Ordering Numbers and Using Inequality Symbols Revision Notes | GCSE AQA Higher Maths

1. Understanding Number Order

Numbers have a natural order, like people queuing by height. Some numbers are bigger, some smaller, and we need precise language to describe these relationships.

The number line is your best friend here. Picture a horizontal line with zero in the middle. Numbers get bigger as you move right, smaller as you move left. This works for all types of numbers: whole numbers, decimals, fractions, and negatives.

Here's where students often stumble: negative numbers work backwards. -10 is actually smaller than -2 because -10 is further left on the number line. Think of it like temperature - minus ten degrees is colder (smaller) than minus two degrees.

The key insight is this: position on the number line determines size, not the digits themselves. -100 might look big because of the 100, but it's actually much smaller than -1 because it's further left.

2. The Language of Comparison

Mathematics gives us precise symbols to compare numbers. These aren't just shortcuts - they're the universal language that mathematicians use worldwide.

Here's a memory trick that works: the symbol always 'eats' the bigger number. The wide mouth opens towards the larger value. So 5 > 3 because the mouth opens towards 5, the bigger number.

The symbols ≤ and ≥ are crucial for exam success. They mean 'or equal to' as well. So x ≤ 5 means x could be 5, or anything smaller than 5. Students often forget this 'or equal' part and lose marks.

**Trap**: Many students confuse the direction. Remember: 7 > 2 is correct, but 2 > 7 is wrong. The arrow always points from the bigger number to the smaller one, like water flowing downhill.

Worked example: Writing Inequality Statements

Write three different inequality statements comparing the numbers -4, 1, and 6

1. On the number line: -4 is leftmost, then 1, then 6 — Always visualise the number line first
2. $-4 < 1 < 6$ — Chain inequalities show the complete order
3. $-4 < 1$, $1 < 6$, $-4 < 6$ — Any pair can be compared directly
4. $6 > 1 > -4$ — Same relationships, symbols flipped

Answer: Multiple valid statements: $-4 < 1 < 6$ or $6 > 1 > -4$ or individual pairs

3. Ordering Mixed Number Types

Real exam questions mix different types of numbers: integers, decimals, and fractions. The secret is converting everything to the same type before comparing.

Decimals are usually easiest to work with. Convert fractions to decimals by dividing: $\frac{3}{4} = 3 \div 4 = 0.75$. For negative fractions, the negative sign applies to the whole fraction: $-\frac{1}{2} = -0.5$.

Here's the systematic approach: convert all numbers to decimals, then order them on the number line. Once you have the correct order, you can write your answer using the original forms.

Worked example: Ordering Mixed Numbers

Put these numbers in ascending order: $\frac{2}{3}$, -1.2, 0.8, $-\frac{1}{4}$, 2

1. $\frac{2}{3} = 2 \div 3 = 0.667...$ $-\frac{1}{4} = -1 \div 4 = -0.25$ — Use long division or calculator
2. 0.667, -1.2, 0.8, -0.25, 2 — Now we can compare directly
3. -1.2, -0.25, 0.667, 0.8, 2 — Negatives first (further left), then positives
4. $-1.2, -\frac{1}{4}, \frac{2}{3}, 0.8, 2$ — Convert back to original notation

Answer: $-1.2, -\frac{1}{4}, \frac{2}{3}, 0.8, 2$

**Trap**: Students often think $\frac{2}{3}$ is bigger than 0.8 because 2 and 3 are bigger numbers than 8. Always convert to decimals first: $\frac{2}{3} ≈ 0.667$ which is smaller than 0.8.

4. Real-World Applications

Number ordering appears everywhere in real life. Temperatures, bank balances, altitudes, and sports scores all use the same principles we've learned.

Bank statements show this clearly. If your account has £50 and you spend £80, your balance becomes -£30 (overdrawn). This is smaller than £0, which is smaller than any positive balance. The bank treats -£30 as 'less money' than -£10.

In geography, altitude works the same way. Sea level is 0 metres. Mount Everest is +8,848m above sea level. The Dead Sea shore is -430m below sea level. So the Dead Sea shore is lower (smaller altitude) than sea level.

Worked example: Temperature Comparison

The temperatures in five cities are: London 3°C, Moscow -12°C, Dubai 28°C, Reykjavik -5°C, Cairo 22°C. List the cities from coldest to warmest.

1. 3, -12, 28, -5, 22 — Focus on the numerical values
2. -12 < -5 < 3 < 22 < 28 — Negatives first, then positives in ascending order
3. Moscow (-12°C), Reykjavik (-5°C), London (3°C), Cairo (22°C), Dubai (28°C) — Coldest to warmest order

Answer: Moscow, Reykjavik, London, Cairo, Dubai

Sports league tables use similar logic. Goal difference can be negative if a team concedes more than they score. A team with -5 goal difference is performing worse than a team with -2 goal difference.

5. Exam Techniques and Common Mistakes

Exam questions on ordering often try to catch you out. The most common trap is negative numbers. Students see -8 and -3, then think -3 is smaller because 3

Another frequent mistake is with fractions and decimals. Don't guess which is bigger - always convert to the same type. Examiners love mixing $\frac{7}{10}$ with 0.65 because they look similar but $\frac{7}{10} = 0.7 > 0.65$.

When writing inequality chains like $a c$ - that's meaningless. If you want to show b is the largest, write $a c > a$.

For exam success, always show your working when ordering mixed numbers. Convert fractions to decimals explicitly - don't do it in your head. Examiners give method marks for clear conversion steps.