👋 Welcome to Structure and Calculation!
Hello! Welcome to the foundation of your GCSE Maths journey. This chapter, "Structure and Calculation," is all about understanding the rules of numbers—how they are built, what different types exist, and the specific order we must follow when we calculate things.
Think of Maths as a language. If you don't know the alphabet (number types) or the grammar rules (order of operations), the sentences (calculations) won't make sense! Mastering this section will make every other Maths topic easier. Don't worry if some concepts like significant figures seem tricky; we’ll break them down step-by-step!
1. The Structure of Numbers: Place Value
What is Place Value?
Place Value simply tells us how much each digit in a number is worth based on its position. This structure applies to both Integers (whole numbers) and Decimals.
Understanding Integer Place Value (Whole Numbers)
Moving left from the decimal point, the value increases by powers of 10.
Example: The number 5,429
- 5 is in the Thousands column (5,000)
- 4 is in the Hundreds column (400)
- 2 is in the Tens column (20)
- 9 is in the Units column (9)
Understanding Decimal Place Value
Moving right from the decimal point, the value decreases. These places are fractions based on powers of 10.
Example: The number 0.367
- 3 is in the Tenths column (\(\frac{3}{10}\) or 0.3)
- 6 is in the Hundredths column (\(\frac{6}{100}\) or 0.06)
- 7 is in the Thousandths column (\(\frac{7}{1000}\) or 0.007)
🔑 Key Takeaway: Every digit's position matters! The structure is based entirely on the number 10.
2. Essential Number Types and Properties
Integers: The Basics
Integers are all the whole numbers—positive, negative, and zero. (e.g., -3, -2, -1, 0, 1, 2, 3...)
Factors and Multiples: Don't Get Confused!
These concepts sound similar but are opposites. Use this simple memory trick:
Multiples (M = Massive/Many)
A Multiple is the result of multiplying a number by an integer. Multiples are generally larger than the original number (unless multiplied by 0 or 1).
- Example: Multiples of 4 are 4, 8, 12, 16, 20...
Factors (F = Fraction/Few)
A Factor is a whole number that divides exactly into another number with no remainder. Factors are usually smaller than the original number.
- Example: Factors of 12 are 1, 2, 3, 4, 6, and 12.
💡 Quick Tip: To find all factors, try dividing the number by 1, then 2, then 3, and so on, stopping when the numbers repeat. (e.g., for 30: 1x30, 2x15, 3x10, 5x6).
Prime Numbers
A Prime Number is a whole number greater than 1 that has exactly two distinct factors: 1 and itself.
- Example Primes: 2, 3, 5, 7, 11, 13, 17, 19...
🧠 Memory Aid: The number 1 is not a prime number. Also, 2 is the only even prime number. All other even numbers are divisible by 2!
3. The Golden Rule of Calculation: Order of Operations (BODMAS)
When you have a long calculation involving different operations (addition, subtraction, multiplication, etc.), the order in which you perform them is absolutely critical. We use the acronym BODMAS (sometimes called BIDMAS or PEMDAS) to remember the correct sequence.
BODMAS Explained Step-by-Step
B - Brackets (or Parentheses): Calculate anything inside brackets first.
O - Orders (or Indices/Exponents): Calculate powers (\(5^2\)) and square roots (\(\sqrt{9}\)).
D - Division and M - Multiplication: Do these next, working from left to right.
A - Addition and S - Subtraction: Do these last, working from left to right.
⚠️ Common Mistake Alert: D and M have equal priority, and A and S have equal priority. If Multiplication comes before Division in the equation (reading left-to-right), you do Multiplication first!
Example Calculation: Solve \(10 + 2 \times (6 - 3)^2\)
- B (Brackets): \(6 - 3 = 3\). The equation becomes: \(10 + 2 \times 3^2\)
- O (Orders): \(3^2 = 9\). The equation becomes: \(10 + 2 \times 9\)
- D/M (Multiplication): \(2 \times 9 = 18\). The equation becomes: \(10 + 18\)
- A/S (Addition): \(10 + 18 = 28\).
The correct answer is 28.
🔑 Key Takeaway: BODMAS is your map! Stick to the order exactly, or your answer will be wrong.
