Year 7 Maths Integers

Numbers Above and Below Zero

Q: What is an integer?

A whole number that is positive, negative or zero, such as -6, 0 or 14.

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Numbers Above and Below Zero - Video - Above and Below Zero Watch

Theory

Numbers above zero are positive and numbers below zero are negative. An integer is a whole number that is positive, negative or zero. This Year 7 maths page shows how to write an integer for a real situation, find the opposite of a number, move up and down past zero, and find the difference between a value above zero and a value below zero.

Numbers greater than zero are positive numbers, such as $6$, $4$ and $6.8$. Numbers less than zero are negative numbers, such as $-7$, $-9.6$ and $-4.2$. Zero is neither positive nor negative.

An integer is a whole number that is positive, negative or zero: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ A positive integer can be written with or without a plus sign: $5$ and $+5$ mean the same thing.

Every integer has a direction (its sign) and a size, also called its magnitude (how far it is from zero). A diver at $-12$ m is below sea level, and the size is $12$. Numbers with a sign are called directed numbers.

The opposite of a number is the same distance from $0$ on the other side. The opposite of $9$ is $-9$, and the opposite of $-9$ is $9$. Opposite words swap the sign: a rise and a fall, a deposit and a withdrawal, north and south.

The minus sign has two jobs. In $-4^\circ\text{C}$ it shows a position below zero. Between two numbers, as in $9 - 4$, it is the operation subtract.

$23^\circ\text{C}$ is $23$ above zero. $-10^\circ\text{C}$ is $10$ below zero.

BCE years are negative and CE years are positive. From $250$ BCE to $150$ CE is $250 + 150 = 400$ years.

Across sea level the two sizes add: $7 + 36 = 43$ m.

Words in a question tell you the sign:

Positive (+)	Negative (−)
above zero, above sea level	below zero, below sea level
rise, up, gain, climb	fall, down, lose, dive
profit, deposit	loss, fee, withdrawal
floors above the ground	car park levels below the ground
CE years	BCE years

\[ \text{opposite of } 9 = -9, \quad \text{opposite of } {-9} = 9 \]

opposite of 9 = - 9, opposite of - 9 = 9

\[ 6 - 9 = 6 - 6 - 3 = -3 \]

6 - 9 = 6 - 6 - 3 = - 3

\[ \text{from } {-11} \text{ up to } 26 = 11 + 26 = 37 \]

from - 11 up to 26 = 11 + 26 = 37

          Rise means add, fall means subtract. To go past zero, go to \(0\) first, then the rest: \(5 - 8 = 5 - 5 - 3 = -3\).
        

          Across zero, the sizes add. From \(-9^\circ\text{C}\) up to \(5^\circ\text{C}\) is \(9\) up to zero, then \(5\) more: \(9 + 5 = 14^\circ\text{C}\).
        

How to write an integer for a situation

Direction: decide if it is above zero (positive) or below zero (negative).
Size: find how far it is from zero.
Integer: write the size with the sign, for example below zero by $8$ is $-8$.

How to find a rise or fall past zero

Start at the first value.
Move to zero first, then move the rest of the way.
Check the sign: below zero is negative.

How to find a difference across zero

Count from the lower value up to $0$.
Count from $0$ up to the higher value.
Add the two sizes.

Example 1 — Writing integers

Write an integer for each situation.
(i) Overnight it was $3$ degrees below zero.
(ii) Sam withdraws $\$60$ from his account.
(iii) A plane climbs $250$ m.

Solution

Below zero and a withdrawal are negative. Climbing is going up, so it is positive.

(i) $-3$
(ii) $-60$
(iii) $250$

Example 2 — Opposites

Write the opposite of each.
(i) $-9$
(ii) $31$
(iii) a rise of $5^\circ\text{C}$
(iv) a deposit of $\$70$

Solution

The opposite is the same distance from $0$ on the other side. Opposite words swap the sign.

(i) $9$
(ii) $-31$
(iii) a fall of $5^\circ\text{C}$
(iv) a withdrawal of $\$70$

Example 3 — Falling past zero

It is $5^\circ\text{C}$ at 9 pm. By 3 am the temperature has fallen by $11^\circ\text{C}$. What is the temperature at 3 am?

