Algebra 2
Complex numbers
Complex plane (geometric representation)
20 practice questions
0 video lessons
Theory + worked examples
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.
\(|3+4i|=5\) is the distance from the origin.
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 always \(\ge 0\) — it is a length.
How to use the complex plane
- Plot \(a+bi\) at \((a,b)\).
- Find the modulus with \(\sqrt{a^2+b^2}\).
- Reflect over the real axis for the conjugate.
- Subtract and take the modulus for a distance.
Example 1 — Plot and find the modulus
Find the modulus of \(3+4i\).
Solution
Use \(|a+bi|=\sqrt{a^2+b^2}\).
| \(|3+4i|\) | \(=\) | \(\sqrt{3^2+4^2}\) |
| \(=\) | \(\sqrt{25}=5\) |
Example 2 — Another modulus
Find \(|-5+12i|\).
Solution
Square the parts and add.
| \(|-5+12i|\) | \(=\) | \(\sqrt{25+144}\) |
| \(=\) | \(\sqrt{169}=13\) |
Example 3 — Distance between points
Find the distance between \(1+2i\) and \(4+6i\).
Solution
Distance is \(|z_1-z_2|\).
| \(|(1+2i)-(4+6i)|\) | \(=\) | \(|-3-4i|\) |
| \(=\) | \(\sqrt{9+16}=5\) |
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.
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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