---
title: "Irrational numbers"
subtitle: "Irrational numbers and index laws · Irrational numbers · Lesson 3"
---

# Irrational numbers

**Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}}

> **Key question:** The diagonal of a 1 m square tile is exactly √2 metres long. You can measure it, but you can never write it down as a fraction or a decimal that ends.


:::teacher-note

**In one sentence:** Students zoom in on √2, √3, π and 1/7 one digit at a time, and sort numbers into rational (fractions, decimals that stop or repeat) and irrational (AC9M8N01, node 1).

**Students should be able to say:** I can say what makes a number rational or irrational. I can sort numbers such as √9, √5, π and 0.333… correctly. I can explain why a rounded value is not the exact number.

**Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet.

:::


## Predict


:::keep-together

**√2 = 1.414… Do its digits ever stop, or settle into a repeating block?**

☐ Neither: they go on forever without repeating

☐ They repeat, like 1/7

☐ They stop after a few more digits

My prediction, and why:

{{lines-3}}

:::


:::teacher-note

**What happens:** Neither. √2 is irrational: no fraction equals it, so its decimal can never stop or repeat.

:::


## Run

Try it on the board in the tutorial. Was your prediction right? What was different?

{{lines-2}}

## Rational or irrational


:::keep-together

### Sort them

Here is the routine.

1. $\text{a fraction?}$ — Yes: rational.
2. $\text{decimal stops or repeats?}$ — Yes: rational (it is a fraction).
3. $\sqrt{n},\ n \text{ a perfect square?}$ — √16 = 4: rational. √15: irrational.
4. $\pi,\ \sqrt{2},\ \sqrt{3}$ — Never stop, never repeat: irrational.

**Rational or irrational?**

Groups: Rational, Irrational

- 1/7: {{field-15}}
- 0.25: {{field-15}}
- √9: {{field-15}}
- √5: {{field-15}}
- π: {{field-15}}
- 0.333…: {{field-15}}
- 22/7: {{field-15}}

:::


:::teacher-note

**Answer:** 1/7 → Rational; 0.25 → Rational; √9 → Rational; √5 → Irrational; π → Irrational; 0.333… → Rational; 22/7 → Rational

Fractions, and decimals that stop or repeat, are rational; √5 and π are not.

:::


:::keep-together

### Find the mistake

**Ana sorts √16. Click the line where her reasoning goes wrong.**

```
 1  Ana sorts √16 into rational or irrational.
 2  √16 is a square root.
 3  Square roots are irrational, so √16 is irrational.
```

Line: {{field-15}}

Why is it wrong?

{{lines-2}}

:::


:::teacher-note

**Answer:** line 3

Line 3: √16 = 4, a whole number, so it is rational. Only roots of numbers that are not perfect squares are irrational.

:::


:::keep-together

### Using them

**Can you calculate with irrational numbers?**

☐ Only with a calculator that has infinite digits

☐ No: they never end

☐ Only if you round them to whole numbers

☐ Yes: use the exact form (√2, π) or a rounded value (1.414, 3.14)

:::


:::teacher-note

**Answer:** Yes: use the exact form (√2, π) or a rounded value (1.414, 3.14)

Irrational does not mean unusable: π and √2 are exact numbers with endless decimals.

:::


:::typical

## Other numbers


**Root 3.** 

**One seventh.** 

**π and 22/7.** 

**Root 2 squared.** 

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

:::


## Rational or irrational


:::keep-together

### Irrational

**Which number is irrational?**

☐ √10

☐ 3/11

☐ 0.121212…

☐ √25

:::


:::teacher-note

**Answer:** √10

10 is not a perfect square, so √10 is irrational.

:::


:::keep-together

### Rational

**Which number is rational?**

☐ √36

☐ √2

☐ √7

☐ π

:::


:::teacher-note

**Answer:** √36

√36 = 6.

:::


:::keep-together

### Never repeats

**A decimal goes on forever without repeating. What kind of number is it?**

☐ Recurring

☐ Irrational

☐ Rational

☐ A whole number

:::


:::teacher-note

**Answer:** Irrational

Never stopping and never repeating means irrational.

:::


:::keep-together

### Review: square roots

**Review: what is √49?**

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** 7

7 × 7 = 49.

:::


## Make: π

Then explain why 3.1416 is not exactly π.

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

## Exit ticket

:::keep-together

**Sort √20, √25, 0.4 and 0.454545… into rational and irrational.**

{{lines-3}}

:::


:::teacher-note

**Answer:** Rational: √25 = 5, 0.4, 0.454545… (= 5/11). Irrational: √20.

:::