Sec 1 · G2 (N(A))

Integers, rational and real numbers

Practise integers, rational and real numbers with step-by-step help. Free account.

1. Negative numbers on the number line

Number lines from −6 to 6. Start at −2 and jump 5 to the right to reach 3. Start at 1 and jump 4 to the left to reach −3.

Rule	Example
adding: move right	$-2 + 5 = 3$
subtracting: move left	$1 - 4 = -3$
subtracting a negative: add	$3 - (-4) = 7$
adding a negative: subtract	$3 + (-4) = -1$
further left is smaller	$-7 < -2$

📖 Worked example

At 6 a.m., the temperature in Harbin was $5^\circ\text{C}$ . By noon, it had fallen by $13^\circ\text{C}$ . Find the temperature at noon.

Step 1 of 1

Write the calculation: the starting temperature, then the change

Temperature at noon

= 5 - 13

= -8

°C

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2. Multiplying and dividing integers

Same signs: the answer is positive

\displaystyle (+) \times (+) = (+) \qquad (-) \times (-) = (+)

e.g.

\displaystyle (-6) \times (-3)

\displaystyle = 18

\displaystyle (-20) \div (-4)

\displaystyle = 5

$\div$ works the same way as $\times$ .

Different signs: the answer is negative

\displaystyle (+) \times (-) = (-) \qquad (-) \times (+) = (-)

e.g.

\displaystyle (-6) \times 3

\displaystyle = -18

\displaystyle (-20) \div 4

\displaystyle = -5

$\div$ works the same way as $\times$ .

A negative number to a power

\displaystyle (-a)^{\text{even}} = (+) \qquad (-a)^{\text{odd}} = (-)

e.g.

\displaystyle (-2)^{4}

\displaystyle = 16

\displaystyle (-2)^{3}

\displaystyle = -8

📖 Worked example

Calculate $(-72) \div 8 \times 4$ .

Step 1 of 2

Signs: an even number of negative signs gives +, an odd number gives −

Number of negative numbers:

1

So the answer is negative.

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3. Order of operations

Order	Do
1	brackets
2	powers and roots
3	$\times$ and $\div$ , from left to right
4	$+$ and $-$ , from left to right

Divide and multiply from left to right

\displaystyle -3 + 12 \div (-4) \times 2

e.g.

\displaystyle -3 + 12 \div (-4) \times 2

\displaystyle = -3 + (-3) \times 2

\displaystyle = -3 + (-6)

\displaystyle = -9

Brackets, then the power

\displaystyle (5 - 8)^{2} - 4

e.g.

\displaystyle (5 - 8)^{2} - 4

\displaystyle = (-3)^{2} - 4

\displaystyle = 9 - 4

\displaystyle = 5

📖 Worked example

Calculate $15 - (4 - 9) \times (-3)$ .

Step 1 of 3

Brackets first

15 - (4 - 9) \times (-3)

= 15 - (-5) \times (-3)

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4. Fractions and decimals

Add: make the denominators the same

\displaystyle \dfrac{2}{3} + \dfrac{3}{4}

e.g.

\displaystyle \dfrac{2}{3} + \dfrac{3}{4}

\displaystyle = \dfrac{8}{12} + \dfrac{9}{12}

\displaystyle = \dfrac{17}{12}

Divide: multiply by the reciprocal

\displaystyle \dfrac{5}{6} \div \dfrac{10}{9}

e.g.

\displaystyle \dfrac{5}{6} \div \dfrac{10}{9}

\displaystyle = \dfrac{5}{6} \times \dfrac{9}{10}

\displaystyle = \dfrac{3}{4}

Mixed numbers: change to improper fractions first

\displaystyle 1\tfrac{1}{2} \times 2\tfrac{2}{3}

e.g.

\displaystyle \dfrac{3}{2} \times \dfrac{8}{3}

\displaystyle = 4

📖 Worked example

Calculate $2\dfrac{2}{3} \div 3\dfrac{2}{3}$ , giving your answer in its simplest form.

Step 1 of 3

Change each mixed number to an improper fraction

2\dfrac{2}{3} \div 3\dfrac{2}{3}

= \dfrac{8}{3} \div \dfrac{11}{3}

📖 Worked example

Without using a calculator, find the value of $1.8 \div 0.06$ .

Step 1 of 2

Multiply both numbers by the same power of 10 so that you divide by a whole number

1.8 \div 0.06 = 180 \div 6

📖 Worked example

Priya spends $\dfrac{3}{4}$ of her savings on a pair of shoes and $\dfrac{5}{6}$ of the remainder on a book. She has $\$10$ left. Find the amount of money she had at first.

Step 1 of 4

The remainder after the first part: 1 take away the first fraction

1 - \dfrac{3}{4} = \dfrac{1}{4}

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5. Rational and real numbers

Nested boxes: whole numbers 0, 1, 2, 3 inside integers (−3, −1) inside rational numbers (2/3, −0.5, 0.25) inside real numbers; irrational numbers such as √2 and π are real but not rational.

Type	Meaning	Example
rational	can be written as $\dfrac{a}{b}$ with integers $a$ , $b$ ( $b \neq 0$ )	$0.25 = \dfrac{1}{4}$
irrational	cannot: the decimal never ends or repeats	$\sqrt{2} = 1.4142\ldots$

📖 Worked example

State whether $\sqrt{10}$ is a rational number or an irrational number.

Step 1 of 2

A square root is rational only when the number under it is a perfect square

10

is not a perfect square, so

\sqrt{10}

is a non-terminating, non-recurring decimal.

📖 Worked example

Mei Ling writes: "The sum of two irrational numbers is always irrational." State whether the statement is true or false. If it is false, give a counterexample. Choose the answer with its reason.

