Welcome to the World of Number!
Hi everyone! This chapter is the foundation of almost everything you will do in Maths. Don't worry if numbers seem confusing sometimes; we're going to break them down into simple, easy-to-manage pieces.
In the "Number" chapter for Specification B, we focus on understanding different types of numbers, how they behave when we multiply them, how to handle massive or tiny numbers (Standard Form), and ensuring our calculations are accurate. Mastering this content makes Algebra and Geometry much easier! Let's get started!
Section 1: Number Types, Factors, and Multiples
1.1 Key Number Classifications
You need to be familiar with the different 'families' numbers belong to:
- Integers: Whole numbers (positive, negative, or zero). Examples: -3, 0, 5, 100.
- Rational Numbers: Numbers that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). This includes all terminating or recurring decimals. Examples: 0.5 (\(\frac{1}{2}\)), 0.333... (\(\frac{1}{3}\)), 4 (\(\frac{4}{1}\)).
- Irrational Numbers: Numbers that cannot be written as a simple fraction. They are non-terminating and non-recurring decimals. The most famous examples are \(\pi\) (Pi) and the square root of non-square numbers, like \(\sqrt{2}\) or \(\sqrt{5}\).
- Real Numbers: All rational and irrational numbers combined. These are the numbers we use every day.
1.2 Prime Numbers, HCF, and LCM
What is a Prime Number?
A Prime Number is a whole number greater than 1 that has exactly two factors: 1 and itself.
- The first few primes are: 2, 3, 5, 7, 11, 13, 17...
- Important Fact: 2 is the only even prime number!
Highest Common Factor (HCF) and Lowest Common Multiple (LCM)
The easiest way to find HCF and LCM, especially for large numbers, is by using Prime Factorisation.
Step-by-Step: Finding HCF and LCM (Example: 12 and 18)
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Prime Factor Tree: Break down each number into its prime factors.
\(12 = 2 \times 2 \times 3 = 2^2 \times 3\)
\(18 = 2 \times 3 \times 3 = 2 \times 3^2\) -
Find HCF (The Shared Factors): Multiply the primes they have in common, using the lowest power for each shared factor.
Common factors are 2 and 3. Lowest power of 2 is \(2^1\). Lowest power of 3 is \(3^1\).
HCF \(= 2 \times 3 = 6\). -
Find LCM (All Factors): Multiply all primes used, using the highest power for each factor.
Highest power of 2 is \(2^2\). Highest power of 3 is \(3^2\).
LCM \(= 2^2 \times 3^2 = 4 \times 9 = 36\).
Key Takeaway: Prime factorization is your best friend for HCF and LCM calculations. HCF uses the lowest shared powers; LCM uses the highest powers of all factors.
Section 2: Indices (Powers and Roots)
Indices (or powers) are just a shorthand way of writing repeated multiplication. Understanding the rules is vital.
2.1 The Laws of Indices
Let \(a\) and \(b\) be non-zero numbers, and \(m\) and \(n\) be integers.
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Multiplication Rule: When multiplying terms with the same base, add the powers.
\(a^m \times a^n = a^{m+n}\)
Example: \(5^3 \times 5^2 = 5^{3+2} = 5^5\) -
Division Rule: When dividing terms with the same base, subtract the powers.
\(a^m \div a^n = a^{m-n}\)
Example: \(7^6 \div 7^2 = 7^{6-2} = 7^4\) -
Power of a Power Rule: When raising a power to another power, multiply the indices.
\((a^m)^n = a^{mn}\)
Example: \((x^4)^3 = x^{4 \times 3} = x^{12}\) -
Zero Index Rule: Anything (except 0) raised to the power of zero equals 1.
\(a^0 = 1\)
Example: \(1,000,000^0 = 1\)
2.2 Negative and Fractional Indices
Negative Indices (The Flip Rule)
A negative index means you take the reciprocal (flip the fraction).
\(a^{-n} = \frac{1}{a^n}\)
Example 1: \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)
Example 2: \(\left(\frac{2}{3}\right)^{-1} = \left(\frac{3}{2}\right)^1 = \frac{3}{2}\)
Memory Aid: A negative sign on the power means the number is unhappy where it is, so it flips sides (goes from top to bottom, or bottom to top) to become positive again!
Fractional Indices (The Root Rule)
Fractional indices relate directly to roots. The denominator of the fraction is the root you take, and the numerator is the power you raise it to.
\(a^{\frac{1}{n}} = \sqrt[n]{a}\)
\(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\)
Example 1: \(25^{\frac{1}{2}} = \sqrt{25} = 5\)
Example 2: \(8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = (2)^2 = 4\)
Tip for Calculation: Always calculate the root part first, as this makes the numbers smaller and easier to manage.
