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Algebra 2 Complex numbers

Complex plane (geometric representation)

20 practice questions 0 video lessons Theory + worked examples

The Complex Plane

California Algebra 2 • Standard N-CN.4 • Complex Numbers

The Complex Plane is a topic in Complex Numbers in the California Common Core State Standards. It is aligned to Standard N-CN.4, which requires students to represent complex numbers on the complex plane and find the modulus and conjugate.

The complex plane plots \(a+bi\) with real part horizontal and imaginary part vertical; the modulus \(\sqrt{a^2+b^2}\) is the distance from the origin.

California Algebra 2 › Complex Numbers › The Complex Plane  —  Standard N-CN.4

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Theory

The complex plane plots \(a+bi\) as the point \((a,b)\):

  • Real axis horizontal, imaginary axis vertical.
  • Modulus \(|a+bi|=\sqrt{a^2+b^2}\) is the distance from the origin.
  • Conjugate \(a-bi\) is the reflection over the real axis.
Distance between two complex numbers is the modulus of their difference.
Modulus on the complex plane The modulus of a complex number is its distance from the origin. Re Im 3+4i |z|=5
\(|3+4i|=5\) is the distance from the origin.
The complex plane The complex plane The complex plane horizontal axis: real part vertical axis: imaginary part modulus |a+bi| = √(a² + b²) conjugate: reflect over real axis
Reading the complex plane.

Modulus and distance:

\[|a+bi|=\sqrt{a^2+b^2},\qquad d(z_1,z_2)=|z_1-z_2|\]
the modulus is the square root of a squared plus b squared
The modulus is always \(\ge 0\) — it is a length.

How to use the complex plane

  1. Plot \(a+bi\) at \((a,b)\).
  2. Find the modulus with \(\sqrt{a^2+b^2}\).
  3. Reflect over the real axis for the conjugate.
  4. Subtract and take the modulus for a distance.
Example 1 — Plot and find the modulus
Find the modulus of \(3+4i\).
Solution

Use \(|a+bi|=\sqrt{a^2+b^2}\).

\(|3+4i|\)\(=\)\(\sqrt{3^2+4^2}\)
\(=\)\(\sqrt{25}=5\)
the modulus is 5
Example 2 — Another modulus
Find \(|-5+12i|\).
Solution

Square the parts and add.

\(|-5+12i|\)\(=\)\(\sqrt{25+144}\)
\(=\)\(\sqrt{169}=13\)
the modulus is 13
Example 3 — Distance between points
Find the distance between \(1+2i\) and \(4+6i\).
Solution

Distance is \(|z_1-z_2|\).

\(|(1+2i)-(4+6i)|\)\(=\)\(|-3-4i|\)
\(=\)\(\sqrt{9+16}=5\)
the distance is 5
Example 4 — Conjugate
Find the conjugate of \(3-2i\) and describe it geometrically.
Solution

Change the sign of the imaginary part.

\(\overline{3-2i}\)\(=\)\(3+2i\)

It is the reflection over the real axis.

the conjugate is 3 plus 2 i, a reflection over the real axis

Common pitfalls

The modulus uses \(+b^2\), not \(-b^2\) — it is a distance.
Conjugate flips the imaginary sign only.
Distance is the modulus of the difference, not the difference of moduli.

Frequently asked questions

What is the complex plane?

A plane plotting \(a+bi\) with real part horizontal and imaginary part vertical.

What is the modulus of a complex number?

Its distance from the origin, \(\sqrt{a^2+b^2}\).

What does the conjugate look like geometrically?

A reflection of the point over the real axis.

How do you find the distance between two complex numbers?

Take the modulus of their difference.

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