Ordering numbers and using inequality symbols

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Every number has a place on the number line.

Numbers get bigger as you move to the right and smaller as you move to the left.

That one picture settles almost every question about which of two numbers is bigger, including negative numbers, decimals and fractions.

You need to be able to "order positive and negative integers, decimals and fractions; use the symbols =, ≠, <, >, ≤, ≥", "including use of a number line".

This unit starts with ordering because every later skill, from rounding to solving inequalities, needs you to place numbers correctly.

Negative numbers on the number line

Here are six temperatures in degrees Celsius, marked on a number line.

A number line from -12 to 8 with six temperatures marked as dots: -11, -4, -1, 0, 3 and 7. The further left a number is, the smaller it is.

$-11$ is to the left of $-4$ , so $-11$ is smaller, even though 11 is bigger than 4.

A temperature of $-11$ °C is colder than $-4$ °C. With negative numbers, the number that looks bigger is the one further from zero, and that makes it smaller.

Zero is neither positive nor negative.

Decimals: compare place by place

To compare decimals, line up the decimal points and compare digit by digit from the left: units, then tenths, then hundredths.

Writing the numbers with the same number of decimal places helps: 0.7 is 0.700 and 0.68 is 0.680, so 0.7 is bigger, because 7 tenths beats 6 tenths.

A longer decimal is not a bigger one.

0.607 has more digits than 0.67, but it has only 0 hundredths where 0.67 has 7 hundredths, so 0.607 is smaller.

The number line below shows the gap.

A number line from 0.60 to 0.72 in steps of 0.01 with four decimals marked: 0.607 just after 0.60, then 0.67, 0.68 and 0.7.

Fractions: a common denominator or a decimal

Two fractions with the same denominator are easy to compare: $\frac{5}{9} < \frac{7}{9}$ , because 5 ninths is less than 7 ninths.

When the denominators differ, either rewrite both with a common denominator or change both to decimals by dividing the numerator by the denominator.

Without a calculator, the common denominator is usually the safer route.

Some fractions give recurring decimals, where a digit repeats for ever.

$\frac{1}{3} = 0.333\ldots$ , written $0.\dot{3}$ with a dot over the digit that repeats.

So $0.\dot{3}$ is bigger than 0.33: in the thousandths place, $0.\dot{3}$ has a 3 and 0.33 has a 0.

The six symbols

Symbol

Meaning

Example

$=$

is equal to

$\frac{1}{2} = 0.5$

$\neq$

is not equal to

$\frac{1}{3} \neq 0.3$

$<$

is less than

$-7 < -2$

$>$

is greater than

$0.7 > 0.68$

$\le$

is less than or equal to

$n \le 2$ means $n$ is 2 or smaller

$\ge$

is greater than or equal to

$x \ge -1$ means $x$ is $-1$ or bigger

The narrow end of $<$ and $>$ points at the smaller number.

$3 < 8$ and $8 > 3$ say the same thing.

An inequality such as $-3 < n \le 2$ describes a whole set of numbers.

Read it from the middle: $n$ is greater than $-3$ and less than or equal to 2.

On a number line, an open circle means the end value is not included ( $<$ or $>$ ) and a filled circle means it is ( $\le$ or $\ge$ ).

A number line from -5 to 4 showing the inequality -3 < n ≤ 2: an open circle at -3, a filled circle at 2, and a line joining them.

If $n$ is an integer (a whole number, positive, negative or zero), the values are $-2, -1, 0, 1, 2$ .

The value $-3$ is left out because of the strict sign $<$ , and 2 is kept because of $\le$ .

Key terms

Integer: a whole number, which can be positive, negative or zero, such as $-4$ , 0 or 17.

Number line: a line on which every number has a position, with numbers increasing to the right.

Inequality: a statement that one quantity is less than, greater than, less than or equal to, or greater than or equal to another, written with $<$ , $>$ , $\le$ or $\ge$ .

Recurring decimal: a decimal in which a digit or a block of digits repeats for ever, such as $0.\dot{3} = 0.333\ldots$

How the marks work

M is a method mark. You earn it for a correct method, even if you then slip with the arithmetic.

A is an accuracy mark for the correct answer from that method. A correct value can imply the method.

B is a mark for an answer or a fact on its own. "dep" means a mark depends on the mark before it.

ft (follow through) means your own earlier wrong value, used correctly later, still earns marks. oe means any equivalent form is accepted.

A correct answer alone usually scores full marks, but a wrong answer alone scores nothing, so always write the method. Where the question says "You must show your working." (or is a Show that or Prove), an answer alone scores 0.

