Welcome to the World of Numbers!
Welcome! Whether you are a math whiz or find numbers a bit intimidating, these notes are for you. This chapter, Number, is the foundation of everything else in your GCSE. We will look at how to order numbers, work with fractions and decimals, and understand the "secret" patterns like primes and powers. Think of these as the "tools" in your mathematical toolkit. Once you master these, the rest of the course becomes much easier!
1. The Basics: Symbols and Ordering
Before we calculate, we need to know how to compare numbers. We use specific symbols to show the relationship between two values:
= means "is equal to"
≠ means "is not equal to"
< means "is less than"
> means "is greater than"
≤ means "is less than or equal to"
≥ means "is greater than or equal to"
Memory Aid: The Alligator
Imagine the < or > symbol is an alligator’s mouth. The alligator is very hungry, so it always wants to eat the larger number.
Example: \( 2 < 10 \) (The mouth opens toward the 10).
Ordering Fractions, Decimals, and Negatives
When you are asked to put numbers in order (ascending means smallest to largest), it helps to turn them all into the same format—usually decimals.
Quick Review: Negatives
Think of a number line. The further left a number is, the "smaller" it is.
Example: \( -10 \) is smaller than \( -2 \). Think of it like being £10 in debt versus only £2 in debt!
Key Takeaway: Always convert fractions to decimals to compare them easily. For example, to compare \( \frac{3}{4} \) and \( 0.8 \), remember that \( \frac{3}{4} = 0.75 \), so \( 0.75 < 0.8 \).
2. The "Four Operations" and Place Value
You need to be comfortable with Addition (+), Subtraction (-), Multiplication (×), and Division (÷) using formal written methods (like column addition or long division). This includes working with integers (whole numbers), decimals, and negative numbers.
Common Mistake to Avoid: When multiplying decimals, students often forget where to put the decimal point.
Trick: Count the total number of decimal places in the question. Your answer must have the same number of decimal places.
Example: \( 0.2 \times 0.3 \). There are two numbers after decimal points in total. \( 2 \times 3 = 6 \), so the answer is \( 0.06 \).
Place Value
Every digit in a number has a value based on its position. For example, in 5,230.1, the '5' represents 5 thousands and the '1' represents 1 tenth. Understanding this is vital when working with very large numbers or tiny decimals.
3. Priority of Operations: BIDMAS
In math, there is a specific order we must follow when a calculation has more than one operation. If you do them in the wrong order, you’ll get the wrong answer!
B – Brackets
I – Indices (Powers like \( 2^2 \) or Roots like \( \sqrt{16} \))
D – Division
M – Multiplication
A – Addition
S – Subtraction
Example: What is \( 2 + 3 \times 4 \)?
If you do \( 2 + 3 \) first, you get 20. This is wrong!
According to BIDMAS, Multiplication comes before Addition.
Correct way: \( 3 \times 4 = 12 \), then \( 12 + 2 = 14 \).
Don't worry if this seems tricky at first! Just take it one step at a time, starting with whatever is in the brackets.
Key Takeaway: Always check for Brackets and Indices before doing any other part of a sum.
4. Factors, Multiples, and Primes
These are the "building blocks" of whole numbers.
Factors: Numbers that fit exactly into another number.
Example: Factors of 10 are 1, 2, 5, and 10.
Multiples: The "times table" of a number.
Example: Multiples of 10 are 10, 20, 30, 40...
Prime Numbers: A number with exactly two factors: 1 and itself.
• 2 is the only even prime number!
• 1 is not a prime number (it only has one factor).
• Primes to remember: 2, 3, 5, 7, 11, 13, 17, 19...
HCF and LCM
HCF (Highest Common Factor): The biggest factor that two numbers share.
LCM (Lowest Common Multiple): The smallest number that appears in the times tables of both numbers.
Step-by-Step: Finding the LCM of 4 and 6
1. List multiples of 4: 4, 8, 12, 16, 20...
2. List multiples of 6: 6, 12, 18, 24...
3. The first one they share is 12. So, \( LCM = 12 \).
5. Powers and Roots
Powers (Indices) tell you how many times to multiply a number by itself.
\( 5^2 \) (5 squared) means \( 5 \times 5 = 25 \).
\( 2^3 \) (2 cubed) means \( 2 \times 2 \times 2 = 8 \).
Roots are the opposite of powers.
The square root of 25 (\( \sqrt{25} \)) is 5, because \( 5 \times 5 = 25 \).
Did you know? You are expected to recognize powers of 2, 3, 4, and 5. For example, \( 2^2=4, 2^3=8, 2^4=16, 2^5=32 \).
Key Takeaway: A "reciprocal" is just \( 1 \) divided by the number. The reciprocal of 4 is \( \frac{1}{4} \).
6. Fractions, Decimals, and Percentages (FDP)
In the Foundation tier, you must be able to switch between these three easily. They are just different ways of saying the same thing!
Fractions to Decimals: Divide the top (numerator) by the bottom (denominator).
Example: \( \frac{3}{8} = 3 \div 8 = 0.375 \).
Percentages as Operators: "Of" in math usually means multiply.
Example: Find 15% of 60.
1. Turn 15% into a decimal: \( 0.15 \).
2. Multiply: \( 0.15 \times 60 = 9 \).
Quick Review Box: Common Conversions
\( \frac{1}{2} = 0.5 = 50\% \)
\( \frac{1}{4} = 0.25 = 25\% \)
\( \frac{1}{10} = 0.1 = 10\% \)
\( \frac{1}{5} = 0.2 = 20\% \)
7. Standard Form
Standard form is used to write very large or very small numbers neatly. It always looks like this: \( A \times 10^n \).
• A must be a number between 1 and 10 (e.g., 3.5).
• n is the number of places the decimal point moves.
Example: 50,000 in standard form is \( 5 \times 10^4 \).
Example: 0.005 in standard form is \( 5 \times 10^{-3} \).
8. Approximation and Accuracy
Sometimes we don't need an exact answer; we just need to be "close enough."
Rounding
Decimal Places (dp): Count how many digits you want after the decimal point.
Significant Figures (sf): Start counting from the first non-zero digit.
Rule: If the next digit is 5 or more, "round up." If it's 4 or less, "keep it the same."
Estimation
To estimate the answer to a calculation, round every number to 1 significant figure first.
Example: Estimate \( 21.4 \times 4.8 \).
Round to \( 20 \times 5 = 100 \). This is much easier than doing the full multiplication!
Limits of Accuracy (Error Intervals)
If a weight is rounded to 70kg (to the nearest 10kg), the actual weight could be anything from 65kg up to (but not including) 75kg. We write this as:
\( 65 \le weight < 75 \)
Key Takeaway: For error intervals, go half-way to the unit above and half-way to the unit below.
Final Encouragement
You've made it through the Number chapter! This section covers the "how-to" of math. If you find a specific part—like fractions or standard form—difficult, don't worry. Practice one small skill at a time, and soon they will all fit together like a puzzle. Keep going!