Number Operations
Revise Number Operations for International Mathematics 0607 (O Level) — revision notes and instant AI marking. Free to start.
Number Operations
Quick Summary
1. Types of Number
Before you can operate on numbers, you need to know what "family" each number belongs to. Think of this section as learning the vocabulary of maths — examiners use these words constantly, and if you don't know what they mean, you can't answer the question even if you know the maths.
Integers and Natural Numbers
Integers are whole numbers — no fractions, no decimals. They can be positive, negative, or zero: ..., −3, −2, −1, 0, 1, 2, 3, ...
Natural numbers are the positive integers, thought of as "counting numbers": 0, 1, 2, 3, 4, ... Notice that 0 is included in this syllabus's definition — don't forget it!
Multiples
A multiple of a number is what you get when you multiply it by an integer. 12 is a multiple of 3 because 12 ÷ 3 = 4 exactly (no remainder). A common multiple is shared between two or more numbers — e.g. 12 is a common multiple of both 4 and 6.
- Even numbers (2, 4, 6, 8, 10, ...) are multiples of 2.
- Odd numbers (1, 3, 5, 7, 9, ...) are not multiples of 2.
- Multiples can be algebraic too — the multiples of k are k, 2k, 3k, 4k, 5k, ...
Factors
A factor of a number divides it exactly, leaving no remainder. For example, 6 is a factor of 18 because 18 ÷ 6 = 3 exactly. Every integer greater than 1 has at least two factors: itself, and 1. A common factor is shared between numbers — 3 is a common factor of both 21 and 18.
Finding factors using factor pairs: Start with 1 and the number itself, then test 2, 3, 4, 5... one at a time until the pairs start repeating.
Answer: Factors of 18 = {1, 2, 3, 6, 9, 18}
Divisibility Tests (no calculator needed!)
| Divisible by | Test |
|---|---|
| 2 | Last digit is even |
| 3 | Sum of digits is a multiple of 3 (e.g. 123 → 1+2+3=6 ✓) |
| 4 | Halving the number twice gives a whole number |
| 5 | Last digit is 0 or 5 |
| 8 | Halving three times gives a whole number |
| 10 | Last digit is 0 |
Prime Numbers
A prime number has exactly two distinct factors: itself and 1. The first ten primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Answer: The factors of 51 are 1, 3, 17, 51 — that's more than two, so 51 is not prime.
Square Numbers & Cube Numbers
A square number is a number multiplied by itself: 1×1=1, 2×2=4, 3×3=9... In algebra: a × a = a².
A cube number is a number multiplied by itself twice: 1×1×1=1, 2×2×2=8... In algebra: a × a × a = a³.
Square numbers (1–15): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Cube numbers (1–5, plus 10): 1, 8, 27, 64, 125 ... and 10³ = 1000
Square roots undo squaring, and cube roots undo cubing. The square root symbol is √, and the cube root symbol is ∛.
- Square roots can be positive AND negative: √25 = 5, but −√25 = −5. If both are wanted, we write ±√25.
- A square root of a non-square integer is called a surd — e.g. √3 is a surd because 3 isn't a square number. Surds are irrational.
- √64 = 8 is rational (it's a whole number). √2 is irrational (2 isn't a square number).
Write down a number which is both a cube number and a square number, and express it two different ways using index notation.
Reciprocals
The reciprocal of a number is 1 divided by that number. Any number multiplied by its reciprocal always equals 1.
Reciprocal of a = 1/a (also written as a⁻¹)
To find the reciprocal of a fraction, just flip it: the reciprocal of 2/3 is 3/2.
Write down a fraction that completes this calculation: 3/7 × ⬜ = 1
2. Rational and Irrational Numbers
What is a Rational Number?
A rational number is any number that can be written as a fraction a/b in its simplest form, where a and b are both integers and b ≠ 0. This is a much bigger category than you might think — it includes:
- All terminating decimals: 0.15 = 15/100
- All recurring decimals: 0.1515151515... = 15/99
- All integers: 5 = 5/1, −3 = −3/1, even 0 = 0/1
What is an Irrational Number?
An irrational number CANNOT be written as a fraction a/b of integers. Its decimal form goes on forever without repeating in a pattern.
π, √2, √3, √5
If you multiply an irrational number by a non-zero rational number, the result is still irrational: 2π, 3√2, and (3/4)√5 are all irrational.
