Modulus

Definition

The modulus of a complex number is its distance from the origin in the complex plane, represented as $|z|$. It is calculated using the formula $|z| = \sqrt{a^2 + b^2}$ where $z = a + bi$.

5 Must Know Facts For Your Next Test

1. The modulus provides the magnitude of a complex number and is always non-negative.
2. If $z = a + bi$, then the modulus $|z|$ is given by $\sqrt{a^2 + b^2}$.
3. The modulus is used when expressing complex numbers in polar form as $r(cos(\theta) + i\sin(\theta))$, where $r$ is the modulus.
4. In trigonometric applications, the modulus helps in converting between rectangular and polar coordinates.
5. The properties of moduli include: $|zw| = |z||w|$ and $|z/w| = |z|/|w|$ for any complex numbers z and w.

Review Questions

• What formula do you use to find the modulus of a complex number?
• How does the modulus relate to converting a complex number into polar form?
• What are some key properties of moduli that can simplify calculations?

Related terms

Complex Number: A number in the form $a + bi$, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit satisfying $i^2 = -1$.

Polar Form: A way to represent complex numbers using modulus and argument: $r(cos(\theta) + i\sin(\theta))$, where 'r' is the modulus and '$\theta$' is the angle.

Argument: The angle formed by the positive real axis and the line representing a complex number in polar coordinates. Denoted as '$\theta$', it can be found using $\tan^{-1}(b/a)$ for a given complex number \(a+bi\).