Solution

$5 - 11$	$=$	$5 - 5 - 6$
	$=$	$-6$

5 - 11 = 5 - 5 - 6 = - 6

A fall means subtract. Go down $5$ to zero, then $6$ more. It is $-6^\circ\text{C}$ at 3 am.

Example 4 — A rise across zero

At dawn it was $-7^\circ\text{C}$. By noon it was $16^\circ\text{C}$. By how much did the temperature rise?

Solution

$7 + 16$

$=$

$23$

7 + 16 = 23

$7$ up to zero, then $16$ more. The temperature rose by $23^\circ\text{C}$.

Example 5 — Years across BCE and CE

(i) How many years are there from $250$ BCE to $150$ CE?
(ii) How many years are there from $310$ BCE to $170$ BCE?

Solution

$250 + 150$	$=$	$400$
$310 - 170$	$=$	$140$

250 + 150 = 400, 310 - 170 = 140

(i) BCE is negative and CE is positive, so the sizes add: $400$ years.
(ii) Both are BCE, on the same side of zero, so subtract the sizes: $140$ years.

Example 6 — A balance below zero

After a fee of $\$15$, Ari’s bank balance is $-\$7$. What was his balance before the fee?

Solution

$-7 + 15$	$=$	$-7 + 7 + 8$
	$=$	$8$

- 7 + 15 = - 7 + 7 + 8 = 8

Undo the fee by adding $15$ back: up $7$ to zero, then $8$ more. Check: $8 - 15 = -7$. He had $\$8$.

Example 7 — Above and below sea level

A lookout is $36$ m above sea level. A snorkeller is $7$ m below sea level.
(i) How far is the lookout above the snorkeller?
(ii) She dives down $6$ m more. Write her new height as an integer.

Solution

$7 + 36$	$=$	$43$
$-7 - 6$	$=$	$-13$

7 + 36 = 43, - 7 - 6 = - 13

(i) $43$ m, as in the third diagram.
(ii) $-13$ m, which is $13$ m below sea level.

Example 8 — A decimal difference

One night the temperature on a mountain was $-3.4^\circ\text{C}$. The next afternoon it was $18.5^\circ\text{C}$. What was the difference?

Solution

$3.4 + 18.5$

$=$

$21.9$

3.4 + 18.5 = 21.9

Across zero the sizes add, even for decimals. The difference is $21.9^\circ\text{C}$.

Common pitfalls

Stopping at zero. From $-9^\circ\text{C}$ up to $5^\circ\text{C}$ is a rise of $14^\circ\text{C}$, not $4^\circ\text{C}$.

A bigger size below zero is colder. $-8^\circ\text{C}$ is colder than $-3^\circ\text{C}$, because it is further below zero.

Mixing up the two jobs of the minus sign. In $-4^\circ\text{C}$ the sign shows a position. In $9 - 4$ it means subtract.

Subtracting sizes across zero. From $250$ BCE to $150$ CE is $250 + 150 = 400$ years, not $100$.

Frequently asked questions

What is an integer?

A whole number that is positive, negative or zero, such as $-6$, $0$ or $14$.

Is zero positive or negative?

Neither. Zero is the point that separates the positive numbers from the negative numbers.

What is the opposite of a number?

The number the same distance from zero on the other side. The opposite of $-9$ is $9$.

How do you find a difference across zero?

Count up to zero, then past it, and add the two sizes. From $-7$ to $16$ is $7 + 16 = 23$.

Why are BCE years negative?

On a timeline, BCE years come before year zero, so they sit to the left of zero like negative numbers.

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Integers on a Number Line

\(5 - 11\)	\(=\)	\(5 - 5 - 6\)
\(\)	\(=\)	\(-6\)

\(250 + 150\)	\(=\)	\(400\)
\(310 - 170\)	\(=\)	\(140\)

\(-7 + 15\)	\(=\)	\(-7 + 7 + 8\)
\(\)	\(=\)	\(8\)

\(7 + 36\)	\(=\)	\(43\)
\(-7 - 6\)	\(=\)	\(-13\)

Numbers Above and Below Zero

📖 Theory

How to write an integer for a situation

How to find a rise or fall past zero

How to find a difference across zero

Common pitfalls

Frequently asked questions

Theory