Rounding too early costs a mark. Wrong further work after a correct answer loses the mark, so stop once the question is answered. Two different answers on the answer line count as a choice and lose the mark.

In a Show that, write every step on its own line, and write the more accurate value before any rounded one. Never join unequal values in one chain of equals signs.

Worked example 1: ordering negative numbers (no calculator)

Here are six temperatures, in °C.

$-4 \quad 3 \quad -11 \quad 0 \quad -1 \quad 7$

Write the temperatures in order, coldest first.

Place them on the number line: the further left, the colder. B1 for all six in the correct order

Answer $-11, -4, -1, 0, 3, 7$

Check the negatives: $-11$ is 11 below zero and $-4$ is only 4 below, so $-11$ comes first.

Worked example 2: ordering decimals (no calculator)

Write these numbers in order of size, smallest first.

$0.7 \quad 0.68 \quad 0.071 \quad 0.607 \quad 0.67$

Give them all three decimal places: 0.700, 0.680, 0.071, 0.607, 0.670

Compare as whole numbers of thousandths: 71, 607, 670, 680, 700 B1 for all five in order

Answer $0.071, 0.607, 0.67, 0.68, 0.7$

0.071 is first because it has 0 tenths.

Every other number has 6 or 7 tenths.

Worked example 3: comparing two fractions (no calculator)

Write $<$ , $>$ or $=$ between $\frac{5}{8}$ and $\frac{7}{11}$ .

Show working to support your answer.

Common denominator 88: $\frac{5}{8} = \frac{55}{88}$ and $\frac{7}{11} = \frac{56}{88}$ M1 (both fractions over a common denominator, or both as decimals)

$55 < 56$ , so $\frac{5}{8} < \frac{7}{11}$ A1

Answer $\frac{5}{8} < \frac{7}{11}$

As decimals, $\frac{5}{8} = 0.625$ and $\frac{7}{11} = 0.6363\ldots$ , which agrees.

The fractions are very close, so a guess from the size of the numbers would be risky.

Worked example 4: symbols with negative decimals and fractions (no calculator)

Write $<$ , $>$ or $=$ in each box.

(a) $-0.6 \;\square\; -\frac{3}{5}$ (b) $-\frac{2}{3} \;\square\; -0.65$ (c) $0.\dot{3} \;\square\; 0.33$

(a) $\frac{3}{5} = 0.6$ , so $-\frac{3}{5} = -0.6$ B1

(b) $-\frac{2}{3} = -0.666\ldots$ , which is further below zero than $-0.65$ , so it is smaller B1

Answer (a) $=$ (b) $<$ (c) $>$

In (b), the two numbers are close.

Think "which is colder?": $-0.666\ldots$ is colder than $-0.65$ .

Worked example 5: integers that satisfy an inequality (no calculator)

$n$ is an integer and $-3 < n \le 2$

Write down all the possible values of $n$ .

$-3$ is not included ( $<$ ) and 2 is included ( $\le$ )

$-2, -1, 0, 1, 2$ B2 (B1 for four correct values and none wrong, or all five with one extra value)

Answer $-2, -1, 0, 1, 2$

Zero is an integer, and it is easy to leave out, so check for it.

Choosing numbers to fit conditions

Recent papers have asked you to choose numbers that fit several conditions at once, written in symbols, in words, or both.

Work through the conditions one at a time, and check your final numbers against every condition.

The usual slip is a pair that gives the right sum but breaks one inequality.

Worked example 6: a pair of numbers from conditions (no calculator)

$a$ and $b$ are integers.

$a + b = 6 \qquad a < -2 \qquad b \le 10$

Write down one possible pair of values for $a$ and $b$ .

Try the largest value $a$ can take: $a = -3$ , so $b = 6 - (-3) = 9$ M1 (a pair with $a + b = 6$ and one inequality satisfied)

Check: $-3 < -2$ is true and $9 \le 10$ is true A1

Answer $a = -3$ , $b = 9$

$a = -4$ , $b = 10$ also works.

$a = -5$ , $b = 11$ does not: the sum is right, but $11 \le 10$ is false.

So there are exactly two pairs.

Worked example 7: a number described in words (no calculator)

Write down a decimal that is greater than $\frac{2}{5}$ and less than $\frac{3}{7}$ .

$\frac{2}{5} = 0.4$ and $\frac{3}{7} = 0.428\ldots$ M1 (both as decimals, or both over a common denominator)

Any decimal between them, for example 0.41 A1

Answer 0.41

Many answers are right (0.42, 0.405, 0.4285).