Key Takeaway: Practice these laws! Negative powers flip the base; fractional powers mean taking a root.
Section 3: Standard Form (Scientific Notation)
3.1 What is Standard Form?
Standard Form is a way to write very large or very small numbers easily, often used in science (hence, Scientific Notation).
It always follows the format: \(A \times 10^n\), where:
- \(A\) is a number between 1 and 10 (it must satisfy \(1 \leq A < 10\)).
- \(n\) is an integer (the power of 10).
3.2 Converting to and from Standard Form
Large Numbers (Positive Power \(n\))
The power \(n\) tells you how many places the decimal point moved to get A.
Example: Convert 45,000,000 to standard form.
- Find A: The decimal point must be after the first non-zero digit. \(A = 4.5\).
- Count the movement: The decimal moved 7 places from the end (right) to sit between 4 and 5.
- Result: \(4.5 \times 10^7\)
Small Numbers (Negative Power \(-n\))
A negative power means the original number was less than 1.
Example: Convert 0.0000021 to standard form.
- Find A: \(A = 2.1\).
- Count the movement: The decimal moved 6 places from the left to sit between 2 and 1.
- Result: \(2.1 \times 10^{-6}\)
Common Mistake: Students sometimes forget that \(A\) must be less than 10. \(35.2 \times 10^4\) is NOT standard form! It should be \(3.52 \times 10^5\).
Key Takeaway: In standard form, the first number must be single-digit (between 1 and 9.999...). A positive exponent means a big number; a negative exponent means a small number.
Section 4: Accuracy, Rounding, and Estimation
4.1 Decimal Places (DP) vs. Significant Figures (SF)
When we round, we must follow simple rules, but the method depends on whether we are asked for Decimal Places or Significant Figures.
Rounding Rules
- Identify the digit you are rounding to.
- Look at the digit immediately to the right (the 'decider' digit).
- If the decider is 5 or more (5, 6, 7, 8, 9), round up the target digit.
- If the decider is 4 or less (0, 1, 2, 3, 4), keep the target digit as it is.
Decimal Places (DP)
DP only counts digits after the decimal point.
Example: Round 34.1748 to 2 decimal places.
Target digit is 7. Decider is 4. (Round down/stay).
Result: 34.17
Significant Figures (SF)
SF counts digits starting from the very first non-zero digit, reading from left to right.
- Leading zeros (0.00...) are NOT significant.
- Zeros between non-zero digits ARE significant (e.g., 305).
Example: Round 0.04508 to 3 significant figures.
- First significant figure is 4.
- Second is 5.
- Third is 0 (it is between 5 and 8, so it counts).
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Decider is 8 (Round up).
Result: 0.0451
Example: Round 23,871 to 1 significant figure.
Target is 2. Decider is 3 (Round down/stay). We must use zeros as placeholders to keep the value big.
Result: 20,000
4.2 Estimation
Estimation means quickly calculating an answer that is close to the real answer. The standard method is to round every number in the calculation to 1 significant figure (1 SF) before doing the calculation.
Example: Estimate the value of \(\frac{7.9 \times 403}{19.5}\)
- Round 7.9 to 1 SF: 8
- Round 403 to 1 SF: 400
- Round 19.5 to 1 SF: 20
- Estimated calculation: \(\frac{8 \times 400}{20} = \frac{3200}{20} = 160\)
Key Takeaway: SF starts counting at the first non-zero digit. DP starts counting after the decimal point. Estimation uses 1 SF for all numbers involved.
Section 5: Fractions, Decimals, and Percentages (FDP)
5.1 Conversions
You must be able to convert fluently between these three forms.
- Fraction to Decimal: Divide the numerator by the denominator. (\(\frac{3}{4} = 3 \div 4 = 0.75\))
- Decimal to Percentage: Multiply by 100. (0.75 \(\times 100 = 75\%\))
- Percentage to Decimal: Divide by 100. (75\% \(\div 100 = 0.75\))
- Decimal to Fraction: Use the place value. (0.6 = \(\frac{6}{10}\), simplify to \(\frac{3}{5}\))
5.2 Calculations with Fractions
Always simplify your answers where possible.
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Adding/Subtracting: Find a common denominator before adding or subtracting the numerators.
Example: \(\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}\) -
Multiplying: Multiply numerators together and denominators together. Simplify before multiplying if possible!
Example: \(\frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}\) -
Dividing: Keep the first fraction, Change the sign to multiply, and Flip the second fraction (KCF or KFC method).
Example: \(\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}\)
5.3 Percentage Change and Reverse Percentages
Percentage Increase/Decrease
Use multipliers for fast calculations.
