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Number and the Number System
Number and the Number System
In Key Stage 3 you extend your understanding of number far beyond whole numbers. You will work with very large and very small numbers, explore the structure of numbers through factors and primes, and express numbers efficiently using standard form and powers.
Place Value and Ordering
The place value system extends in both directions from the units column:
| Millions | Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Units | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|---|---|
| 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 | 0.1 | 0.01 | 0.001 |
To order a set of numbers — including decimals and negatives — compare digit by digit from the left or place them on a number line.
Example: Order these from smallest to largest: −3.5, 2.07, −0.9, 2.7, 0 Answer: −3.5, −0.9, 0, 2.07, 2.7
Positive and Negative Integers
Rules for the four operations with negatives:
| Operation | Rule | Example |
|---|---|---|
| Adding a negative | Same as subtracting | 5 + (−3) = 2 |
| Subtracting a negative | Same as adding | 5 − (−3) = 8 |
| Multiplying/dividing — same signs | Positive result | (−4) × (−3) = 12 |
| Multiplying/dividing — different signs | Negative result | (−4) × 3 = −12 |
Factors, Multiples and Primes
- Factor: a whole number that divides exactly into another. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- Multiple: result of multiplying a number by a positive integer. Multiples of 7: 7, 14, 21, 28, …
- Prime number: has exactly two factors — 1 and itself. First primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, …
- 1 is NOT a prime number (it has only one factor).
Highest Common Factor (HCF): the largest factor shared by two or more numbers. HCF of 36 and 48: factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}; factors of 48 = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48} → HCF = 12
Lowest Common Multiple (LCM): the smallest multiple shared by two or more numbers. LCM of 4 and 6: multiples of 4 = 4, 8, 12, …; multiples of 6 = 6, 12, … → LCM = 12
Prime Factorisation
Every whole number greater than 1 can be written as a product of prime factors (the Fundamental Theorem of Arithmetic). Use a factor tree or repeated division.
Example: Express 360 as a product of prime factors using repeated division:
360 ÷ 2 = 180 → 180 ÷ 2 = 90 → 90 ÷ 2 = 45 → 45 ÷ 3 = 15 → 15 ÷ 3 = 5 → 5 ÷ 5 = 1
So 360 = 2³ × 3² × 5
To find HCF and LCM from prime factorisation:
- HCF: multiply the lowest powers of all shared prime factors.
- LCM: multiply the highest powers of all prime factors.
Powers and Roots
| Notation | Meaning | Example |
|---|---|---|
| a² | a squared (a × a) | 7² = 49 |
| a³ | a cubed (a × a × a) | 4³ = 64 |
| aⁿ | a to the power n | 2⁵ = 32 |
| √a | square root of a | √144 = 12 |
| ∛a | cube root of a | ∛27 = 3 |
Key square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 Key cube numbers: 1, 8, 27, 64, 125, 216, 343
Laws of indices (for the same base):
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1
Standard Form
Standard form writes very large or very small numbers as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
| Number | Standard Form |
|---|---|
| 4,700,000 | 4.7 × 10⁶ |
| 0.00035 | 3.5 × 10⁻⁴ |
| 870 | 8.7 × 10² |
| 0.0009 | 9 × 10⁻⁴ |
Converting to standard form:
- Write the digits with a decimal point after the first significant figure.
- Count how many places the point moves — that is the power of 10.
- Moving left → positive power; moving right → negative power.
Rounding and Estimation
Rounding to decimal places (d.p.): look at the next digit; if 5 or more, round up. 3.14159 to 2 d.p. → 3.14
Rounding to significant figures (s.f.): count from the first non-zero digit. 0.004736 to 2 s.f. → 0.0047
Estimation: round all values to 1 s.f. to obtain a rough answer quickly. Example: 4.87 × 19.3 ≈ 5 × 20 = 100
Priority of Operations (BIDMAS)
Brackets → Indices (powers/roots) → Division → Multiplication → Addition → Subtraction
Division and multiplication are equal priority (left to right); same for addition and subtraction.
Example: 3 + 4² ÷ 2 = 3 + 16 ÷ 2 = 3 + 8 = 11
Practice
- Write 2,340,000 in standard form.
- Write 6.05 × 10⁻³ as an ordinary number.
- Find the HCF and LCM of 60 and 84.
- Express 180 as a product of its prime factors.
- Calculate (−3)² + √49 − 2³.
- Round 0.08475 to 3 significant figures.
- Evaluate 5 + 3 × (2² − 1) using BIDMAS.
- Order from smallest to largest: −1.5, 0.3, −0.03, 1, −1.
- Find all the factor pairs of 72.
- A number written in standard form is 2.4 × 10⁴. What is this number?