Year 9 Mathematics: Number Skills and Index Laws Study Guide

Unit Overview: Number Skills and Index Laws

• Subject: Year 9 Mathematics, Semester One Term One.

• Objective: Students will review and expand their study of number skills, specifically learning to solve problems involving:

  • All real numbers (rational and irrational).

  • Exponents (Index Laws).

  • Very large or small measurements expressed in scientific notation.

• Resources: LearnON Exercise 1.1 Pre-Test.

Integer Operations (Component 1)

• Integer Definition: An integer is any number from the set of numbers that includes positive whole numbers, negative whole numbers, and zero.

• Rules for Multiplying and Dividing Integers:

  • Multiplying or dividing a positive and a negative number results in a negative number.

  • Multiplying or dividing two positive or two negative numbers results in a positive number.

• BIDMAS (Order of Operations):

  • B: Brackets.

  • I: Indices (including radicals/roots).

  • D: Division.

  • M: Multiplication.

  • A: Addition.

  • S: Subtraction.

BIDMAS Flowchart and Evaluation Process

• Step-by-Step Logic:

  1. Check for Brackets: If present, inspect context. If there are nested brackets, evaluate the innermost first. Return to the start of BIDMAS for the internal terms.

  2. Evaluate Indices and Radicals: Process these from left to right as they appear.

  3. Evaluate Multiplication and Division: Process from left to right as they appear.

  4. Evaluate Addition and Subtraction: Process from left to right as they appear.

• Example 1: Evaluate $72 \div \sqrt{16} + (9-8)(3^3) - 5^2$

  • Step 1 (Brackets): $(9-8) = 1$ and $(3^3) = 27$. Expression becomes: $72 \div \sqrt{16} + (1)(27) - 5^2$.

  • Step 2 (Indices/Radicals): $\sqrt{16} = 4$ and $5^2 = 25$. Expression becomes: $72 \div 4 + (1)(27) - 25$.

  • Step 3 (Division/Multiplication): $72 \div 4 = 18$ and $(1)(27) = 27$. Expression becomes: $18 + 27 - 25$.

  • Step 4 (Addition/Subtraction): $18 + 27 - 25 = 20$.

Multiples, Factors, and Prime Numbers

• Multiples: A multiple of a given number is the product of that number and an integer. They are the results of skip counting or multiplication tables.

  • Example: First six multiples of $3$ are $3, 6, 9, 12, 15, 18$.

• Lowest Common Multiple (LCM): The smallest multiple shared between two or more numbers.

  • Method 1 (Listing): List multiples until the first common value is found.

  • Method 2 (Prime Factorisation):

    1. Express all numbers in prime factor form.

    2. Multiply all distinct primes. For common primes, use the highest power found.

    3. Example: Find LCM of $24$ and $60$.

       • $24 = 2^3 \times 3$

       • $60 = 2^2 \times 3 \times 5$

       • $LCM = 2^3 \times 3 \times 5 = 120$.

• Factors: A natural number that divides exactly into another natural number.

  • Example: Factors of $15$ are $1, 3, 5, 15$.

• Highest Common Factor (HCF): The largest factor shared between at least two numbers.

  • Method (Prime Factorisation): The HCF is the product of all common prime factors, using their lowest index.

  • Example: HCF of $532$ and $114$.

• Prime Numbers: A prime number is any number for which its only factors are $1$ and itself. A prime number must have exactly two factors.

Real Numbers and Classification

• Real Numbers ($\mathbb{R}$): The group of numbers normally encountered in measurement/quantifying, comprised of rational and irrational numbers.

• Rational Numbers ($\mathbb{Q}$): Any number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers.

  • Includes Integers ($\mathbb{Z}$): Positive whole numbers, negative whole numbers, and zero.

  • Includes Natural Numbers ($\mathbb{N}$): Counting numbers ($1, 2, 3...$), excluding zero.

  • Includes Terminating Decimals: Decimals that end (e.g., $1.2$, $8$, $-6.123$).

  • Includes Recurring Decimals: Decimals that go on forever but have repeating sequences (e.g., $0.333...$ written as $0.\dot{3}$, $0.55...$, $-1.44...$).

• Irrational Numbers ($\mathbb{I}$): Numbers that are non-terminating AND non-recurring.

  • Examples: $\sqrt{2} \approx 1.414213...$, $\pi \approx 3.14159...$, and $\text{phi } (\phi) \approx 1.618033...$ (The Golden Ratio).

Rounding and Significant Figures

• Purpose of Rounding: To make calculations easier, improve understandability, and avoid over-precision in measurements.

• Significant Figure Rules:

  • The first non-zero digit is always significant.

  • Zeros between non-zero digits are significant.

  • Zeros trailing to the right of a decimal point are significant.