- Increase by 15%: Multiplier is \(1 + 0.15 = 1.15\).
- Decrease by 20%: Multiplier is \(1 - 0.20 = 0.80\).
Example: Find the cost of a \$400 item after a 10% VAT increase.
\(400 \times 1.10 = \$440\)
Reverse Percentages (The "Working Backwards" Trick)
This is when you are given the final amount after a percentage change and need to find the original amount.
Rule: Final Amount \(\div\) Multiplier = Original Amount.
Example: A coat costs \$120 after a 20% discount. What was the original price?
- The new price (\$120) represents \(100\% - 20\% = 80\%\) of the original price.
- The multiplier is 0.80.
- Original Price = \(120 \div 0.80 = \$150\).
Key Takeaway: Use common denominators for addition/subtraction. Use KCF for division. Use multipliers for fast percentage changes.
Section 6: Ratio and Proportion
6.1 Ratios
Ratios are used to compare quantities. Ratios should always be simplified to their lowest integer terms.
Dividing a Quantity by a Ratio
Example: Share \$70 in the ratio 3:2.
- Find the total parts: \(3 + 2 = 5\) parts.
- Find the value of one part: \(70 \div 5 = \$14\) per part.
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Distribute:
Person 1: \(3 \times 14 = \$42\)
Person 2: \(2 \times 14 = \$28\)
6.2 Proportion
Direct Proportion
If two quantities are in Direct Proportion, as one increases, the other increases at the same rate. (If you buy twice as many apples, you pay twice the price.)
You can often solve these using the unitary method (finding the value of 1).
Example: If 3 pencils cost \$1.50, how much do 7 pencils cost?
- Find cost of 1 pencil: \(1.50 \div 3 = \$0.50\)
- Find cost of 7 pencils: \(7 \times 0.50 = \$3.50\)
Inverse Proportion
If two quantities are in Inverse Proportion, as one increases, the other decreases. (If you double the number of workers, the time taken to complete the job halves.)
Rule: The total quantity (Product of the two values) remains constant.
Example: It takes 4 workers 6 hours to paint a fence. How long would it take 3 workers?
- Total work (in worker-hours): \(4 \times 6 = 24\) worker-hours.
- Divide the total work by the new number of workers: \(24 \div 3 = 8\) hours.
It takes longer (8 hours) because fewer people are working.
Key Takeaway: Direct means both go up/down together. Inverse means when one goes up, the other goes down.
Section 7: Upper and Lower Bounds (Error Intervals)
When a measurement is rounded, we need to know the possible range of its original, true value. This range is defined by the Lower Bound (LB) and the Upper Bound (UB).
7.1 Calculating Bounds for a Single Value
If a number is rounded to the nearest 'unit' (like nearest 10, nearest 0.1, or nearest integer), the error is half of that unit.
Error = \(\frac{1}{2}\) \(\times\) Precision
- Lower Bound (LB): Rounded Value - Error
- Upper Bound (UB): Rounded Value + Error
Example: A length \(L\) is 15 cm, rounded to the nearest cm.
- Precision = Nearest 1 cm. Error = \(1 \div 2 = 0.5\) cm.
- LB: \(15 - 0.5 = 14.5\) cm.
- UB: \(15 + 0.5 = 15.5\) cm.
The true value \(L\) lies in the interval: \(14.5 \leq L < 15.5\). (Note: The UB is always written as 'less than' because 15.5 itself would round up to 16, not 15).
7.2 Calculations with Bounds
When calculating with two or more rounded values (A and B), you must use the bounds to find the maximum and minimum possible results.
| Operation | To Find Maximum Result | To Find Minimum Result |
|---|---|---|
| Addition (A + B) | Max A + Max B | Min A + Min B |
| Subtraction (A - B) | Max A - Min B | Min A - Max B |
| Multiplication (A \(\times\) B) | Max A \(\times\) Max B | Min A \(\times\) Min B |
| Division (A \(\div\) B) | Max A \(\div\) Min B | Min A \(\div\) Max B |
Analogy: To get the biggest possible division result (Max), you need the biggest possible numerator and the smallest possible denominator.
Example: \(A = 50\) (nearest 10), \(B = 5\) (nearest integer). Find the maximum of \(A \div B\).
- Bounds for A: LB = 45, UB = 55
- Bounds for B: LB = 4.5, UB = 5.5
- Max \(A \div B\) = Max A \(\div\) Min B = \(55 \div 4.5 \approx 12.22\)
Key Takeaway: Error is half the measurement unit. Remember the UB is exclusive (<). For calculations, mixing Max and Min strategically is essential, especially for subtraction and division.