Functions, complex numbers, and consecutive integers

Functions

Definitions

Function: A rule that assigns exactly one output to each input.

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 $f (x) = x^{3} + 4 x$ .

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

$f (x) f (2) f (2) f (2) = x^{3} + 4 x = (2)^{3} + 4 (2) = 8 + 8 = 16$

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 whose square is negative; it is not found on the real number line
Complex number: A number that can be written as $a + bi$ , where $a$ and $b$ are real numbers (either part may be 0).

The imaginary unit, written as $i$ , is defined as $- 1$ . 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.

$- 16 = 16 \cdot - 1 = 4 \cdot i = 4 i$

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

For example:

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

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

Example:

Write each number in $a + bi$ form:

$- 7$

$3 i$

$- 7 = - 7 + 0 i$

$3 i = 0 + 3 i$

Steps taken:

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

Knowledge check:

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

$7 + 2 i$

$5 i$

$- 3$

(spoiler)

All of them.

Any real number can be written as $a + 0 i$ , 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:

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

Step 1: Distribute the negative.

$10 + 3 i + 7 - i$

Step 2: Combine like terms.

$10 + 7 3 i - i = 17 = 2 i$

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

$17 + 2 i$

Knowledge check:

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

(spoiler)

$9 + i$

Multiplying complex numbers

When multiplying complex numbers, use the fact that $i^{2} = - 1$ .

Starting from $i = - 1$ :

$i i^{2} i^{2} = - 1 = (- 1)^{2} = - 1$

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

Example:

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

Step 1: FOIL and combine like terms.

$(2) (7) + (2) (4 i) + (- 5 i) (7) + (- 5 i) (4 i) = 14 + 8 i - 35 i - 20 i^{2} = 14 - 27 i - 20 i^{2}$

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

$14 - 27 i - 20 i^{2} = 14 - 27 i - 20 (- 1) = 14 - 27 i + 20$

Note: replacing $i^{2}$ with $- 1$ turns $- 20 i^{2}$ into $- 20 \times (- 1) = + 20$ . Forgetting that the sign flips here is the step students most often get wrong.

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

$34 - 27 i$

Knowledge check: What is

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

(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.

For consecutive even or consecutive odd integers, add $2$ each time: $n, n + 2, n + 4, \dots$ (a common mistake is using $n + 1$ , which switches between odd and even).

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:

$4 n + 6 = 22$

$4 n = 16$

$n = 4$

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)

$10$

Functions

Definitions

Function: A rule that assigns exactly one output to each input.

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 $f (x) = x^{3} + 4 x$ .

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

$f (x) f (2) f (2) f (2) = x^{3} + 4 x = (2)^{3} + 4 (2) = 8 + 8 = 16$

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 whose square is negative; it is not found on the real number line
Complex number: A number that can be written as $a + bi$ , where $a$ and $b$ are real numbers (either part may be 0).

The imaginary unit, written as $i$ , is defined as $- 1$ . 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.

$- 16 = 16 \cdot - 1 = 4 \cdot i = 4 i$

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

For example:

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

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

Example:

Write each number in $a + bi$ form:

$- 7$

$3 i$

$- 7 = - 7 + 0 i$

$3 i = 0 + 3 i$

Steps taken:

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

Knowledge check:

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

$7 + 2 i$

$5 i$

$- 3$

(spoiler)

All of them.

Any real number can be written as $a + 0 i$ , 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:

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

Step 1: Distribute the negative.

$10 + 3 i + 7 - i$

Step 2: Combine like terms.

$10 + 7 3 i - i = 17 = 2 i$

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

$17 + 2 i$

Knowledge check:

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

(spoiler)

$9 + i$

Multiplying complex numbers

When multiplying complex numbers, use the fact that $i^{2} = - 1$ .

Starting from $i = - 1$ :

$i i^{2} i^{2} = - 1 = (- 1)^{2} = - 1$

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

Example:

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

Step 1: FOIL and combine like terms.

$(2) (7) + (2) (4 i) + (- 5 i) (7) + (- 5 i) (4 i) = 14 + 8 i - 35 i - 20 i^{2} = 14 - 27 i - 20 i^{2}$

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

$14 - 27 i - 20 i^{2} = 14 - 27 i - 20 (- 1) = 14 - 27 i + 20$

Note: replacing $i^{2}$ with $- 1$ turns $- 20 i^{2}$ into $- 20 \times (- 1) = + 20$ . Forgetting that the sign flips here is the step students most often get wrong.

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

$34 - 27 i$

Knowledge check: What is

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

(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.

For consecutive even or consecutive odd integers, add $2$ each time: $n, n + 2, n + 4, \dots$ (a common mistake is using $n + 1$ , which switches between odd and even).

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:

$4 n + 6 = 22$

$4 n = 16$

$n = 4$

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)

$10$