• Rounding Algorithm to Significant Figures:

  1. Identify the first non-zero digit (this is the 1st Sig Fig).

  2. Count digits until the required number of rounded digits is reached.

  3. Look at the digit to the right (‘critical digit’).

  4. If the digit is $≥ 5$, round the last digit up. If < 5, keep it the same.

  5. For non-decimals: convert trailing digits to $0$. For decimals: truncate/remove following numbers.

• Numerical Examples:

  • Australia Population 2024: $27,204,809$.

  • Australia Population 2014: $23,625,600$.

Index Laws Summary

• Key Components: Base numeral and index/exponent (e.g., $6^3$ is read as "6 to the power of 3").

• First Law (Multiplication): When multiplying same bases, add the indices: $a^m \times a^n = a^{m+n}$.

• Second Law (Division): When dividing same bases, subtract the indices: $a^m \div a^n = a^{m-n}$.

• Multiplicative Identity and the Zero Index:

  • The multiplicative identity is $1$. Multiplied by any number, the result is that number ($1 \times 72 = 72$).

  • Third Law: Any base (except zero) raised to power $0$ is $1$: $a^0 = 1$.

  • The case of $0^0$: This is indeterminate. It can be interpreted as $1$ by the zero-index rule or undefined because $0 \div 0$ is undefined.

• Fourth Law (Power to a Power): $(a^m)^n = a^{m \times n}$.

• Fifth Law (Brackets with Products): $(a \times b)^n = a^n \times b^n$. All terms inside are raised to the power.

• Sixth Law (Brackets with Fractions): $(\frac{a}{b})^n = \frac{a^n}{b^n}$. Both numerator and denominator are raised to the power.

• Seventh Law (Negative Indices): Negative indices represent the inverse of repeated multiplication, which is repeated division. They are reciprocals of positive counterparts: $a^{-n} = \frac{1}{a^n}$, where $a \neq 0$.

  • Example: $2^{-3} = 1 \div 2 \div 2 \div 2 = \frac{1}{2^3} = \frac{1}{8}$.

Scientific Notation

• Form: $a \times 10^b$.

• Coefficient ($a$): Must satisfy 1 \le |a| < 10. (Positive version: 1 \le a < 10; Negative version: -10 < a \le -1).

• Index ($b$):

  • If the absolute value of the original number is $≥ 1$, $b$ is positive.

  • If the absolute value of the original number is < 1, $b$ is negative.

• Real World Examples:

  • US Government Debt (Feb 7, 2025): $\$3.6 \times 10^{13}$ (or $\$36,000,000,000,000$).

  • Diameter of a red blood cell: $7.0 \times 10^{-6}\,\text{m}$ (or $0.000007\,\text{m}$).

• Writing Technique:

  1. Move decimal point between the first and second significant figures.

  2. Count moving places (this is the magnitude of the exponent).

  3. Move Left = Positive exponent (large numbers); Move Right = Negative exponent (small numbers).

Square Roots and Cube Roots

• Radical Notation: Contains a radical symbol, an index (e.g., $3$ for cube root), and a radicand (the number inside).

• Root Definition: The number that multiplies by itself as many times as the index to equal the radicand.

  • Example: $\sqrt[3]{27} = 3$ because $3 \times 3 \times 3 = 27$.

• Positive and Negative Roots: Every positive number has two square roots (positive and negative). For example, $\sqrt{9} = \pm 3$ because $3^2 = 9$ and $(-3)^2 = 9$.

• Fractional Indices: Roots are the inverse of powers. A root can be written as the reciprocal of its index index:

  • Square root: $\sqrt{4} = 4^{1/2}$.

  • Cube root: $\sqrt[3]{27} = 27^{1/3}$.

  • General form: $\sqrt[n]{x} = x^{1/n}$.

Questions & Discussion

• Question: Why is $1$ not a prime number?

• Response: A prime number must have exactly two factors: itself and $1$. The number $1$ only has one factor ($1$).

• Question: Is $0$ a prime number?

• Response: No, $0$ is not prime. It does not fit the definition of having exactly two factors.

• Question: Do negative prime numbers exist?

• Response: (Context denotes discussion point: traditionally, prime numbers are defined within the set of natural numbers/positive integers).

• Question: List the first 5 prime numbers.

• Response: $2, 3, 5, 7, 11$.

• Question: What are the results for the first six multiples of $3$?

• Response: $1 \times 3 = 3$, $2 \times 3 = 6$, $3 \times 3 = 9$, $4 \times 3 = 12$, $5 \times 3 = 15$, $6 \times 3 = 18$.

• Question: How many people were added to the Australian population between 2014 ($23,625,600$) and 2024 ($27,204,809$)?

• Response: Approximately $3,579,209$ people.