Integer Operations
Every question with a fully worked solution.
- Integer Operations - Video - HCF (Highest Common Factor) or GCD (Greatest Common Divisor) Watch
- Integer Operations - Video - The Order of Operations with Integers Watch
- Integer Operations - Video - What are Square Roots? | Exponents | Best Square Root Tricks Watch
- Integer Operations - Video - What is the Cube Root of a Number? Watch
Theory
Integers are whole numbers that can be positive, negative or zero. This Year 9 maths page revises the Year 8 work on directed numbers: adding and subtracting negatives, the sign rules for multiplying and dividing, the order of operations, and using prime factors to find a highest common factor, a square root or a cube root.
An integer is a whole number that may be positive, negative or zero:
On a number line, negative numbers sit to the left of zero.
When two signs sit together, rewrite them as one sign before doing anything else. Adding a negative is the same as subtracting: \(5 + (-2) = 5 - 2\). Subtracting a negative is the same as adding: \(5 - (-2) = 5 + 2\).
To multiply or divide, work out the size first, then the sign. If the two signs are the same, the answer is positive. If they are different, the answer is negative.
The order of operations is the same as always: brackets, then powers, then \(\times\) and \(\div\) from left to right, then \(+\) and \(-\) from left to right.
A prime number has exactly two factors, \(1\) and itself. Every whole number bigger than \(1\) can be written as a product of prime factors, such as \(60 = 2^2 \times 3 \times 5\). Prime factors show the highest common factor (HCF) of two numbers, and they show square roots and cube roots.
Real situations become directed number sentences. Below zero, down, owing and colder are negative. Above zero, up, a deposit and warmer are positive.
Two signs together become one:
The sign rules for multiplying and dividing:
| Signs | Multiply | Divide | Answer |
|---|---|---|---|
| same | \((+) \times (+)\), \((-) \times (-)\) | \((+) \div (+)\), \((-) \div (-)\) | positive |
| different | \((+) \times (-)\), \((-) \times (+)\) | \((+) \div (-)\), \((-) \div (+)\) | negative |
Roots can be split over a product, but not over a sum:
How to evaluate an integer expression
- Rewrite any pair of signs as a single sign.
- Work inside the brackets first, then any powers.
- Multiply and divide from left to right. For each step, find the size, then use the sign rule.
- Add and subtract from left to right. A number line helps: adding moves right, subtracting moves left.
- Check the sign: count the negatives in a multiplying chain, or estimate on a number line.
How to use prime factors
- Split each number into prime factors with a factor tree or by repeated division.
- HCF: multiply the prime factors that appear in both numbers, using the lower power each time.
- Root: share the prime factors into \(2\) equal groups for a square root, or \(3\) for a cube root. One group is the root.
- Check by multiplying back.
(i) \(-7 + (-5)\)
(ii) \(-4 - (-9)\)
(iii) At 6 am the temperature is \(-4^\circ\text{C}\). By 1 pm it has risen \(11\) degrees. What is the temperature at 1 pm?
(i) Rewrite the two signs as one: \(+(-5) = -5\).
| \(-7 + (-5)\) | \(=\) | \(-7 - 5\) |
| \(\) | \(=\) | \(\mathbf{-12}\) |
(ii) Subtracting a negative is adding: \(-(-9) = +9\).
| \(-4 - (-9)\) | \(=\) | \(-4 + 9\) |
| \(\) | \(=\) | \(\mathbf{5}\) |
(iii) Write a directed number sentence. Below zero is negative and a rise is a plus:
| \(-4 + 11\) | \(=\) | \(\mathbf{7}\) |
The temperature at 1 pm is \(7^\circ\text{C}\).
(i) \(-8 \times 7\)
(ii) \(-2 \times (-5) \times (-6)\)
(iii) \(60 \div (-5) \div (-3)\)
(i) Different signs give a negative:
| \(-8 \times 7\) | \(=\) | \(\mathbf{-56}\) |
(ii) Work one pair at a time, left to right, deciding the sign each time:
| \(-2 \times (-5) \times (-6)\) | \(=\) | \(10 \times (-6)\) |
| \(\) | \(=\) | \(\mathbf{-60}\) |
Check: there are three negatives, an odd number, so the answer is negative.
(iii) The same rules work for division:
| \(60 \div (-5) \div (-3)\) | \(=\) | \(-12 \div (-3)\) |
| \(\) | \(=\) | \(\mathbf{4}\) |
(i) \(-20 + 6 \times (-3)\)
(ii) \(\dfrac{-18 + 4 \times 3}{10 - 2 \times 8}\)
(i) Multiplication comes before addition, so do it first. Different signs give a negative:
| \(6 \times (-3)\) | \(=\) | \(-18\) |
| \(-20 + (-18)\) | \(=\) | \(-20 - 18\) |
| \(\) | \(=\) | \(\mathbf{-38}\) |
Working left to right instead would give \(-14 \times (-3) = 42\), which is wrong. The order of operations does not change because numbers are negative.
(ii) The fraction bar works like brackets. Work out the top and the bottom separately, multiplying before adding or subtracting:
| \(\text{top}\) | \(=\) | \(-18 + 12 = -6\) |
| \(\text{bottom}\) | \(=\) | \(10 - 16 = -6\) |
| \(\dfrac{-6}{-6}\) | \(=\) | \(\mathbf{1}\) |
Same signs, so the answer is positive.
(i) the HCF of \(36\) and \(60\)
(ii) \(\sqrt[3]{216}\)
(iii) \(\sqrt{196}\)
(i) Write each number as a product of primes:
| \(36\) | \(=\) | \(2 \times 2 \times 3 \times 3\) |
| \(60\) | \(=\) | \(2 \times 2 \times 3 \times 5\) |
Both share \(2 \times 2 \times 3\):
| \(\text{HCF}\) | \(=\) | \(2 \times 2 \times 3 = \mathbf{12}\) |
(ii) Share the prime factors into three equal groups:
| \(216\) | \(=\) | \((2 \times 3) \times (2 \times 3) \times (2 \times 3)\) |
| \(\sqrt[3]{216}\) | \(=\) | \(2 \times 3 = \mathbf{6}\) |
Check: \(6 \times 6 \times 6 = 216\).
(iii) Split \(196\) into two perfect squares, then take each root:
| \(\sqrt{196}\) | \(=\) | \(\sqrt{49 \times 4}\) |
| \(\) | \(=\) | \(\sqrt{49} \times \sqrt{4}\) |
| \(\) | \(=\) | \(7 \times 2 = \mathbf{14}\) |
Common pitfalls
Frequently asked questions
What is a negative times a negative?
A positive. When the two signs are the same, the answer is positive. For example, \(-4 \times (-6) = 24\).
What happens when you subtract a negative number?
Subtracting a negative is the same as adding the positive. For example, \(2 - (-7) = 2 + 7 = 9\).
Is negative 3 squared positive or negative?
It depends on the brackets. \((-3)^2 = 9\), but \(-3^2 = -9\), because without brackets only the \(3\) is squared.
What is an integer?
An integer is a whole number. It can be positive, negative or zero, such as \(-8\), \(0\) or \(15\). Fractions and decimals such as \(2.5\) are not integers.
How do you add a negative and a positive number?
Think of a number line. Start at the first number and move right for a positive or left for a negative. For example, \(-9 + 4 = -5\).
How do you find the HCF using prime factors?
Write both numbers as products of primes and multiply the primes they share. \(36 = 2 \times 2 \times 3 \times 3\) and \(60 = 2 \times 2 \times 3 \times 5\) share \(2 \times 2 \times 3\), so the HCF is \(12\).