Place Value, Ordering & the Number Line

SPEC N1 • NUMBER

The foundation for every calculation. Comparing decimals, negative numbers, and the inequality symbols — without falling into the traps AQA sets every year.

What is Place Value?

Each column in a number is worth 10× the column to its right. Moving left gets bigger. Moving right — across the decimal point — gets smaller.

ThousandsHundredsTensUnits•TenthsHundredthsThousandths

1000100101•0.10.010.001

3045•07

The number above is 3045.07. Each digit's value depends entirely on the column it sits in.

The Core Rule

The value of a digit = its column heading × the digit itself. In 3045.07, the 7 is in the hundredths column, so its value is 0.07 — not 7.

Three Types of Number

Before ordering numbers, you need to know what kind of numbers you're dealing with.

Integer

A whole number — positive, negative, or zero. No fractions, no decimal part.

Examples: −7, −1, 0, 4, 258

Rational Number

A number that can be written as p / q where p and q are integers (and q ≠ 0). This includes every integer, every terminating decimal, and every recurring decimal.

Examples: ½, 0.75, −3, 0.3̇ (= ⅓)

Irrational Number

A number that cannot be written as p/q. Its decimal expansion never terminates and never recurs.

Examples: √2, √3, π, e

Classifying a Number

In the exam, AQA may give you a list of numbers and ask which one is irrational (or rational). Use this table as your cheat-sheet.

NumberInteger?Rational?Irrational?Why

−5✔✔✘Whole number; can write as −5/1

0.75✘✔✘Terminates → can write as 3/4

0.3̇✘✔✘Recurring → equal to 1/3

√9✔✔✘= 3. A root isn't automatically irrational.

√2✘✘✔Decimal never terminates or repeats

π✘✘✔Non-terminating, non-recurring

Quick Tip

A square root is only irrational if the positive integer inside isn't a perfect square. √9, √16, √25 are all rational. √2, √3, √5, √7 are all irrational.

Ordering Decimals

To order decimals, line up the decimal points and compare digit by digit from the left. The number of digits after the point doesn't tell you how big it is.

The Classic Trap — Which is bigger: 0.45 or 0.5?

0.45 0.50

Compare the tenths column first: 4 vs 5. Since 4 < 5, we get 0.45 < 0.5.

The trap

Students see "45" and compare it to "5", decide 45 is bigger, and write 0.45 > 0.5. Wrong. Always compare column by column, starting with the tenths.

Worked Example 1 MARK01

Which is larger: 0.305 or 0.35?

1

Pad to equal decimal places: 0.305 vs 0.350

2

Compare tenths: both are 3. Tied.

3

Compare hundredths: 0 vs 5. Since 0 < 5, the second number is larger.

Answer: 0.35 is larger.

Writing 0.35 as 0.350 makes the columns line up — this is the safest method on Paper 1.

Negatives on the Number Line

A number line runs left to right. Further right = larger. Further left = smaller. With negatives, the digit can trick you.

The Rule

−2 > −7 because −2 is to the right of −7. The digit 2 is smaller than 7, but the number −2 is bigger than −7.

The Negative-Decimal Trap

When the numbers are negative and decimals, two traps combine. Here's how a weak answer and a strong answer differ on the same question.

Which is bigger: −0.3 or −0.33?

Weak student

"33 is bigger than 3, so −0.33 must be bigger than −0.3."

Conclusion: −0.33 > −0.3. ✘

Ignored the negative sign. Compared digits as if positive.

Strong student

"Sketch a number line. −0.3 sits at tenth 3. −0.33 sits slightly further left (more negative)."

Conclusion: −0.33 < −0.3. ✔

Used the number line to visualise. Further left = smaller.

−0.33 is to the left of −0.3 → so −0.33 < −0.3.

Inequality Symbols

Six symbols cover everything you need for GCSE comparison statements.

SymbolMeaningExample

<less than−5 < −2 ("−5 is less than −2")

>greater than0.5 > 0.45 ("0.5 is greater than 0.45")

≤less than or equal tox ≤ 3 (x could be 3 or anything smaller)

≥greater than or equal tox ≥ 0 (x is 0 or positive)

≠not equal tox ≠ 0 (x can be any number except 0)

≈approximately equal toπ ≈ 3.14

Memory hook

The "open mouth" of the < or > always faces the larger number. Think of a crocodile that wants to eat the bigger meal: 3 < 5 — jaws open towards 5.