Classify each of the following as rational or irrational: (a) √16 (b) √7 (c) 0.333... (d) 2 + π
3. Negative Numbers
Negative numbers trip people up mostly because of sign rules — but there's really only two rules to remember, and they never change.
Multiplying and Dividing
Same signs → POSITIVE result | Different signs → NEGATIVE result
Adding and Subtracting
Subtracting a negative = Adding the positive
Adding a negative = Subtracting the positive
Real-Life Contexts
Temperature: if it's 3°C and cools by 5°C, the new temperature is 3 − 5 = −2°C. If it's −4°C and warms by 6°C, that's (−4) + 6 = 2°C.
Money and debt: a negative sign means you owe money. If someone has a £200 debt and borrows another £400, their total debt is (−200) + (−400) = −£600.
−3² gives −9, but (−3)² correctly gives 9. The second one is right — always bracket it.
Work out: (a) (−8) − (−6) (b) (−9) × (−2) (c) (−10) ÷ 5
4. Mathematical Symbols
You'll be expected to read and use these symbols fluently in exam questions — mixing them up costs easy marks.
| Symbol | Meaning | Example |
|---|---|---|
| = | Equal to | 3x + 7 = 19 |
| ≠ | Not equal to | 2 − 5 ≠ 5 − 2 |
| ≈ | Approximately equal to | π ≈ 3.14 |
| ≡ | Identical / equivalent to | 12x + 6 ≡ 3(4x + 2) |
| > | Greater than | 5 > −2 |
| < | Less than | 1/2 < 2/3 |
| ⩾ | Greater than or equal to | — |
| ⩽ | Less than or equal to | — |
| ( ) | Brackets — group symbols | (2x + 4) − 3(x + 7) |
| ± | Plus minus — two distinct answers | x = 3 ± 2.5 → x = 5.5 or x = 0.5 |
| π | Pi — ratio of circumference to diameter | π ≈ 3.14159... |
5. Order of Operations (BIDMAS / BODMAS)
Here's the problem BIDMAS solves: if I give you 2 + 3 × 4, do you add first (getting 5 × 4 = 20) or multiply first (getting 2 + 12 = 14)? Without an agreed order, everyone would get different answers to the same sum. BIDMAS is that agreement.
B — Brackets → I — Indices (powers/roots) → DM — Division & Multiplication (left to right) → AS — Addition & Subtraction (left to right)
(BODMAS uses "Order" instead of "Indices" and "Of" instead of "Multiplication" — same rule, different letters.)
Hidden Brackets: Fractions and Roots
This is where most marks are lost. A fraction line and a root symbol both act like invisible brackets around everything on the top (or under the root).
Answer: 100
Work out: 3 + (12 − 4) ÷ 2²
6. Addition & Subtraction (Without a Calculator)
Column Method for Addition
Line the digits up by place value (units under units, tens under tens...). Add each column from right to left. If a column totals 10 or more, write the units digit and "carry" the tens digit to the top of the next column.
Answer: 10 352
Column Method for Subtraction
Same idea, but now if the top digit is smaller than the bottom digit in a column, you "borrow" 10 from the column to its left.
The 9 becomes 8 (borrowed 10 for the units column, which turns 2 into 12; 12 − 8 = 4). Then 8 − 2 = 6, and 3 − 0 = 3.
Answer: 364
Alternative: Counting Up
To find 673 − 289, start at 289 and count up to 673 in convenient jumps:
So 673 − 289 = 384. This method is great for mental maths, especially with money and time.
Find the difference between 803 and 456 using the column method (no calculator).
7. Multiplication & Division (Without a Calculator)
You only need to be confident with one method for each — but it helps to see a few, so pick the one that clicks for you.
Grid Method (Multiplication)
Best for keeping place value visible. Split each number by place value, multiply every pair, then add all the results.
| 3000 | 500 | 10 | 6 | |
|---|---|---|---|---|
| 7 | 21000 | 3500 | 70 | 42 |
21000 + 3500 + 70 + 42 = 24 612
Column Method (Multiplication)
Multiply the top number by each digit of the bottom number (right to left), using 0s as placeholders, then add the rows.
Repeated Addition ("Chunking")
Build the answer from doubling facts you already know. To find 13 × 23:
Short Division ("Bus Stop" Method)
The most efficient method when dividing by a single digit. Work left to right, carrying any remainder to the next digit.
