Integers, Decimals and Place Value

This lesson covers the foundational number skills required for the Edexcel GCSE Mathematics (1MA1) specification. Understanding place value, ordering numbers, rounding to various degrees of accuracy, truncation, bounds and estimation underpins almost every other topic across all three papers. These skills are tested on both Paper 1 (non-calculator) and Papers 2 and 3 (calculator).

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Place Value in Integers

Every digit in a whole number has a place value determined by its position. Our number system is a base-10 (decimal) system where each column is worth ten times the column to its right.

Millions	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Units
1,000,000	100,000	10,000	1,000	100	10	1

For example, in the number 5,274,809:

• The 5 is worth 5,000,000 (five million)
• The 2 is worth 200,000 (two hundred thousand)
• The 7 is worth 70,000 (seventy thousand)
• The 4 is worth 4,000 (four thousand)
• The 8 is worth 800 (eight hundred)
• The 0 is worth 0 (zero tens)
• The 9 is worth 9 (nine units)

Edexcel Exam Tip: When a question asks "what is the value of the digit 8 in 5,274,809?", the answer is 800 — give the full place value, not just the column name.

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Place Value in Decimals

The place value system extends to the right of the decimal point. Each column is one-tenth of the column to its left.

Units	.	Tenths	Hundredths	Thousandths	Ten-thousandths
1	.	0.1	0.01	0.001	0.0001

In the number 3.4087:

• The 3 is worth 3 units
• The 4 is worth 4 tenths (0.4)
• The 0 is worth 0 hundredths (0.00)
• The 8 is worth 8 thousandths (0.008)
• The 7 is worth 7 ten-thousandths (0.0007)

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Ordering Numbers

To order numbers — including decimals and negative numbers — compare digit by digit from the highest place value.

Worked Example 1: Ordering Decimals

Put these numbers in ascending order: 0.72, 0.7, 0.072, 0.702

Step 1: Write each number with the same number of decimal places by adding trailing zeros:

Number	Rewritten
0.72	0.720
0.7	0.700
0.072	0.072
0.702	0.702

Step 2: Compare as whole numbers: 72, 700, 720, 702 → rearranging: 72, 700, 702, 720

Answer: 0.072, 0.7, 0.702, 0.72

Ordering Negative Numbers

Remember that on a number line, numbers increase from left to right. A negative number further from zero is smaller.

Order from smallest to largest: -4, 2, -9, 7, -1

On a number line: -9 is furthest left, then -4, then -1, then 2, then 7.

Answer: -9, -4, -1, 2, 7

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Rounding to Decimal Places

To round to a given number of decimal places (d.p.):

1. Count that many digits after the decimal point.
2. Look at the next digit (the "decider").
3. If the decider is 5 or more, round up. If it is less than 5, round down.

Worked Example 2

Round 3.4762 to 2 decimal places.

• Two decimal places gives 3.47...
• The decider (third decimal place) is 6 (which is ≥ 5), so round up.

Answer: 3.48

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Rounding to Significant Figures

Significant figures (s.f.) count from the first non-zero digit.

Rules for Identifying Significant Figures

Rule	Example
All non-zero digits are significant	345 has 3 s.f.
Zeros between non-zero digits are significant	3045 has 4 s.f.
Leading zeros are NOT significant	0.0052 has 2 s.f.
Trailing zeros after a decimal point ARE significant	2.50 has 3 s.f.
Trailing zeros in a whole number may or may not be significant	3400 could be 2, 3 or 4 s.f.

Worked Example 3

Round 0.004567 to 2 significant figures.

• The first significant figure is 4 (ignore the leading zeros).
• The second significant figure is 5.
• The decider is 6 (≥ 5), so round up.

Answer: 0.0046

Worked Example 4

Round 27,849 to 3 significant figures.

• First three significant figures: 2, 7, 8
• Decider: 4 (< 5), so round down.
• Replace remaining digits with zeros.

Answer: 27,800

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Truncation

Truncation means cutting off digits without rounding — simply removing them.

Worked Example 5

Truncate 4.6789 to 2 decimal places.

• Keep only the first 2 decimal places: 4.67
• Do NOT round — we just remove the remaining digits.

Answer: 4.67

Key Difference: Rounding 4.6789 to 2 d.p. gives 4.68 (rounded up), but truncating gives 4.67.

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Estimation

Estimation means rounding each number to one significant figure (unless told otherwise) to make a calculation simpler.

Worked Example 6

Estimate the value of (4.87 × 21.3) / 0.053

Step 1: Round each number to 1 significant figure:

• 4.87 ≈ 5
• 21.3 ≈ 20
• 0.053 ≈ 0.05

Step 2: Calculate:

• (5 × 20) / 0.05 = 100 / 0.05 = 2000

Answer: Approximately 2000

Edexcel Exam Tip: On Paper 1 (non-calculator), estimation questions always appear. Show each rounded value clearly — you earn method marks for showing your 1 s.f. approximations. The command word "estimate" tells you to round first.

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Bounds from Rounding

When a number has been rounded, the lower bound is the smallest value that would round to the given number, and the upper bound is the smallest value that would round to the next number up.

Worked Example 7

A length is 4.7 cm, correct to 1 decimal place. Find the lower and upper bounds.

• The number has been rounded to 1 d.p., so the precision is 0.1 cm.
• Half the precision = 0.05 cm.
• Lower bound = 4.7 - 0.05 = 4.65 cm
• Upper bound = 4.7 + 0.05 = 4.75 cm

We write: 4.65 ≤ length < 4.75

Note: the lower bound is included (≤) but the upper bound is excluded (<), because a value of exactly 4.75 would round up to 4.8.

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Common Mistakes and Misconceptions

Mistake	Correction
Confusing "value of a digit" with "position name"	Always give the full value (e.g. 5000, not "thousands")
Adding trailing zeros changes a number	0.5 = 0.50 = 0.500 — the value is the same
Rounding 4.95 to 1 d.p. gives 4.9	The decider is 5, so round UP to 5.0
Truncation and rounding are the same	They are not — truncation simply removes digits
Leading zeros are significant	They are NOT — 0.003 has only 1 s.f.
Upper bound is included in an error interval	The upper bound uses < (strictly less than)

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Practice Problems

1. Write down the value of the digit 6 in 2,365,841.
2. Put these in order from smallest to largest: 0.305, 0.35, 0.035, 0.503
3. Round 45,678 to 2 significant figures.
4. Round 0.03456 to 3 significant figures.
5. Truncate 7.896 to 1 decimal place.
6. Estimate the value of (19.8 × 3.14) / 0.48.
7. A mass is 3.45 kg, correct to 2 decimal places. Write down the error interval.
8. Round 0.9951 to 2 decimal places.

Answers

1. 60,000
2. 0.035, 0.305, 0.35, 0.503
3. 46,000
4. 0.0346
5. 7.8
6. 20 × 3 / 0.5 = 60 / 0.5 = 120
7. 3.445 ≤ mass < 3.455
8. 1.00

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Summary

• Place value is the value of a digit based on its position in a number.
• Ordering numbers requires careful comparison from the highest place value.
• Rounding to decimal places or significant figures uses the "decider" digit.
• Truncation removes digits without rounding.
• Estimation rounds to 1 s.f. to simplify calculations.
• Bounds give the range of possible values before rounding; use ≤ for the lower bound and < for the upper bound.