Functions and complex numbers

Functions

Definitions

Functions

A method of writing equations with an input and an output

A function describes a relationship between an input value and an output value. One way to picture it is like a machine: you put in an $x$-value, and you get out a $y$-value.

Most of the time, functions are written like this:

$$f(x)=expression$$

Here:

• $f$ is the name of the function
• $x$ is the input
• $f(x)$ is the output (often the same value you’d call $y$)

Example:

We have the function $f(x)=x^3+4x$.

To find $f(2)$, replace every $x$ with $2$ and simplify.

$$\begin{array}{rl}f(x)& =x^3+4x\\ f(2)& =(2{)}^3+4(2)\\ f(2)& =8+8\\ f(2)& =16\end{array}$$

So the input was $2$ and the output was $16$. You can also write this as $x=2$ and $y=16$, or as the coordinate pair $(2,16)$.

Steps taken:

• Substitute $2$ for $x$
• Simplify

Imaginary and complex numbers

Definitions

Imaginary number

A number that does not exist on the coordinate plane

Complex number

A number that includes both a real and an imaginary number

The imaginary unit, written as $i$, is defined as $\sqrt{-1}$. No real number multiplied by itself gives a negative result, so $i$ is not a real number.

For instance, $1\cdot 1=1$ and $-1\cdot -1=1$. So no real number times itself will ever be negative.

Even though $i$ is imaginary, it is useful because it allows us to rewrite negative square roots.

$$\begin{array}{rl}\sqrt{-16}& =\sqrt{16}\cdot \sqrt{-1}\\ & =4\cdot i\\ & =4i\end{array}$$

A complex number is written in the form $a+bi$, where $a$ is the real part and $bi$ is the imaginary part.

For example:

• $4i$ can be written as $0+4i$
• $19$ can be written as $19+0i$

This means all real numbers can also be written as complex numbers.

Example:

Write each number in $a+bi$ form:

$-7$
$3i$

$-7=-7+0i$
$3i=0+3i$

Steps taken:

• Identify real and imaginary parts
• Rewrite in $a+bi$ form

Knowledge check:

Which of the following is written in the form $a+bi$?

$$7+2i$$

$$5i$$

$$-3$$

(spoiler)

All of them.

Any real number can be written as $a+0i$, and any imaginary number can be written as $0+bi$.

Adding and subtracting complex numbers

To add or subtract complex numbers, treat $i$ like a variable and combine like terms. Let’s walk through an example step-by-step!

Example:

$$(10+3i)-(-7+i)$$

Step 1: Distribute the negative.

$$10+3i+7-i$$

Step 2: Combine like terms.

$$\begin{array}{rl}10+7& =17\\ 3i-i& =2i\end{array}$$

Step 3: Write the final answer in $a+bi$ form.

$$17+2i$$

Steps taken:

• Distribute negative
• Combine like terms
• Write in $a+bi$ form

Knowledge check:

$$(6+2i)+(3-i)$$

(spoiler)

$$9+i$$

Multiplying and dividing complex numbers

When multiplying and dividing complex numbers, it helps to use the fact that $i^2=-1$.

Starting from $i=\sqrt{-1}$:

$$\begin{array}{rl}i& =\sqrt{-1}\\ i^2& =(\sqrt{-1}{)}^2\\ i^2& =-1\end{array}$$

Let’s walk through an example step-by-step.

Example:

$$(2-5i)(7+4i)$$

Step 1: FOIL and combine like terms.

$$\begin{array}{rl}(2)(7)+(2)(4i)+(-5i)(7)+(-5i)(4i)& \\ & =14+8i-35i-20i^2\\ & =14-27i-20i^2\end{array}$$

Step 2: Replace $i^2$ with $-1$.

$$\begin{array}{rl}14-27i-20i^2& \\ & =14-27i-20(-1)\\ & =14-27i+20\end{array}$$

Step 3: Combine like terms and write in $a+bi$ form.

$$34-27i$$

Steps taken:

• FOIL
• Combine like terms
• Substitute $i^2=-1$
• Simplify

Knowledge check: What is

$$(3+6i)(2-4i)?$$

(spoiler)

$$30$$

(The imaginary parts cancel out, and $i^2=-1$.)

Consecutive numbers

Definitions

Consecutive numbers

Terms that are next to each other on the number line

Consecutive numbers follow each other in order. We can represent unknown numbers using a variable so we can build an equation.

• Consecutive integers starting at $7$ are $7,8,9,10,11,$ etc.
• Consecutive even integers could be $6,8,10,12,$ etc.
• Consecutive odd integers could be $1,3,5,7,$ and so on.

Let’s take a look at an example question about consecutive numbers and solve it step-by-step.

Example:

What is the last number in a set of four consecutive integers that sum to $22$?

Let’s start by setting up an equation. Each consecutive integer increases by $1$, so let the first number be $n$.

The next numbers are:

$n+1,n+2,n+3$

We know their sum is $22$, so we write:

$$n+(n+1)+(n+2)+(n+3)=22$$

Simplify:

$$4n+6=22$$

$$4n=16$$

$$n=4$$

This means the four integers are:

$4,5,6,7$

The question asks for the last number, so the answer is:

$$7$$

Steps taken:

• Let the first number be $n$
• Write the next consecutive numbers ($n+1,n+2,n+3$)
• Set up an equation using their sum
• Solve for $n$
• List all numbers and identify the final answer

Knowledge check:

What is the last number in a set of three consecutive integers that sum to $27$?

(spoiler)

$$10$$