3 into 1 → 0 remainder 1 → carry the 1 to make 17 → 3 into 17 → 5 remainder 2 → carry the 2 to make 24 → 3 into 24 → 8 remainder 0
Answer: 174 ÷ 3 = 58
Factorising & Cancelling (Division)
Treat the division like simplifying a fraction — cancel down before doing the hard work.
Dividing by Powers of 10
This is just shifting digits along the place value columns. Dividing by 10 shifts one column, by 100 shifts two columns, by 1000 shifts three columns.
Divide 568 by 8 using short division, and check your answer using estimation.
8. Operations with Decimals
Adding & Subtracting Decimals
Use the same column methods as whole numbers — the trick is making sure every decimal point lines up in the same column. Add "placeholder" zeros to keep the columns tidy.
Estimate check: 33 − 2 = 31, close to 30.76 ✓ Answer: £30.76
Multiplying Decimals
The cleanest approach: turn the decimals into whole numbers, do the easy multiplication, then undo what you did to get back to the real answer.
1. Multiply each decimal by a power of 10 to make it a whole number.
2. Do the multiplication normally.
3. Divide the result by the same powers of 10 you multiplied by.
Answer: 10.025
Alternative method: ignore the decimal points completely, multiply as whole numbers, then use estimation to work out where the decimal point should go in the final answer.
Dividing Decimals
Same idea in reverse. Write the division as a fraction, multiply top AND bottom by a power of 10 to clear the decimals, then divide normally.
Answer: 150
Without a calculator, find 4.68 ÷ 6, using estimation to place the decimal point.
What to Memorise
| Term / Fact | What it means |
|---|---|
| Integer | Whole number: positive, negative, or zero |
| Natural number | Positive integers including 0 (counting numbers) |
| Prime number | Exactly two distinct factors: itself and 1 (1 is NOT prime; 2 is the only even prime) |
| First 10 primes | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 |
| First 15 square numbers | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 |
| First 5 cube numbers | 1, 8, 27, 64, 125 (and 10³ = 1000) |
| Reciprocal | 1 ÷ number, or a → 1/a. Flip fractions to find reciprocals. |
| Rational number | Can be written as a/b (integers, b≠0) — includes all terminating/recurring decimals |
| Irrational number | Cannot be written as a/b — decimal never terminates or repeats. Know: π, √2, √3, √5 |
| Surd | Square root of a non-square integer — always irrational |
| Negative × / ÷ rule | Same signs → positive. Different signs → negative. |
| Negative + / − rule | Subtracting a negative = adding positive. Adding a negative = subtracting positive. |
| BIDMAS order | Brackets → Indices → Division/Multiplication (L→R) → Addition/Subtraction (L→R) |
| Hidden brackets | Fraction lines and root signs act as invisible brackets around everything inside |
| Multiplying/dividing decimals | Convert to whole numbers using powers of 10, calculate, then undo the conversion |
Concepts Checklist
Tick off each concept as you feel confident with it. Come back and revisit anything unticked before your exam!
Exam Tips & Common Mistakes
Don't call 1 a prime number
This is one of the most common marks lost in "list the primes" or "explain why X is/isn't prime" questions. 1 has only ONE factor, not two.
Watch for hidden brackets under roots and fraction lines
Examiners love testing whether you realise √(9+16) means take the root of the WHOLE sum, not √9 + 16. Same logic applies to fractions — work out the top and bottom fully before dividing.
Bracket negative numbers on your calculator
−3² and (−3)² give different answers on most calculators. If a question involves squaring a negative number, always type it with brackets.
Always estimate first
Rounding to 1 significant figure and doing a rough calculation in your head catches silly errors before you write your final answer — examiners reward answers that are "sensible."
Learn the worded vocabulary
"Sum," "total," "plus" = add. "Difference," "take away" = subtract. "Product," "times," "lots of" = multiply. "Quotient," "share," "per" = divide. Worded questions test whether you know these, not just whether you can calculate.
√2 × √2 ≠ irrational
Don't assume "irrational × irrational = always irrational." Check the actual value — sometimes it simplifies neatly to a rational number.
- 6. Addition & Subtraction (Without a Calculator)
- 7. Multiplication & Division (Without a Calculator)
- Exam Tips & Common Mistakes
- Square Numbers & Cube Numbers
- Factorising & Cancelling (Division)
- Adding & Subtracting Decimals
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