Review of Number Concepts

1. Types of Numbers

Classifications

  • Natural Numbers ($\mathbb{N}$): Counting numbers starting from 1 (1, 2, 3, …).
  • Integers ($\mathbb{Z}$): Whole numbers, including negatives and zero (…, -2, -1, 0, 1, 2, …).
  • Prime Numbers: Natural numbers greater than 1 with exactly two factors: 1 and themselves (2, 3, 5, 7, 11, …). Note: 1 is not prime; 2 is the only even prime.
  • Square Numbers: Result of multiplying an integer by itself ($1, 4, 9, 16, 25, \dots$).
  • Cube Numbers: Result of multiplying an integer by itself twice ($1, 8, 27, 64, 125, \dots$).
  • Rational Numbers ($\mathbb{Q}$): Numbers that can be written as a fraction $\frac{a}{b}$ where $a, b$ are integers and $b \neq 0$. Includes terminating and recurring decimals.
  • Irrational Numbers: Numbers that cannot be written as simple fractions (e.g., $\pi, \sqrt{2}, \sqrt{3}$). Their decimal expansions are non-terminating and non-recurring.
  • Reciprocals: The reciprocal of a number $x$ is $\frac{1}{x}$. The product of a number and its reciprocal is always 1.

Number System Hierarchy

2. Prime Factorization, HCF, and LCM

  • Prime Factorization: Expressing a composite number as a product of its prime factors using a factor tree or division method.
  • Highest Common Factor (HCF): The largest factor shared by two or more numbers. Found by multiplying the lowest powers of common prime factors.
  • Lowest Common Multiple (LCM): The smallest multiple shared by two or more numbers. Found by multiplying the highest powers of all prime factors present.

3. Powers, Roots, and Indices

  • Multiplication: $a^m \times a^n = a^{m+n}$
  • Division: $a^m \div a^n = a^{m-n}$
  • Power of a Power: $(a^m)^n = a^{mn}$
  • Zero Index: $a^0 = 1$ (for $a \neq 0$)
  • Negative Index: $a^{-n} = \frac{1}{a^n}$
  • Fractional Indices: $a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = (\sqrt[n]{a})^m$

Laws of Indices Summary

4. Order of Operations (BIDMAS/BODMAS)

Calculations must follow this priority:

  1. Brackets
  2. Indices (Powers/Roots)
  3. Division and Multiplication (Left to right)
  4. Addition and Subtraction (Left to right)

5. Rounding and Estimation

Accuracy

  • Decimal Places (dp): Rounding to a specific number of digits after the decimal point.
  • Significant Figures (sf): Rounding based on the first non-zero digit.
  • Estimation: Rounding numbers to 1 significant figure before calculating to find an approximate answer.

Bounds

  • Lower Bound: The smallest possible value a number could have been before rounding.
  • Upper Bound: The smallest value that would round up to the next unit (effectively the ceiling).
  • When calculating with bounds:
    • Max value of $A + B = \text{Upper } A + \text{Upper } B$
    • Min value of $A + B = \text{Lower } A + \text{Lower } B$
    • Max value of $A - B = \text{Upper } A - \text{Lower } B$
    • Min value of $A - B = \text{Lower } A - \text{Upper } B$

6. Directed Numbers

  • Addition/Subtraction: Use a number line. Adding a negative is the same as subtracting; subtracting a negative is the same as adding.
  • Multiplication/Division:
    • Same signs $\rightarrow$ Positive result $(+ \times + = +$ or $- \times - = +)$
    • Different signs $\rightarrow$ Negative result $(+ \times - = -$ or $- \times + = -)$