Number & Calculation

Introduction to Number & Operations

Welcome to the foundation of Cambridge O Level Mathematics. Numbers are the language of science, commerce, and daily life, from calculating the price of groceries at a market in Lahore to understanding cricket statistics. A strong command over different types of numbers and the rules that govern their operations is essential for success in all other areas of mathematics.

1. The Family of Numbers

To work with numbers, we must first understand their classifications. Think of them as a large, extended family with different branches.

• Natural Numbers (N): These are the positive counting numbers you first learned: {1, 2, 3, 4, ...}.
• Whole Numbers (W): This set includes all natural numbers plus zero: {0, 1, 2, 3, ...}.
• Integers (Z): This set expands to include all whole numbers and their negative counterparts: {..., -3, -2, -1, 0, 1, 2, 3, ...}.
• Rational Numbers (Q): Any number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. This includes all integers, terminating decimals (like 0.75 = 3/4), and recurring decimals (like 0.333... = 1/3).
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal representation goes on forever without repeating. Famous examples include **π (pi)** and the square roots of non-square numbers like **√2** and **√3**.
• Real Numbers (R): This is the set of all rational and irrational numbers combined.

Within these sets, we have special categories:

• Prime Numbers: A natural number greater than 1 that has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17...
• Composite Numbers: A natural number greater than 1 that is not prime (i.e., it has more than two factors). Examples: 4, 6, 8, 9, 10...

2. Deconstructing Numbers: HCF and LCM

Understanding the building blocks of numbers allows us to see how they relate to each other. The core skill here is prime factorisation.

Prime Factorisation: This is the process of expressing a composite number as a product of its prime factors. The 'ladder method' is a common technique.

Example: Find the prime factors of 60.

1. Start with the smallest prime number, 2. `60 ÷ 2 = 30`
2. Divide by 2 again. `30 ÷ 2 = 15`
3. 2 doesn't work, so try the next prime, 3. `15 ÷ 3 = 5`
4. 5 is a prime number, so we stop. `5 ÷ 5 = 1`

So, the prime factorisation of 60 is 2 × 2 × 3 × 5 or 2² × 3 × 5.

Highest Common Factor (HCF): The largest number that is a factor of two or more given numbers.

Lowest Common Multiple (LCM): The smallest number that is a multiple of two or more given numbers.

Method using Prime Factors (for HCF and LCM):

1. Find the prime factorisation of each number.
2. For HCF: Multiply the lowest power of each common prime factor.
3. For LCM: Multiply the highest power of each prime factor that appears in any of the numbers.

Example: Find the HCF and LCM of 36 and 90.

• Prime factorisation of 36 = 2² × 3²
• Prime factorisation of 90 = 2 × 3² × 5
• HCF: The common primes are 2 and 3. The lowest power of 2 is 2¹ and the lowest power of 3 is 3². So, HCF = 2 × 3² = 2 × 9 = **18**.
• LCM: The primes involved are 2, 3, and 5. The highest power of 2 is 2², the highest power of 3 is 3², and the highest power of 5 is 5¹. So, LCM = 2² × 3² × 5 = 4 × 9 × 5 = **180**.

3. Order of Operations: BODMAS/BIDMAS

To ensure everyone gets the same answer for a calculation, we follow a strict order.

• Brackets
• Orders (or Indices) - powers and square roots
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Example: Calculate 5 + (6 - 2)² ÷ 8 × 3

1. Brackets: (6 - 2) = 4 → `5 + 4² ÷ 8 × 3`
2. Orders: 4² = 16 → `5 + 16 ÷ 8 × 3`
3. Division/Multiplication (left to right):

• `16 ÷ 8 = 2` → `5 + 2 × 3`
• `2 × 3 = 6` → `5 + 6`

1. Addition/Subtraction: `5 + 6 = 11`

Common Exam Trap: Students often forget that Division/Multiplication and Addition/Subtraction have equal priority and must be done from left to right. For example, in `10 - 4 + 3`, you must do `10 - 4` first, not `4 + 3`.

4. Accuracy and Approximation

In the real world, measurements are never perfectly exact. We often need to round numbers. The Cambridge exam is very strict about accuracy.

Rounding to Decimal Places (d.p.):

• Look at the digit to the right of the required decimal place.
• If it's 5 or more, round up the last digit.
• If it's 4 or less, keep the last digit as it is.

Example: 4.578 to 2 d.p. is 4.58. 4.572 to 2 d.p. is 4.57.

Rounding to Significant Figures (s.f.):

• The first non-zero digit (from the left) is the first significant figure.
• Apply the same rounding rule as for decimal places.

Example: 0.05281 to 3 s.f. is 0.0528. 2,389,000 PKR to 2 s.f. is 2,400,000 PKR.

Practical Application: When calculating the total cost of items in a store, prices in Pakistani Rupees (PKR) are often given to two decimal places (paisa), but the final bill is rounded to the nearest Rupee.

Exam Tip: In multi-step calculations, **never use a rounded value for the next step**. Keep the full calculator display value and only round your final answer to the required degree of accuracy (usually 3 s.f. if not specified).