4. Accuracy and Estimation: Rounding
Sometimes, we don't need the exact answer, or we need to shorten a long decimal. This is where Rounding comes in. The key rule for rounding is:
"5 or more, raise the score. 4 or less, let it rest."
Rounding to Decimal Places (d.p.)
This involves looking at the digits after the decimal point.
Step 1: Identify the position you are rounding to (the "target digit").
Step 2: Look at the digit immediately to the right (the "deciding digit").
Step 3: If the deciding digit is 5 or more, round the target digit up. If it is 4 or less, keep the target digit the same.
Example: Round 4.738 to 2 decimal places.
- Target digit is 3 (in the hundredths position).
- Deciding digit is 8.
- Since 8 is 5 or more, we round 3 up to 4.
- Answer: 4.74
Rounding to Significant Figures (s.f.)
Significant Figures (s.f.) are used to describe the accuracy of a number, regardless of where the decimal point is.
How to find the First Significant Figure:
- Start from the left of the number.
- The first digit that is not zero is the 1st significant figure.
- Zeros used as placeholders at the start of a number are not significant.
Examples of the 1st significant figure:
- In 5,489, the 1st s.f. is 5.
- In 0.0072, the 1st s.f. is 7.
Process for Rounding to s.f.:
Apply the "5 or more, raise the score" rule to the digit immediately following the desired significant figure.
Example: Round 23,809 to 3 significant figures.
- 1st s.f. = 2, 2nd s.f. = 3, 3rd s.f. = 8 (Target digit).
- The deciding digit (to the right of 8) is 0.
- Since 0 is 4 or less, 8 stays the same.
- We must replace the remaining digits (09) with zeros to keep the place value correct.
- Answer: 23,800
Example: Round 0.04519 to 2 significant figures.
- 1st s.f. = 4. 2nd s.f. = 5 (Target digit).
- The deciding digit (to the right of 5) is 1.
- Since 1 is 4 or less, 5 stays the same.
- We do not add zeros at the end of decimals unless they are significant.
- Answer: 0.045
Estimation
Estimation means finding a quick, rough answer to a calculation. The standard way to estimate is to round every number in the calculation to 1 significant figure before performing the operation.
Example: Estimate the answer to \(48.7 \times 11.2\).
- Round 48.7 to 1 s.f.: 50
- Round 11.2 to 1 s.f.: 10
- Perform the estimate: \(50 \times 10 = 500\).
This is useful for checking if your calculator answer (which should be close to 545.44) is reasonable!
5. Working with Very Large and Very Small Numbers: Standard Form
In science and real-world calculation, we often deal with numbers that are too massive (like the distance to a star) or too tiny (like the size of an atom). Standard Form (also known as Scientific Notation) is a concise way to write these numbers.
The Rule of Standard Form
A number written in standard form must follow the structure:
\(A \times 10^n\)
Where:
- A (the coefficient) must be a number between 1 and 10 (it can be 1, but must be less than 10). We write this as \(1 \le A < 10\).
- n (the index or exponent) is an integer (a whole number).
Converting to Standard Form
1. Large Numbers (Positive Index)
Example: Write 65,000,000 in standard form.
- Find A: Move the decimal point so that only one non-zero digit is in front of it: 6.5
- Find n: Count how many places you moved the decimal point (from the end of the original number to its new position). We moved it 7 places to the left.
- Answer: \(6.5 \times 10^7\)
2. Small Numbers (Negative Index)
A negative index indicates a number less than 1 (a small decimal).
Example: Write 0.000412 in standard form.
- Find A: Move the decimal point so it is after the first non-zero digit: 4.12
- Find n: Count how many places you moved the decimal point. We moved it 4 places to the right.
- Answer: \(4.12 \times 10^{-4}\)
🧠 Memory Aid:
If the original number is Big (over 10), the exponent is Positive.
If the original number is Small (under 1), the exponent is Negative.
Quick Review Box: Structure and Calculation
Place Value: Columns determine value (Units, Tens, Tenths, etc.).
BODMAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction. Do D/M and A/S left-to-right!
Rounding: Look at the next digit. 5 or more, round up.
Standard Form: Must be \(A \times 10^n\) where \(1 \le A < 10\).