BOUND — Locate a number that is not a fraction

DISCOVER

Start with a sketch and a prediction.

Square 1.414 and 1.415. Which is below 2? Could a longer decimal make $\sqrt{2}$ an exact fraction?

Try on paper first. Use the model to test your idea before accepting the explanation.

Understand

Squaring positive numbers preserves their order. Successive decimal bounds locate $\sqrt{2}$ without making a finite decimal exact. Irrationality requires a proof: a non-terminating decimal alone need not be irrational.

Key Ideas

1.414 < $\sqrt{2}$ < 1.415

Write $\sqrt{2}$ ≈ 1.414, never $\sqrt{2}$ = 1.414. Repeating decimals such as $\frac{1}{3}$ are rational.

Visualise

Method

Compare squares with 2. For irrationality suppose $\sqrt{2}$ = $\frac{m}{n}$ for positive integers. Then $m^2$ = $2n^2$. Prime 2 occurs an even number of times in $m^2$ but an odd number in $2n^2$, a contradiction.

Worked Example

AIMS Method

1. A — Attempt independently
   Sketch and predict before using a formula.
2. I — Investigate the idea
   Separate given facts from visual assumptions.
3. M — Make connections
   Connect area, length, algebra and geometric constructions.
4. S — Scrutinise your reasoning
   Check each hypothesis and justify the conclusion.

Think first. Use AI purposefully. Verify always.

Focused Practice

Check & Feedback

Textbook connection: Ganita Prakash · Grade 8 · Part II · Chapter 2 — The Baudhāyana–Pythagoras Theorem · printed page 39