Basic Numbers and Calculations

You'll work with different types of numbers every day in maths. Integers are your whole numbers $like -3, 0, 92$, whilst decimals have that crucial decimal point (3.7, 0.94, -24.07). Don't forget that negative numbers are simply anything less than zero - they can be whole numbers or decimals too.

The four basic operations are your mathematical toolkit. Addition finds totals, subtraction finds differences, multiplication is repeated addition, and division splits things into equal groups. Remember that division can leave a remainder - that's just the bit left over when numbers don't divide exactly.

BIDMAS is absolutely crucial for getting calculations right. It stands for Brackets, Indices, Division, Multiplication, Addition, Subtraction - that's the exact order you must follow. Without it, 6 + 3 × 5 would wrongly give you 45 instead of the correct answer of 21.

Quick Tip: When you see strings of multiplication/division or addition/subtraction with no brackets, just work from left to right!

Recurring decimals pop up when division doesn't end neatly. You'll spot them by the dots above repeating digits, like 1/3 = 0.3̇ (that dot shows the 3 repeats forever).

Factors and Multiples

Multiples are simply the times tables of any number - so the multiples of 7 are 7, 14, 21, 28, 35, and so on. Factors work the opposite way - they're numbers that divide exactly into your original number without leaving a remainder.

Finding the Lowest Common Multiple (LCM) means spotting the smallest number that appears in multiple times tables. The Highest Common Factor (HCF) is the biggest number that divides into two or more numbers exactly. These concepts are essential for fraction work later on.

Prime numbers have exactly two factors - themselves and one. The number 1 isn't prime because it only has one factor. The first few primes (2, 3, 5, 7, 11, 13) are worth memorising as they appear constantly in maths.

Remember: Every number can be broken down into prime factors - use a factor tree to find them systematically!

Prime factorisation shows any number as a product of prime numbers. For example, 36 = 2² × 3². This technique becomes incredibly useful for simplifying fractions and finding HCFs and LCMs efficiently.

Accuracy and Approximation

Place value determines what each digit in a number is actually worth. In 726, that 2 represents 20 because it sits in the tens column. Understanding this concept is fundamental for all number work and rounding.

Rounding makes numbers simpler whilst keeping them close to their original value. Look at the digit to the right of where you're rounding - if it's 5 or more, round up; if it's less than 5, round down. You might round to the nearest ten, hundred, or to specific decimal places.

Significant figures show which digits actually matter in a number. The first significant figure can never be zero, and trailing zeros after a decimal point don't count. When rounding 19357 to 3 significant figures, you get 19400 - those zeros keep the digits in the right place value columns.

Money Tip: Always write £27.40, never £27.4 - those trailing zeros matter for currency!

Estimation helps you check if answers make sense. Round each number to 1 significant figure, then calculate. So 348 + 692 becomes roughly 300 + 700 = 1000. This catches silly mistakes before they cost you marks.

Understanding Fractions

Fractions represent division - the numerator (top number) divided by the denominator (bottom number). Unit fractions like 1/2, 1/3, 1/4 have 1 as the numerator and are particularly useful building blocks.

The reciprocal of any number is 1 divided by that number. When you multiply a number by its reciprocal, you always get 1. This becomes crucial when dividing fractions later. Mixed numbers combine whole numbers with fractions, like 2¾.

Simplifying fractions means dividing both top and bottom by their highest common factor. Equivalent fractions represent the same value but look different - like 2/5 = 4/10 = 20/50. You'll use these constantly when adding and subtracting fractions.

To compare fractions, convert them to equivalent fractions with the same denominator. Then you can easily arrange them in ascending (smallest to biggest) or descending (biggest to smallest) order.

Quick Method: To find a fraction of an amount, divide by the bottom, then multiply by the top!

When adding or subtracting fractions, find the LCM of the denominators first. Convert each fraction to this common denominator, then add or subtract the numerators whilst keeping the denominator the same.

Multiplying and Dividing Fractions

Multiplying fractions is surprisingly straightforward - just multiply the numerators together and multiply the denominators together. So 3/8 × 2/9 = 6/72 = 1/12. Always simplify your final answer if possible.

Dividing fractions uses the memorable "Keep it, Flip it, Change it" method (KFC). Keep the first fraction unchanged, flip the second fraction upside down (find its reciprocal), then change the division sign to multiplication. This transforms division into the easier multiplication process.

Remember: Dividing by a fraction is the same as multiplying by its reciprocal - that's why the KFC method works!

Basic Percentages and Conversions

Percentages simply mean "parts per 100" - so 31% means 31/100. To find 10% of anything, divide by 10. To find 1%, divide by 100. These form the basis for calculating any percentage.

Percentage change compares the difference to the original amount using the formula: (Difference ÷ Original) × 100%. This shows up constantly in real-world situations like price changes and statistics.

Converting between fractions, decimals, and percentages is a core skill. To convert fractions to decimals, divide the numerator by the denominator. For decimals to fractions, write over 10, 100, or 1000 and simplify.

Conversion Shortcut: To convert decimals to percentages, multiply by 100. To go the other way, divide by 100.

For percentages to fractions, write the percentage over 100 and simplify. Going from fractions to percentages, make the denominator 100 using equivalent fractions, or use a calculator to multiply the fraction by 100.

Calculating with Percentages

Increasing or decreasing by a percentage can be done two ways. Without a calculator, find the percentage amount and add or subtract it. With a calculator, find the percentage multiplier and multiply directly.

Percentage multipliers make calculations faster. To increase by 20%, multiply by 1.2 $that's 100$. To decrease by 15%, multiply by 0.85 $that's 100$.

Reverse percentages work backwards from a final amount to find the original. If something costs £48.60 after a 10% reduction, you know that £48.60 represents 90% of the original price. Find 1%, then multiply by 100 to get the original amount.

Key Words: Look out for "before", "original", or "was" - these signal reverse percentage problems!

Simple interest calculates interest as a percentage of the original amount each year. £1000 at 10% simple interest for 3 years gives £100 interest per year, totalling £300 interest.

Indices and Powers

Square numbers come from multiplying a number by itself - like 5² = 25. Square roots reverse this process, so √25 = 5. Remember that equations like x² = 25 have two solutions: x = 5 or x = -5 $written as x = ±5$.

Cube numbers involve multiplying a number by itself twice more - so 2³ = 8. Cube roots reverse cubing, and unlike square roots, they only have one solution.

The index laws make working with powers much easier. When multiplying powers with the same base, add the indices $aᵐ × aⁿ = aᵐ⁺ⁿ$. When dividing, subtract the indices $aᵐ ÷ aⁿ = aᵐ⁻ⁿ$. When raising a power to another power, multiply the indices $(aᵛ)ⁿ = aᵛⁿ$.

Don't Forget: Any number to the power 0 equals 1, and any number to the power 1 is just itself!

These laws apply whether you're working with numbers or algebra, making them incredibly versatile tools for simplifying expressions and solving equations.

Standard Form

Standard form expresses very large or very small numbers neatly as A × 10ᵇ, where A is between 1 and 10, and b is an integer. This format makes scientific calculations much more manageable.

For large numbers like 8400, move the decimal point left until you have one digit before it: 8.4 × 10³. For small numbers like 0.00036, move the decimal point right: 3.6 × 10⁻⁴. The negative power shows the number is smaller than 1.

Quick Check: Positive powers mean big numbers, negative powers mean small numbers!