parts and equal imaginary coefficients.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.