1. Integers

Before we talk about integers, let’s think about the numbers we first learned as children.

Natural Numbers

The natural numbers arise from counting. We can have

no objects: 0

one object: 1

two objects: 2

... etc.

We can represent the natural numbers on a one-dimensional number line. Here is a graph of the first 4 natural numbers 0, 1, 2, and 3:

Number line showing natural numbers

We put a dot on those numbers that are included. In this case, we have graphed 0, 1, 2, and 3, but we have not included 4 to illustrate the point.

We can also write the natural numbers as a set:

{0, 1, 2, 3, 4, ...}

The natural numbers are sometimes called whole numbers or counting numbers. There is some disagreement among mathematicians about whether or not zero should be included. [See Is 0 a Natural Number?]

Less Than and Greater Than

On the number line, LEFT is LESS. We use the "<" (less than) sign to indicate one number is smaller than another number.

So for example:

`1 < 3` (We say this as "one is less than three".)

We use the ">" (greater than) sign to indicate one number is bigger than another.

`4 > 0` (We say "four is greater than zero".)

Negative Numbers

One of the biggest problems people have in mathematics is with negative numbers. Take great care with negatives!

A negative number is any number whose value is less than zero. We write a minus in front of negative numbers.

Examples of Negative Numbers

(a) ` −3.97`

(b) `-pi`

(c) `−2`

(d) `−3/5`

The negative numbers are represented on the left half of the number line. The four negative numbers above are indicated on the following number line like this:

number line negative numbers examples

Once again, we have put a dot on the numbers which are included in the set.

Uses of Negative Numbers

Now we are ready to talk about integers.

Integers

The integers are defined as:

Notice that the set of integers does not include decimals or fractions − just whole numbers.

We represent integers on the number line like this:

integers number line

Recall from above that on the number line, LEFT is LESS.

Examples of Comparing Integers

(a) `−4 < 2` (we say "negative 4 is less than 2")

(b) `−103 < −45`

Similarly, if a number is to the RIGHT of second number, it is greater than the second number.

(c) `2 > −2` (we say "2 is greater than negative 2")

(d) `3 > −150`

Absolute Value

The distance from 0 to a number is its absolute value (written using vertical line brackets around the number).

Examples of Absolute Value

(a) `|-4| = 4`

The distance from `−4` to `0` is `4` units:

absolute value is distance from origin

Similarly:

(b) `|3|=3`

(c) `|-12.85| = 12.85`

In each example, the number on the right is the distance from 0.

Opposite of an Integer

The opposite of an integer is obtained by changing its sign. (That is, change − to + or change + to −).

Examples of Opposite of an Integer

(a) The opposite of `−3` is `3` and

(b) The opposite of `4` is ` −4`.

Notice that opposite is not the same as absolute value.

Integer Addition

Use the number line and think of "journeys".

Examples of Integer Addition

(a) ` −2 + 5` means "start at `−2` and go `5` in the positive direction"

So we have:

adding integers using the number line

Our answer is:

`−2 + 5 = 3`

Similarly

(b) `3 + −7` means "start at `3` and go `7` in the negative direction".

Our answer is: `3 + −7 = −4`

(c) ` −5 + 12` means "start at ` −5` and go `12` in the positive direction"

The answer is ` −5 + 12 = 7`

Practical Example of Integer Addition

It is −4° and snowing. The forecast for tomorrow is for a rise in temperature of 6°. What will it be tomorrow?

Answer: `−4 + 6 = 2`. It will be 2° tomorrow.

Integer Subtraction

We can change the subtraction into a more familiar addition by realising that subtracting an integer is the same as adding its opposite.

Examples of Integer Subtraction

(a) ` −4 − (−3) = −4 + (+3) = −1 `

(We added +3 because the opposite of −3 is +3.)

(b) `5 − (+7) = 5 + (−7) = −2.`

(We added −7 because the opposite of +7 is −7. Eventually you’ll see the question is the same as "5 − 7" and we can do this as a journey: Start at 5 and move 7 units to the left. Answer: −2.)

Integer Multiplication

When multiplying integers, we can think of multiplying "blocks" of negative numbers.

