# Complex Number Help: Find Modulus & Principle Argument

• Wardlaw
In summary, complex numbers are numbers in the form a + bi, with a and b being real numbers and i being the imaginary unit. The modulus of a complex number is its distance from the origin on the complex plane, calculated by taking the absolute value. The principle argument is the angle it forms with the positive real axis, found using arctan of the imaginary part divided by the real part. They are useful in various fields and can be operated on by adding, subtracting, multiplying using the FOIL method, and dividing by rationalizing the denominator or converting to polar form for more complex operations.
Wardlaw
Complex number help!

## Homework Statement

Find the modulus and principle argument of 1/(-sqrt(3)+i)

## The Attempt at a Solution

I attempted this solution by using the complex conjugate, and as i^2=-1, i eventually ended up with 4 in the denominator. Any suggestions?

$$(-\sqrt{3}+j)(-\sqrt{3}-j)=3-1=2$$

So you should get $$\frac{-\sqrt{3}-j}{2}$$

Before you make use of the previous post, multiply numerator and denominator by the complex conjugate of the denominator. Then make use of Yungman's post. From there it's a standard manipulation.

## What are complex numbers?

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

## What is the modulus of a complex number?

The modulus of a complex number is the distance of the number from the origin on the complex plane. It is calculated by taking the absolute value of the number, which is the square root of the sum of the squares of the real and imaginary parts.

## How do I find the principle argument of a complex number?

The principle argument of a complex number is the angle it forms with the positive real axis on the complex plane. It can be found using the inverse tangent function (arctan) of the imaginary part divided by the real part.

## Why are complex numbers useful?

Complex numbers are useful in many areas of mathematics and science, such as in electrical engineering, signal processing, and quantum mechanics. They allow us to represent and manipulate quantities that have both a real and imaginary component.

## How do I perform operations on complex numbers?

To add or subtract complex numbers, simply add or subtract the real and imaginary parts separately. To multiply, use the FOIL method (First, Outer, Inner, Last). To divide, rationalize the denominator by multiplying by the conjugate of the denominator. For more complex operations, it is helpful to convert the numbers to polar form.

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