Year 8 Extension — Study Notes (Non‑Calculator Section) Summary & Study Notes

🧾 Overview

Scope: These notes cover the non‑calculator section (Section 1) topics: integer arithmetic, signs and order of operations, multiplication and division (including negatives), classifying numbers (rational/irrational), converting between fractions and decimals (including repeating decimals), ordering numbers, basic fraction arithmetic, and simple equation solving. Use written working for full marks.

➕ Integer addition/subtraction (Signs & order)

Key idea: Remove brackets carefully and follow left‑to‑right when operations are the same precedence. When subtracting a negative, it becomes addition. Small worked examples:

• $(-11)-(-25)-40 = -11 + 25 - 40 = 14 - 40 = -26$.
• $8 + (-12) - (-7) + 5 = 8 - 12 + 7 + 5 = 8$.

Tip: Convert consecutive plus/minus signs into single actions (e.g. $-(-x)=+x$). Keep a column for signs when mixing many terms.

× Multiplying & dividing with negatives

Rule: Negative × negative = positive; negative × positive = negative. Same for division.

• Example: $-99 \times -8.75 = 99 \times 8.75 = 866.25$.

🔁 Repeating (recurring) decimals and rational numbers

Definition: Any repeating decimal is rational because it can be expressed as a fraction of integers.

• Example: $0.125\overline{ }$ (if repeating) is rational.

Converting a simple repeating decimal to a fraction (method):

1. Let $x$ equal the repeating decimal, e.g. $x = 0.\overline{5}$.
2. Multiply by an appropriate power of 10 to shift one full repeat: $10x = 5.\overline{5}$.
3. Subtract: $10x - x = 9x = 5$ so $x = \frac{5}{9}$.

Example: $0.\overline{5} = \frac{5}{9}$. Example: $1.\overline{21}$: let $x=1.212121\dots$, then $100x=121.2121\dots$, subtract $99x=120\Rightarrow x=\frac{120}{99}=\frac{40}{33}$.

🔢 Converting terminating decimals to fractions (and simplify)

Rule: Count decimal places, put over $10^n$, then simplify.

• $0.65 = \frac{65}{100} = \frac{13}{20}$.
• $0.075 = \frac{75}{1000} = \frac{3}{40}$.
• $4.125 = 4 + \frac{125}{1000} = 4 + \frac{1}{8} = 4\frac{1}{8} = \frac{33}{8}$.

📈 Ordering numbers (including repeating decimals)

Strategy: Convert each number into either a common form (all decimals to the same number of places) or compare stepwise by integer part then decimal places. For repeating decimals, convert to fraction if necessary.

• Example method: To compare $4$, $4.\overline{6}$, $4.2$, $-5$ — compare integer parts first. Negative numbers are always less than positives.

Tip: When decimals are equal up to a point, look further digits or convert to fractions for exact comparison.

🔄 Fraction to decimal (terminating vs repeating)

Rule: A fraction $\frac{a}{b}$ will terminate in base 10 iff $b$ (after simplifying with $a$) has prime factors only $2$ and/or $5$. Otherwise it repeats.

• Example: $\frac{1}{8}=0.125$ (terminates). $\frac{1}{3}=0.\overline{3}$ (repeats).

Long division: Use repeated subtraction (or long division) to generate decimal digits; a remainder that repeats indicates a repeating block.

➗ Fraction multiplication & division (basic reminders)

Multiply: Multiply numerators and denominators, then simplify.

• Example: $\left(-\frac{1}{2}\right) \times \frac{11}{1} = -\frac{11}{2}$.

Divide: Multiply by the reciprocal.

• Example: $\frac{1}{3} \div \frac{6}{1} = \frac{1}{3} \times \frac{1}{6} = \frac{1}{18}$.

(Always simplify to lowest terms.)

❓ Solving for an unknown integer (linear equations)

Strategy: Follow inverse operations; keep the equation balanced. Check your answer by substituting.

• Example structure: $-(4-10) - \Box \times 5 = 36$. First simplify brackets: $-( -6 ) - 5\Box = 36 \Rightarrow 6 - 5\Box = 36 \Rightarrow -5\Box = 30 \Rightarrow \Box = -6$.

Tip: Watch negative signs and distribution carefully when removing brackets.

√ Irrational numbers — closure properties (common pitfalls)

Important: The sum/difference/product of irrationals can be either rational or irrational — it depends.

• Statements like “if $a$ and $b$ are positive irrationals then $a+b$ is irrational” are not always true. Provide counterexample: choose $a=\sqrt{2}$ and $b=2-\sqrt{2}$ (note $b$ is irrational?), better: choose $a=\sqrt{2}$ and $b=2-\sqrt{2}$ — here $b$ may be irrational but $a+b=2$ rational. Simpler counterexample for difference: let $a=\sqrt{2}$ and $b=\sqrt{2}$, then $-a+b=0$ (rational). So be careful with universal claims.

Key conclusions:

• The sum of irrationals can be rational (counterexample above).
• The difference of irrationals can be rational.
• The product of irrationals can be rational (e.g. $\sqrt{2}\times\sqrt{2}=2$).

✔ Checklist for exam (non‑calculator)

• Show full working for every step.
• Simplify fractions fully.
• Convert repeating decimals using the algebraic method (multiply, subtract, solve).
• For ordering, convert to the same representation where possible.
• When solving equations, carefully expand and isolate the unknown.

📌 Worked quick reference (selected answers & methods)

• $(-11)-(-25)-40 = -26$.
• $8 + (-12) - (-7) + 5 = 8$.
• $-99 \times -8.75 = 866.25$.
• $-( -25 + 250 ) = -(225) = -225$.
• Classify without calculation: repeating decimals and surds — repeating decimals = rational; square roots of perfect squares produce rationals (e.g. $\sqrt{16}/\sqrt{81}=4/9$); sums involving $\sqrt{2}$ plus a non‑special rational are typically irrational.

Keep these approaches in mind when doing the non‑calculator section. Work neatly and check sign mistakes first.

📝 Scope note (User instruction)

Source instruction: The user requested notes for "just the non calc section". These study notes therefore focus exclusively on Section 1 topics: integer arithmetic, fractions/decimals conversions (including recurring decimals), ordering numbers, fraction arithmetic and simple linear equations — all without a calculator.

🔎 How this affects study

• Practice long division and converting repeating decimals by hand.
• Emphasise sign rules, fraction simplification, and showing full working.
• Do not rely on calculator shortcuts for repeating decimal recognition or fraction simplification; practice the algebraic conversion method instead.