Step 1 of 2

A statement that says always is false if ONE example breaks it (a counterexample); one example that works proves nothing

The statement is false.

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Sec 1 · G2 (N(A))

Integers, rational and real numbers

Practise integers, rational and real numbers with step-by-step help. Free account.

Start free

1. Negative numbers on the number line

Rule	Example
adding: move right	$-2 + 5 = 3$
subtracting: move left	$1 - 4 = -3$
subtracting a negative: add	$3 - (-4) = 7$
adding a negative: subtract	$3 + (-4) = -1$
further left is smaller	$-7 < -2$

📖 Worked example

At 6 a.m., the temperature in Harbin was $5^\circ\text{C}$ . By noon, it had fallen by $13^\circ\text{C}$ . Find the temperature at noon.

Step 1 of 1

Write the calculation: the starting temperature, then the change

Temperature at noon

= 5 - 13

= -8

°C

Practise this →

2. Multiplying and dividing integers

Same signs: the answer is positive

\displaystyle (+) \times (+) = (+) \qquad (-) \times (-) = (+)

e.g.

\displaystyle (-6) \times (-3)

\displaystyle = 18

\displaystyle (-20) \div (-4)

\displaystyle = 5

$\div$ works the same way as $\times$ .

Different signs: the answer is negative

\displaystyle (+) \times (-) = (-) \qquad (-) \times (+) = (-)

e.g.

\displaystyle (-6) \times 3

\displaystyle = -18

\displaystyle (-20) \div 4

\displaystyle = -5

$\div$ works the same way as $\times$ .

A negative number to a power

\displaystyle (-a)^{\text{even}} = (+) \qquad (-a)^{\text{odd}} = (-)

e.g.

\displaystyle (-2)^{4}

\displaystyle = 16

\displaystyle (-2)^{3}

\displaystyle = -8

📖 Worked example

Calculate $(-72) \div 8 \times 4$ .

Step 1 of 2

Signs: an even number of negative signs gives +, an odd number gives −

Number of negative numbers:

1

So the answer is negative.

Practise this →

3. Order of operations

Order	Do
1	brackets
2	powers and roots
3	$\times$ and $\div$ , from left to right
4	$+$ and $-$ , from left to right

Divide and multiply from left to right

\displaystyle -3 + 12 \div (-4) \times 2

e.g.

\displaystyle -3 + 12 \div (-4) \times 2

\displaystyle = -3 + (-3) \times 2

\displaystyle = -3 + (-6)

\displaystyle = -9

Brackets, then the power

\displaystyle (5 - 8)^{2} - 4

e.g.

\displaystyle (5 - 8)^{2} - 4

\displaystyle = (-3)^{2} - 4

\displaystyle = 9 - 4

\displaystyle = 5

📖 Worked example

Calculate $15 - (4 - 9) \times (-3)$ .

Step 1 of 3

Brackets first

15 - (4 - 9) \times (-3)

= 15 - (-5) \times (-3)

Practise this →

4. Fractions and decimals

Add: make the denominators the same

\displaystyle \dfrac{2}{3} + \dfrac{3}{4}

e.g.

\displaystyle \dfrac{2}{3} + \dfrac{3}{4}

\displaystyle = \dfrac{8}{12} + \dfrac{9}{12}

\displaystyle = \dfrac{17}{12}

Divide: multiply by the reciprocal

\displaystyle \dfrac{5}{6} \div \dfrac{10}{9}

e.g.

\displaystyle \dfrac{5}{6} \div \dfrac{10}{9}

\displaystyle = \dfrac{5}{6} \times \dfrac{9}{10}

\displaystyle = \dfrac{3}{4}

Mixed numbers: change to improper fractions first

\displaystyle 1\tfrac{1}{2} \times 2\tfrac{2}{3}

e.g.

\displaystyle \dfrac{3}{2} \times \dfrac{8}{3}

\displaystyle = 4

📖 Worked example

Calculate $2\dfrac{2}{3} \div 3\dfrac{2}{3}$ , giving your answer in its simplest form.

Step 1 of 3

Change each mixed number to an improper fraction

2\dfrac{2}{3} \div 3\dfrac{2}{3}

= \dfrac{8}{3} \div \dfrac{11}{3}

📖 Worked example

Without using a calculator, find the value of $1.8 \div 0.06$ .

Step 1 of 2

Multiply both numbers by the same power of 10 so that you divide by a whole number

1.8 \div 0.06 = 180 \div 6

📖 Worked example

Priya spends $\dfrac{3}{4}$ of her savings on a pair of shoes and $\dfrac{5}{6}$ of the remainder on a book. She has $\$10$ left. Find the amount of money she had at first.

Step 1 of 4

The remainder after the first part: 1 take away the first fraction

1 - \dfrac{3}{4} = \dfrac{1}{4}

Practise this →

5. Rational and real numbers

Type	Meaning	Example
rational	can be written as $\dfrac{a}{b}$ with integers $a$ , $b$ ( $b \neq 0$ )	$0.25 = \dfrac{1}{4}$
irrational	cannot: the decimal never ends or repeats	$\sqrt{2} = 1.4142\ldots$

📖 Worked example

State whether $\sqrt{10}$ is a rational number or an irrational number.

Step 1 of 2

A square root is rational only when the number under it is a perfect square

10

is not a perfect square, so

\sqrt{10}

is a non-terminating, non-recurring decimal.

📖 Worked example

Mei Ling writes: "The sum of two irrational numbers is always irrational." State whether the statement is true or false. If it is false, give a counterexample. Choose the answer with its reason.

Step 1 of 2

A statement that says always is false if ONE example breaks it (a counterexample); one example that works proves nothing

The statement is false.

Practise this →

Practise integers, rational and real numbers with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free