Main content

### Course: Precalculus > Unit 3

Lesson 9: The fundamental theorem of algebra# Complex numbers: FAQ

## What is the complex plane?

The complex plane is a way of visually representing complex numbers. We plot the real component of the complex number on the x-axis, and the imaginary component on the y-axis.

## How do we find the distance and midpoint of complex numbers?

The distance and midpoint of complex numbers are similar to the distance and midpoint of points on a coordinate plane. The distance is how far apart two complex numbers are, and the midpoint is the point that is exactly in between them.

To find the distance between two complex numbers, we use the Pythagorean theorem.

To find the midpoint between two complex numbers, we use the average of their real parts and the average of their imaginary parts.

## What is a complex conjugate?

The complex conjugate of a complex number is what we get when we change the sign of the imaginary component. For example, the complex conjugate of $2+3i$ is $2-3i$ .

## Why do we divide complex numbers by using their complex conjugates?

When we multiply a complex number by its complex conjugate, we get a real number. This makes it easier to divide one complex number by another, since we can multiply both the numerator and the denominator by the complex conjugate of the denominator. This way, we make the denominator a real number that we can more easily divide.

## What are some identities with complex numbers?

Identities are equations that are always true, no matter what values we plug in for the variables. They are useful for simplifying expressions and solving problems. Some common identities with complex numbers are:

${i}^{2}=-1$ ${i}^{3}=-i$ ${i}^{4}=1$ $(a+bi{)}^{2}={a}^{2}+2abi-{b}^{2}$ $(a+bi{)}^{3}={a}^{3}+3{a}^{2}bi-3a{b}^{2}-{b}^{3}i$ $(a+bi{)}^{4}={a}^{4}+4{a}^{3}bi-6{a}^{2}{b}^{2}-4a{b}^{3}i+{b}^{4}$ (this is the complex conjugate)$\stackrel{\u2015}{a+bi}=a-bi$ (this is the modulus or absolute value)$|a+bi|=\sqrt{{a}^{2}+{b}^{2}}$ $(a+bi)(a-bi)={a}^{2}+{b}^{2}$ $\stackrel{\u2015}{(a+bi)(c+di)}=\stackrel{\u2015}{(a+bi)}\cdot \stackrel{\u2015}{(c+di)}$ $|a+bi|\cdot |c+di|=|(a+bi)(c+di)|$

We can use these identities to simplify complex numbers and find their properties, like modulus, argument, and polar form.

## What are the modulus and argument of a complex number?

The modulus of a complex number is the distance from the origin to the complex number on the complex plane. The argument is the angle the complex number makes with the positive x-axis.

## Why do we multiply and divide complex numbers in polar form?

It's often easier to multiply and divide complex numbers when they are in polar form, rather than in rectangular form. In polar form, we can multiply two complex numbers by multiplying their moduli and adding their arguments, and we can divide two complex numbers by dividing their moduli and subtracting their arguments.

## What is the fundamental theorem of algebra?

The fundamental theorem of algebra is a very important and beautiful result that tells us something amazing about complex numbers and polynomials. The fundamental theorem of algebra says that every polynomial of degree $n$ , where $n$ is a positive whole number, has exactly $n$ complex roots, or solutions. A root of a polynomial is a value that makes the polynomial equal to zero, like $x=2$ for ${x}^{2}-4=0$ . A complex root is a root that is a complex number, like $x=1+i$ for ${x}^{2}-2x+2=0$ .

The fundamental theorem of algebra is amazing because it tells us that complex numbers are essential for solving any polynomial equation, no matter how complicated or simple it is. It also tells us that complex numbers are complete, meaning that there is no other type of number that we need to invent to find the roots of any polynomial. It also tells us that if a complex number is a root of a polynomial, then so is its complex conjugate. For example, if $1+i$ is a root of ${x}^{2}-2x+2=0$ , then so is $1-i$ .

The fundamental theorem of algebra has many applications in mathematics, science, engineering, and art, where polynomials are used to model and study various phenomena and patterns. For example, polynomials can be used to describe the shape of curves, the motion of objects, the behavior of waves, the distribution of data, the encryption of information, and the creation of fractals.

## Want to join the conversation?

- The complex conjugate of (a+bi)(c+di)≠ The complex conjugate of (a+bic+di).

It is equal to the [Complex conjugate of ac-bd+i(da+bc)]=ac-bd-adi-bci.(16 votes)- Good eye, Abhinav! I fixed a formatting issue -- see the updated identity. It's a cool one...it says that the conjugate of the product of the two complex numbers is equal to the product of their conjugates. Thank you for your comment!(9 votes)

- I think technically below is not correct:

a + bi = a − bi

(a - bi) is a conjugate of (a + bi) but it does not equal it.

Maybe rephrase that bullet point to:

(a - bi) is a complex conjugate of (a + bi)

To avoid confusion.(2 votes)- I thought so too at first, but then I noticed a bar symbol above a+b, which means "conjugate." But yeah, a little confusing at first because my eye skipped over the bar symbol like it was a graphical separator.(7 votes)

- when they mentioned that n should be a positive whole number, I wondered what the graph of x^1.5 or sqrt(x^3) would look like? does anybody know how to plot it? I tried desmos but it wouldn't show anything below the x-axis(2 votes)
- What is the argument of a complex number?(1 vote)
- If you plot the number in the complex plane, then draw a segment from the number to the origin, the argument is the angle between the positive real axis and the segment.

So positive real numbers have an argument of 0, negative reals have an argument of π, i has an argument of π/2, and 1+i has an argument of π/4.(1 vote)