02MCE Worked Example 2 MARKS

Put the following in order, starting with the smallest:

−0.3, 0.03, −0.33, 0.303, 0.033

•

MMethod. Split into negatives and positives. Negatives come first (most negative = smallest). Within each group, line up decimal places to compare.

•

CCalculate.

Negatives: −0.33 (further left) < −0.30

Positives, padded to 3 d.p.: 0.030, 0.033, 0.303

So 0.030 < 0.033 < 0.303

•

EEvaluate. Combine: negatives first (smallest to largest), then positives (smallest to largest). Answer:−0.33, −0.3, 0.03, 0.033, 0.303

Mark scheme note: On 1-mark ordering questions, AQA awards the mark only if the whole ordering is correct in the direction stated.

Examiner Alert: Negatives

What examiners report every year

On ordering questions involving negative decimals, students treat the digits as if they were positive. They write −0.3 < −0.33 because "3 is smaller than 33." Mark lost every time.

What to write instead

Sketch a quick number line in the margin. Place each number. Read left-to-right — left is smaller, right is bigger. No exceptions.

The one-line rule

For negatives: the closer to zero = the larger the number.

Common Errors

Students often writeCorrect versionWhy it matters

−5 > −2−2 > −5 (−2 is to the right of −5)Loses the comparison mark

0.45 > 0.5 (comparing "45" to "5")0.45 < 0.5 (compare tenths: 4 < 5)Decimal ordering error

Calling √9 irrational√9 = 3 → integer and rationalClassification mark lost

Using ≤ where < is needed (e.g. in error intervals)Check if the endpoint is included or excludedEndpoint marks are lost on bounds questions

Practice Question 1

Circle the smallest number.1 MARK

4.31 4.3̇ 4.301 4.33

(Style based on AQA 8300/3H June 2022 Q1)

1

Convert each to the same decimal places for fair comparison:

4.31 = 4.310

4.3̇ = 4.333... (the 3 recurs)

4.301 = 4.301

4.33 = 4.330

2

All share 4.3 at the tenths level. Compare the hundredths: 1, 3, 0, 3.

3

4.301 has a 0 in the hundredths — the smallest.

Answer: 4.301

Practice Question 2

Write these numbers in order of size, starting with the smallest.1 MARK

0.45 −0.3 0.3 −0.45

1

Split into negatives and positives.

Negatives: −0.3, −0.45

Positives: 0.45, 0.3

2

Order the negatives. −0.45 is further left on the number line → smaller.

So: −0.45, then −0.3.

3

Order the positives. 0.3 < 0.45 (tenths: 3 vs 4).

Answer: −0.45, −0.3, 0.3, 0.45

Practice Question 3

Insert the correct symbol (<, >, or =) in each box.3 MARKS

(a) −7 ☐ −4

(b) 0.608 ☐ 0.61

(c) √16 ☐ 4

(a)

−7 is further left than −4 on the number line → smaller. Answer: −7 < −4.

(b)

Pad: 0.608 vs 0.610. Compare hundredths: 0 vs 1. Answer: 0.608 < 0.61.

(c)

√16 = 4 exactly. Answer: √16 = 4.

AQA tip: do not write "4 = √16 because a square root" — that's ambiguous. Just the symbol is enough for the mark.

Summary

Key Definitions

Integer: a whole number (positive, negative, or zero)

Rational: can be written as p/q — includes all terminating and recurring decimals

Irrational: cannot be written as p/q (e.g. √2, π, e)

Ordering Rules

Decimals: line up the decimal point; compare column by column (tenths first)

Pad with trailing zeros if needed: 0.35 = 0.350

Negatives: further left on the number line = smaller

Closer to zero = larger (when both are negative)

Inequality Symbols

The open mouth of < / > faces the larger number.

Examiner Alert — the #1 trap

With negatives, the digit and the number aren't the same. −0.33 has a bigger digit but is a smaller number than −0.3. Always sketch a number line when in doubt.

Ready for 1.02

Next up: Four Operations & BIDMAS — the priority order, non-calculator techniques, and why (−3)² and −3² give different answers.