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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.
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!
Split the number into groups of three from the right:
You must be able to go both ways:
Watch out: Do not forget zeros. The number 1,209,050 has a zero in the hundreds place — make sure you include it!
| Symbol | Meaning | Example |
|---|---|---|
| > | Greater than | 5,432 > 3,987 |
| < | Less than | 2,100 < 2,110 |
| = | Equal to | 500 = 500 |
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:
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 means replacing a number with a simpler number that is close in value.
| 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 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
|---|---|---|---|---|---|---|---|---|---|
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 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 of a number are found by multiplying it by 1, 2, 3, 4, 5 ...
Multiples of 7: 7, 14, 21, 28, 35, 42, 49 ...
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.
The HCF is the largest number that is a factor of two or more numbers.
Find the HCF of 12 and 18:
The LCM is the smallest number that is a multiple of two or more numbers.
Find the LCM of 4 and 6:
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 |
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 |
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 |
Convert MCMXLIV to a number: M = 1,000 | CM = 900 | XL = 40 | IV = 4 1,000 + 900 + 40 + 4 = 1,944
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.