Examples of Integer Multiplication

(a) Here is an integer times an integer:

`5 × −3 = −15`

Why is this so? We are simply taking "5 lots of −3", like this:

Multiplying negative numbers

Notice that we were multiplying a positive number by a negative number and our result was negative.

Similarly, we can show:

(b) ` −6 × 2 = −12` (negative times positive gives negative)

and

(c) `−3 × −7 = 21` (multiplying 2 negatives gives a positive)

Practical example of Integer Multiplication

The acceleration due to gravity (1 g) on the earth is about −10 ms−2. (It’s negative because it acts downwards. This is a convention.)

On the planet Jupiter, the acceleration due to gravity is 2.65 times the acceleration due to gravity on the Earth.

On Jupiter, the acceleration is around:

`2.65 × −10 "ms"^-2 ≈ −26.5\ "ms"^-2`

Summary

We can summarise the results for multiplying integers:

positive × positive = positive

+ × + = +

Signs the same: positive answer

Example: 2 × 3 = 6

negative × positive = negative

× + = −

Signs different: negative answer

Example: −8 × 2 = −16

positive × negative = negative

+ × − = −

Signs different: negative answer

Example: 5 × −2 = −10

negative × negative = positive

× − = +

Signs the same: positive answer

Example: −5 × −3 = 15

Challenge: Integer Multiplication

What is a practical application for −3 × −7 = 21?

Integer Division

When we divide with negative numbers, we have similar results to those for multiplication:

positive ÷ positive = positive

+ ÷ + = +

Signs the same: positive answer

Example: 15 ÷ 3 = 5

negative ÷ positive = negative

÷ + = −

Signs different: negative answer

Example: −8 ÷ 2 = −4

positive ÷ negative = negative

+ ÷ − = −

Signs different: negative answer

Example: 21 ÷ −7 = −3

negative ÷ negative = positive

÷ − = +

Signs the same: positive answer

Example: −50 ÷ −5 = 10

We can rewrite division problems as multiplication problems, as in the following example.

Example of Integer Division

`−32 ÷ 4`

is the same question as

4 × what? = −32

Challenge: Integer Division

What is a practical application for ` −10 ÷ 5 = −2`?

Integer Properties

The set of integers is closed, commutative, associative and has an identity under both addition and multiplication.

The following table gives examples and explains what this means in plain English.

  Addition Multiplication
Closed

`3 + −7 = −4`

When we add 2 integers, we get an integer.

`−5 × −3 = 15`

When we multiply 2 integers, we get an integer.

Commutative

`4 + −5 = −5 + 4`

It doesn’t matter what order we add integers, we get the same answer.

`2 × −5 = −5 × 2`

It doesn’t matter what order we multiply integers, we get the same answer.

Associative

`(4 + −2) + −5 = 4 + (−2 + −5)`

When adding 3 integers, it doesn’t matter if we start by adding the first pair or the last pair; the answer is the same.

`(4 × −2) × −5 = 4 × (−2 × −5)`

When multiplying 3 integers, it doesn’t matter if we start by multiplying the first pair or the last pair; the answer is the same.

Identity

`−5 + 0 = 0 + −5 = −5`

Zero is the identity element for addition. By adding zero on either side, we don’t change the number.

`−3 × 1 = 1 × −3 = −3`

One is the identity element for multiplication. By multiplying by 1 on either side, we don’t change the number.

The Distributive Law over addition and subtraction holds for integers:

  Addition Subtraction
Distributive

`3(2 + −4)`
`= 3 × 2 + 3 × (−4)`

We multiply each number inside the brackets by the number outside, retaining the plus in the middle.

`−2(5 − 7) `
`= (−2 × 5) − (−2 × 7)`

We multiply each number inside the brackets by the number outside, retaining the minus in the middle.

Game

Magic Square

In a magic square, all the rows, all the columns and the 2 diagonals must add to the same number.

1. Complete the magic square, using only the positive integers 1 to 9:

Magic square using integers

2. Complete the magic square, using only the integers:

−10, −8, −6, −4, 0, 2, 4, 6

magic square involving integers

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