"Greater than" and "less than" both mean the end values themselves do not count, so 0.4 is wrong.

Why it works (not examined)

The number line is built by reflecting the positive numbers in zero.

Reflecting reverses order: 4 is less than 11, so the mirror image $-4$ is greater than $-11$ .

That is why making a number negative flips the inequality between two numbers, and you will meet the same flip in the algebra unit when you multiply an inequality by a negative number.

Connections

The symbols here are the language of inequalities in algebra, where you solve $2x + 1 < 9$ and show the answer on a number line.

Ordering decimals is the first step of rounding and of error intervals in the next unit.

In statistics, ordering a list comes before every median.

Exam tip

When you put numbers in order, write the original numbers on the answer line, in their original form.

If you changed $\frac{3}{7}$ to 0.4285 to compare it, the answer line still needs $\frac{3}{7}$ , and check whether the question wants smallest first or largest first.

Common mistakes

Thinking a longer decimal is bigger. 0.607 is less than 0.67. Give the decimals the same number of places and compare.

Ordering negatives by their digits. $-11 < -4$ . A negative number with bigger digits is further left, so it is smaller.

Comparing fractions by their denominators. $\frac{1}{8}$ is smaller than $\frac{1}{5}$ , because the whole is cut into more pieces. Use a common denominator.

Mixing up open and filled circles. An open circle means the end value is not included; a filled circle means it is.

Leaving out zero when listing the integers in an inequality.

Numbers that fit some conditions but not all. Check your final answer against every condition, not only the sum.

Changing the form on the answer line. Write the numbers as they were given.

Check your understanding

1. Write $<$ , $>$ or $=$ in each box.

(a) $-9 \;\square\; -12$

[1]

(b) $-0.2 \;\square\; -0.25$

[1]

2. Write these numbers in order of size, smallest first.

$0.305 \quad 0.35 \quad 0.053 \quad 0.5 \quad 0.3$

Answer ______

[1]

3. Here are four fractions.

$\frac{3}{4} \quad \frac{5}{7} \quad \frac{7}{10} \quad \frac{2}{3}$

Write them in order of size, smallest first.

Answer ______

[2]

4. $n$ is an integer and $-4 \le n < 1$

Write down all the possible values of $n$ .

Answer ______

[2]

5. For each statement, tick the correct box.

Statement

True

False

$-7 < -3$

☐

$\frac{3}{8} > 0.4$

☐

$-\frac{1}{2} \neq -0.5$

☐

[3]

6. Priya says, "0.45 is bigger than 0.5, because 45 is bigger than 5."

What mistake has Priya made?

[1]

7. $x$ and $y$ are integers.

$x - y = 7 \qquad x < 10 \qquad y > 0$

Write down all the possible pairs of values of $x$ and $y$ .

[2]

8. Write down a fraction that is greater than 0.6 and less than 0.65

Answer ______

[1]

Answers

1. (a) $>$ B1 ( $-9$ is to the right of $-12$ )

(b) $>$ B1 ( $-0.20$ is closer to zero than $-0.25$ )

2. $0.053, 0.3, 0.305, 0.35, 0.5$ B1

3. As decimals: $0.75$ , $0.714\ldots$ , $0.7$ , $0.666\ldots$ (or over 420: $\frac{315}{420}, \frac{300}{420}, \frac{294}{420}, \frac{280}{420}$ ) M1

$\frac{2}{3}, \frac{7}{10}, \frac{5}{7}, \frac{3}{4}$ A1

4. $-4, -3, -2, -1, 0$ B2 (B1 for four correct values and none wrong, or all five with one extra value). $-4$ is included ( $\le$ ) and 1 is not ( $<$ ).

5. True; False ( $\frac{3}{8} = 0.375$ ); False ( $-\frac{1}{2}$ is equal to $-0.5$ ) B3 (B1 for each correct row)

6. She compared 45 with 5 as whole numbers (oe). 0.5 is 5 tenths and 0.45 is only 4 tenths, so 0.5 is bigger (or 0.5 = 0.50 and 50 hundredths is more than 45 hundredths). B1

7. $x = 8, y = 1$ and $x = 9, y = 2$ B2 (B1 for one correct pair with no wrong pair). $x = 7$ would need $y = 0$ , which breaks $y > 0$ .

8. Any correct fraction, for example $\frac{5}{8}$ ( $= 0.625$ ) or $\frac{31}{50}$ ( $= 0.62$ ) B1

Ordering numbers and using inequality symbols

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