Number Skills and Index Laws - Year 9 Mathematics

Integer Operations and BIDMAS

• Integers: A set including positive whole numbers, negative whole numbers, and zero.
• Sign Rules:
  • Multiplying or dividing two numbers with the same sign results in a positive number.
  • Multiplying or dividing two numbers with different signs results in a negative number.
• BIDMAS Order of Operations:
  1. Brackets
  2. Indices (including radicals/roots)
  3. Division and Multiplication (left to right)
  4. Addition and Subtraction (left to right)
• Example Expression: $"72 / \sqrt{16} + (1)(27) - 25 = 18 + 27 - 25 = 20$"

Multiples, Factors, and Prime Numbers

• Multiples: The product of a given number and an integer (skip counting).
• Lowest Common Multiple (LCM): The smallest multiple shared by two or more numbers.
  • Find via Prime Factorisation: Multiply the highest power of every prime factor present in the numbers.
  • Example: For $24 = 2^3 \times 3$ and $60 = 2^2 \times 3 \times 5$, the $LCM = 2^3 \times 3 \times 5 = 120$.
• Factors: Natural numbers that divide exactly into another natural number.
• Highest Common Factor (HCF): The largest shared factor between numbers.
  • Find via Prime Factorisation: Multiply only the common prime factors using their lowest index.
• Prime Numbers: Numbers with exactly two factors (1 and itself).
  • 1 is not a prime number.
  • 0 is not a prime number.

The Real Number System

• Real Numbers ($R$): The set of all rational and irrational numbers.
• Natural Numbers ($N$): Counting numbers (positive integers, excluding 0).
• Integers ($Z$): Positive/negative whole numbers and zero.
• Rational Numbers ($Q$): Numbers expressible as a fraction $\frac{a}{b}$. Includes terminating decimals and recurring decimals (e.g., $0.\dot{3} = \frac{1}{3}$).
• Irrational Numbers ($I$): Non-terminating and non-recurring decimals (e.g., $\pi$, $\sqrt{2}$, and the Golden Ratio $\phi \approx 1.618$).

Rounding and Significant Figures

• Rounding Rule: If the digit to the right of the rounding place is $\geq 5$, round up; otherwise, keep it the same.
• Significant Figures (SF):
  • The first non-zero digit is the first SF.
  • Zeros between non-zero digits are significant.
  • Trailing zeros to the right of a decimal point are significant.
• Large Numbers: When rounding high-value integers, convert trailing digits to 0 to maintain place value.

Index Laws

• First Law: $a^m \times a^n = a^{m+n}$
• Second Law: $a^m \div a^n = a^{m-n}$
• Third Law (Zero Index): $a^0 = 1$ (where $a \neq 0$). Note that $0^0$ is indeterminate.
• Fourth Law (Power to a Power): $(a^m)^n = a^{m \times n}$
• Fifth Law: $(ab)^n = a^n b^n$
• Sixth Law: $(\frac{a}{b})^n = \frac{a^n}{b^n}$
• Seventh Law (Negative Indices): $a^{-n} = \frac{1}{a^n}$. Negative indices represent repeated division or the reciprocal of the positive power.

Scientific Notation

• Standard Form: $a \times 10^b$
  • Coefficient $(a)$: Must be $1 \leq |a| < 10$.
  • Exponent $(b)$: Positive for values with an absolute value $\geq 1$; negative for values $< 1$.
• Real-world Examples:
  • US Government Debt (Feb 2025): $\$36,000,000,000,000 = \$3.6 \times 10^{13}$
  • Human Red Blood Cell diameter: $0.000007\,m = 7.0 \times 10^{-6}\,m$

Radicals and Roots

• Definition: Finding a root is the inverse operation of raising a number to a power.
• Radical Symbols: Comprised of the radical symbol, the index (e.g., 3 for cube root), and the radicand (the number inside).
• Square Roots: Every positive number has a positive and negative square root (e.g., $\sqrt{9} = \pm 3$), though primary focus is usually the positive root.
• Fractional Indices:
  • Square root: $\sqrt{x} = x^{\frac{1}{2}}$
  • Cube root: $\sqrt[3]{x} = x^{\frac{1}{3}}$
  • General form: $\sqrt[n]{x} = x^{\frac{1}{n}}$" , "title": "Number Skills and Index Laws - Year 9 Mathematics" }