GANITA MANJARI · GRADE 9 · PART I · CHAPTER 6

APPROXIMATE — Understanding π

Think first. Use AI purposefully. Verify always.

DISCOVER

Start with a question.

Are $\frac{22}{7}$ , $\frac{355}{113}$ and 3.1416 exactly the same number? Which is closest to π?

Understand

π cannot be written as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating. This is a theorem, not a conclusion proved by inspecting a few digits.

Key Ideas

π \approx \frac{22}{7}

π = $\frac{22}{7}$ is false. An observed lack of repetition in a finite decimal display is not a proof of irrationality.

Visualise

Which approximation is closer? A finite displayed decimal cannot establish irrationality.

Choose an approximation

The markers use a common scale from 3.14 to 3.17; nearby values can overlap.

Method

Separate an exact symbol from an approximation. Use the value and rounding requested by the problem. Retain extra working digits before rounding the final result.

Worked Example

Given

Compare $\frac{22}{7}$ and $\frac{355}{113}$ with a numerical reference for π.

To Find

Identify the closer approximation.

Formula / Property / Theorem / Reasoning

Compare absolute errors, not the number of digits written.

Solution

$\frac{22}{7}$ ≈ 3.14285714; its error is about 0.00126449.
$\frac{355}{113}$ ≈ 3.14159292; its error is about 0.00000027.

Result

$\frac{355}{113}$ is closer; neither is π. Based on the approximations discussed on pages 122–124.

AIMS Method

A — Attempt independently. Make a prediction or start a sketch.
I — Investigate the idea. Identify the boundary, region and given conditions.
M — Make connections. Relate the figure to a known property or rearrangement.
S — Scrutinise your reasoning. Check conditions, units and whether the result is exact.

Think first. Use AI purposefully. Verify always.

Focused Practice

A student writes π = 3.14 because the first three digits agree. Correct the statement and explain.

Your reasoning

Check & Feedback

Can a fraction closer than $\frac{355}{113}$ to π exist?

Explain your conclusion

Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.3 · π Is Irrational · printed pages 123–125.