Number Toolkit

Types of Numbers

• **Integers** are whole numbers (positive, negative, zero). **Natural numbers** are positive integers (1,2,3,…).
• **Multiples**: a multiple of an integer is formed by multiplying it by another positive integer (e.g., 12 is a multiple of 3).
• **Factors**: a factor divides a number exactly (e.g., 6 is a factor of 18). Find factors using factor pairs or divisibility tests.
• **Prime numbers** have exactly two distinct factors: 1 and itself. First ten primes: 2,3,5,7,11,13,17,19,23,29. 1 is not prime; 2 is the only even prime.
• **Square numbers**: result of multiplying a number by itself (e.g., 1,4,9,16,25,…). **Cube numbers**: result of multiplying a number by itself twice (e.g., 1,8,27,64,125).
• **Square root** (√) is the inverse of squaring; **cube root** (∛) is the inverse of cubing. Square roots of non-square numbers are **surds** (irrational).
• **Reciprocal** of a number is 1 divided by that number; product of a number and its reciprocal equals 1.

Irrational Numbers

• A **rational number** can be written as a fraction a/b (a,b integers, $b\ne 0)$. Includes integers, terminating and recurring decimals.
• An **irrational number** cannot be written as a simple fraction; its decimal is non-terminating and non-recurring.
• Common irrationals: $\pi , \sqrt{2}, \sqrt{3}, \sqrt{5}$, etc. Multiplying an irrational by a non-zero rational gives an irrational (e.g., 2π).
• $\sqrt{n}$ is irrational if n is not a perfect square. $\sqrt{4}=2$ is rational; $\sqrt{2}$ is irrational.

Negative Numbers

• **Multiplying/dividing**: same signs → positive; different signs → negative. E.g., (-12)÷(-4)=3; $(-6)\times 4=-24$.
• **Adding/subtracting**: subtracting a negative = adding the positive (e.g., 5-(-3)=8); adding a negative = subtracting the positive $(e.g., 7+(-3)=4)$.
• Real-life contexts: temperature (e.g., 3°C cooling by 5°C gives -2°C) and $money/debt (e.g., -200 + (-400) = -600)$.
• Use brackets on calculators for negative numbers: $(-3)^{2}=9$, $but -3^{2}=-9$.

Mathematical Symbols

• **Basic operations**: $+ (addition), - (subtraction), \times (multiplication), \div (division)$.
• **Equals and inequality**: $= (equal), \ne (not equal)$, ≈ (approximately), ≡ $(identical), > (greater), < (less), \ge (greater$ or $equal), \le (less$ or equal).
• **Other symbols**: ( ) brackets, $^ or$ superscript for powers, $\sqrt{square root}$, ∛ $(cube root), \pm (plus/minus), \pi (\pi )$.

Order of Operations (BIDMAS/BODMAS)

• **BIDMAS/BODMAS**: Brackets, Indices/Orders (powers, roots), Division/Multiplication (left to right), Addition/Subtraction (left to right).
• Fractions have **invisible brackets** around numerator and denominator (e.g., (2+5)/(7-2)). Roots also have invisible brackets $(e.g., \sqrt{9+16})$.
• Worked example: $(5-3)+2\times 7^{2} = 2+2\times 49 = 2+98 = 100$.

Addition & Subtraction

• **Column method**: line up digits by place value, add/subtract right to left, carry/borrow as needed.
• For subtraction, if top digit is smaller, borrow 10 from the next column to the left.
• Alternative subtraction: **counting up** (e.g., 673-289: add 1→290, +10→300, +300→600, +73→673; total 384).
• Key words: sum/total (addition), difference/take away (subtraction). Estimate to check answers.

Multiplication & Division

• **Column method**: multiply each digit of bottom number by top number, use zeros as placeholders, then add results.
• **Lattice method**: draw grid with diagonals, multiply digit pairs, sum diagonals.
• **Grid method**: split numbers by place value, multiply in grid cells, sum all cells.
• **Short division (bus stop)**: for dividing by single digit; work left to right, carry remainders.
• Dividing by powers of 10 shifts digits right $(e.g., 380\div 10=38)$. Factorising can simplify division $(e.g., \frac{1008}{28} = \frac{252}{7} = 36)$.

Operations with Decimals

• **Add/subtract decimals**: line up decimal points, use zeros as placeholders, then use column method.
• **Multiply decimals**: convert to integers by multiplying by powers of 10, multiply, then divide by same powers. Or ignore decimal, multiply, then place decimal using estimation.
• **Divide decimals**: write as fraction, multiply numerator and denominator by powers of 10 to make integers, divide, then adjust by dividing/multiplying back.
• Always **estimate** to check decimal placement $(e.g., 1.57\times 0.78$ ≈ $2\times 1=2$, so answer ~1.2246).

Types of Numbers Venn Diagram

Order of Operations (BIDMAS)

Column Method for Addition

Multiplying Decimals (Method 1)