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Pre-Calculus Complex numbers (advanced)

Complex plane (rectangular form review)

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Theory

A complex number \(z=a+bi\) has a real part \(a\) and an imaginary part \(b\), where \(i=\sqrt{-1}\) so \(i^2=-1\).

It is plotted on the complex plane: the horizontal axis is real, the vertical axis imaginary. Two features matter most:

  • Modulus \(|z|=\sqrt{a^2+b^2}\) — the distance from the origin.
  • Conjugate \(\bar z=a-bi\) — the reflection across the real axis.

Add and subtract by combining like parts; multiply with FOIL, replacing \(i^2=-1\).

\(z\bar z\) is always real: \(z\bar z=a^2+b^2=|z|^2\). This is the key to dividing complex numbers.
A complex number on the plane The complex number 3 plus 2 i is plotted 3 right and 2 up; its modulus is the distance to the origin. Re Im 3+2i
\(z=3+2i\): 3 along the real axis, 2 up the imaginary axis.
A complex number and its conjugate The conjugate reflects a complex number across the real axis, negating the imaginary part. Re Im 3+2i 3−2i (conj.)
The conjugate \(\bar z\) reflects \(z\) across the real axis.

Modulus, conjugate, and the basic operations:

\[|a+bi|=\sqrt{a^2+b^2},\qquad \overline{a+bi}=a-bi\]
\[(a+bi)+(c+di)=(a+c)+(b+d)i,\qquad i^2=-1\]
modulus is the square root of a squared plus b squared; conjugate negates the imaginary part; i squared is negative one
To divide, multiply numerator and denominator by the conjugate of the denominator to make it real.

How to work in rectangular form

  1. Plot \(a+bi\) at \((a,b)\).
  2. Add/subtract: combine real parts and imaginary parts.
  3. Multiply: FOIL, then replace \(i^2=-1\).
  4. Divide: multiply by the conjugate of the denominator.
Example 1 — Plot and find the modulus
Plot \(z=3+2i\) and find \(|z|\).
Solution

Move 3 right (real) and 2 up (imaginary). The modulus is the distance to the origin.

\(|z|\)\(=\)\(\sqrt{3^2+2^2}\)
\(=\)\(\sqrt{13}\approx 3.61\)
modulus is the square root of 13, about 3.61
Example 2 — Add complex numbers
Compute \((3+2i)+(1-5i)\).
Solution

Add real parts and imaginary parts separately.

\(=\)\((3+1)+(2-5)i\)
\(=\)\(4-3i\)
sum is 4 minus 3 i
Example 3 — Multiply, using i squared
Compute \((2+3i)(1-4i)\).
Solution

Expand with FOIL, then replace \(i^2=-1\).

\(=\)\(2-8i+3i-12i^2\)
\(=\)\(2-5i-12(-1)\)
\(=\)\(14-5i\)
product is 14 minus 5 i
Example 4 — The conjugate
Find the conjugate of \(z=3+2i\) and compute \(z\bar z\).
Solution

The conjugate flips the sign of the imaginary part; the product is real.

\(\bar z\)\(=\)\(3-2i\)
\(z\bar z\)\(=\)\(3^2+2^2=13\)

Note \(z\bar z=|z|^2\).

conjugate is 3 minus 2 i; the product is 13

Common pitfalls

\(i^2=-1\). Forgetting to substitute it is the most common multiplication error.
Keep parts separate when adding. Reals with reals, imaginaries with imaginaries.
The modulus is never negative. It is a distance, \(\sqrt{a^2+b^2}\).

Frequently asked questions

What is a complex number?

A number \(a+bi\) with a real part \(a\) and imaginary part \(b\), where \(i=\sqrt{-1}\).

What is the modulus of a complex number?

Its distance from the origin on the complex plane: \(|a+bi|=\sqrt{a^2+b^2}\).

What is the complex conjugate?

\(\overline{a+bi}=a-bi\); it reflects the number across the real axis and makes \(z\bar z=|z|^2\) real.

How do you multiply complex numbers?

Use FOIL as with binomials, then replace \(i^2\) with \(-1\) and combine like terms.

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