Place Value & Number

Understanding place value is the foundation of all mathematics. In the GL 11+ exam, you will be expected to read, write, compare, and manipulate numbers with confidence. This lesson will help you master the number system so that you can tackle place-value questions quickly and accurately.

What Is Place Value?

Every digit in a number has a value that depends on its position (or place) in the number.

Millions	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Ones	.	Tenths	Hundredths
1,000,000	100,000	10,000	1,000	100	10	1	.	0.1	0.01

For example, in the number 4,732,518:

Digit	Place	Value
4	Millions	4,000,000
7	Hundred Thousands	700,000
3	Ten Thousands	30,000
2	Thousands	2,000
5	Hundreds	500
1	Tens	10
8	Ones	8

Top Tip: When the question asks "What is the value of the digit 3 in 4,732,518?", the answer is 30,000 — not just 3!

Reading and Writing Large Numbers

How to read a large number

Split the number into groups of three from the right:

4,732,518 is read as "four million, seven hundred and thirty-two thousand, five hundred and eighteen."

Writing numbers in words and figures

You must be able to go both ways:

Words to figures: "Three hundred and forty-five thousand, six hundred and twelve" = 345,612
Figures to words: 1,209,050 = "One million, two hundred and nine thousand and fifty"

Watch out: Do not forget zeros. The number 1,209,050 has a zero in the hundreds place — make sure you include it!

Ordering and Comparing Numbers

Using inequality symbols

Symbol	Meaning	Example
>	Greater than	5,432 > 3,987
<	Less than	2,100 < 2,110
=	Equal to	500 = 500

Worked Example

Put these numbers in order from smallest to largest: 34,521 — 34,251 — 35,142 — 34,512

Step 1: Compare the ten-thousands digit. All start with 3, so look at the thousands digit. Three numbers have 4 in the thousands place, one has 5.

Step 2: The number with 5 in the thousands place (35,142) is the largest.

Step 3: For the three numbers starting 34,xxx compare the hundreds digit:

34,251 (hundreds = 2)
34,512 (hundreds = 5)
34,521 (hundreds = 5)

Step 4: 34,251 is smaller than the other two. For 34,512 and 34,521, compare the tens digit: 1 < 2.

Answer: 34,251, 34,512, 34,521, 35,142

Rounding Numbers

Rounding means replacing a number with a simpler number that is close in value.

Rules for rounding

Find the digit in the place you are rounding to.
Look at the digit one place to the right.
If that digit is 5 or more, round up.
If that digit is less than 5, round down (keep the same).
Replace all digits to the right with zeros.

Worked Examples

Number	Round to nearest 10	Round to nearest 100	Round to nearest 1,000
3,847	3,850	3,800	4,000
12,345	12,350	12,300	12,000
6,951	6,950	7,000	7,000
99,999	100,000	100,000	100,000

Exam Tip: Questions may ask you to round to the nearest 10, 100, 1,000 or even 10,000. Always read the question carefully!

Negative Numbers

Negative numbers are numbers less than zero. They appear on a number line to the left of zero.

  -5  -4  -3  -2  -1   0   1   2   3   4   5
   |---|---|---|---|---|---|---|---|---|---|

Key facts about negative numbers

-3 is less than -1 (it is further from zero on the left).
The further left a number is on the number line, the smaller it is.
Counting backwards past zero: 3, 2, 1, 0, -1, -2, -3 ...

Worked Example

The temperature at midnight was -4°C. By noon it had risen by 9°C. What was the temperature at noon?

Start at -4. Count up 9: -4 → -3 → -2 → -1 → 0 → 1 → 2 → 3 → 4 → 5.

Answer: 5°C

Factors, Multiples, and Primes

Factors

Factors of a number are whole numbers that divide into it exactly (with no remainder).

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

How to find factors: Work in pairs. 1 × 24, 2 × 12, 3 × 8, 4 × 6. Stop when the pairs meet.

Multiples

Multiples of a number are found by multiplying it by 1, 2, 3, 4, 5 ...

Multiples of 7: 7, 14, 21, 28, 35, 42, 49 ...

Prime Numbers

A prime number has exactly two factors: 1 and itself.

Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Remember: 1 is not a prime number (it only has one factor). 2 is the only even prime number.

Highest Common Factor (HCF)

The HCF is the largest number that is a factor of two or more numbers.

Find the HCF of 12 and 18:

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
HCF = 6

Lowest Common Multiple (LCM)

The LCM is the smallest number that is a multiple of two or more numbers.

Find the LCM of 4 and 6:

Multiples of 4: 4, 8, 12, 16, 20, 24 ...
Multiples of 6: 6, 12, 18, 24 ...
LCM = 12

Square Numbers and Cube Numbers

Square numbers

A square number is a number multiplied by itself.

Number	Square	Written as
1	1	1²
2	4	2²
3	9	3²
4	16	4²
5	25	5²
6	36	6²
7	49	7²
8	64	8²
9	81	9²
10	100	10²
11	121	11²
12	144	12²

Cube numbers

A cube number is a number multiplied by itself three times.

Number	Cube	Written as
1	1	1³
2	8	2³
3	27	3³
4	64	4³
5	125	5³
10	1,000	10³

Roman Numerals

You may be tested on Roman numerals up to 1,000.

Symbol	Value
I	1
V	5
X	10
L	50
C	100
D	500
M	1,000

Rules

Symbols are added when placed after a larger symbol: VI = 5 + 1 = 6
Symbols are subtracted when placed before a larger symbol: IV = 5 - 1 = 4
Common subtraction pairs: IV (4), IX (9), XL (40), XC (90), CD (400), CM (900)

Worked Example

Convert MCMXLIV to a number: M = 1,000 | CM = 900 | XL = 40 | IV = 4 1,000 + 900 + 40 + 4 = 1,944

Practice Checklist

I can identify the value of any digit in a number up to 10 million
I can read and write numbers in words and figures
I can order numbers, including negative numbers
I can round numbers to the nearest 10, 100, 1,000 or 10,000
I can find factors, multiples, HCF, and LCM
I know all prime numbers up to 50
I know square numbers up to 12² and cube numbers up to 5³
I can read and write Roman numerals

Summary

Place value is the building block of everything in mathematics. Being able to manipulate numbers quickly — reading, writing, ordering, rounding, and finding factors and multiples — will save you valuable time in the exam and help you tackle harder multi-step problems with confidence.