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# Irrational numbers

Irrational numbers and index laws · Irrational numbers

<!-- Speaker note: 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). About 35 minutes (Predict and Run 8 min · Investigate 10 min · Modify 8 min · Practise and Make 9 min). -->

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## Today

**We are learning to** a rational number is a fraction: its decimal stops or repeats. An irrational number cannot be written as a fraction: its decimal never stops and never repeats. √2, √3 and π are irrational; √16 = 4 is not.

- 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.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need recurring decimals (previous unit) and square roots (Number Foundations). The one idea: rational numbers are fractions, so their decimals stop or repeat; irrational numbers never stop and never repeat. The common mistake is calling every square root irrational (Ana's mistake). The board checks each zoom with squares for square roots, and with "too big, too small" for π. Proving √2 is irrational is beyond Year 8; the board is evidence, not proof (say so). Wider: historical approximations of π (Egyptian, Chinese, Indian, Greek), the golden ratio. -->

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## Why it matters

> 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.

<!-- Speaker note: A **rational** number can be written as a fraction $\frac{a}{b}$ of integers (whole numbers or their negatives, with $b \ne 0$): 0.25, $\frac{1}{7}$, −3, $0.\overline{3}$. Its decimal **terminates** or **recurs**. An **irrational** number cannot be written as a fraction. Its decimal goes on **forever without repeating**. Examples: $\sqrt{2} = 1.41421356\ldots$, $\sqrt{3}$, $\sqrt{5}$, and $\pi = 3.14159265\ldots$ The square root of a whole number is irrational unless the number is a perfect square: $\sqrt{16} = 4$ is rational. Irrational numbers are ordinary numbers you can calculate with; we use the exact form ($\sqrt{2}$, $\pi$) or a rounded value (1.414, 3.14). -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**√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

<!-- Speaker note: What happens: Neither. √2 is irrational: no fraction equals it, so its decimal can never stop or repeat. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what a repeating √2 would mean (it would be a fraction). -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Freal-numbers%2Firrational-numbers%2Fstep-2&code=widget%3A+zoomline%0Anumber%3A+sqrt2%0Aplaces%3A+3%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/irrational-numbers/?full=step-2#step-2)

<!-- Speaker note: Press the piece of the line that √2 is in. A wrong piece shows you the squares of its ends. -->

<!-- Speaker note: Teacher: ask how they know 1.4 is too small. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

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

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Sort them

Here is the routine.

$$\text{a fraction?}$$

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

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write a fraction? and say "Yes: rational." Line 1: write decimal stops or repeats? and say "Yes: rational (it is a fraction)." Line 2: write n,n a perfect square? and say "√16 = 4: rational. √15: irrational." Line 3: write ,2,3 and say "Never stop, never repeat: irrational." -->

<!-- Speaker note: Teacher: ask Ana to work out √16. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Sort them

**Rational or irrational?**

Groups: Rational · Irrational

1/7 · 0.25 · √9 · √5 · π · 0.333… · 22/7

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

<!-- Speaker note: Fractions, and decimals that stop or repeat, are rational; √5 and π are not. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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.
```

* Answer: line 3

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

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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)

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

<!-- Speaker note: Irrational does not mean unusable: π and √2 are exact numbers with endless decimals. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Other numbers



- **Root 3:** 
- **One seventh:** 
- **π and 22/7:** 
- **Root 2 squared:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/irrational-numbers/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Zooming in on √3 and 1/7 compares an irrational number with a repeating one; π and 22/7 differ; irrationals can combine to a rational. Hints, in order: 1.7² = 2.89 and 1.8² = 3.24. For students who need more support: A card with two columns: 'stops or repeats: rational' and 'never stops, never repeats: irrational'. For early finishers: Find how ancient Egyptian, Chinese and Indian mathematicians approximated π. Which approximation was closest? The hard part, on purpose: Believing a decimal can go on forever without repeating. Let the zooming carry on past where students expect it to settle. When students get stuck: A student calls a long calculator decimal irrational. Ask: Does it repeat? Is it a fraction? Fix: Repeating means rational; only endless and non-repeating is irrational. / A student calls every root irrational. Ask: What is √16? Fix: Perfect squares have rational roots. / A student thinks π is 22/7. Ask: What is 22 ÷ 7 to 4 places? Fix: 3.1428… against 3.1415…: close, not equal. -->

<!-- Speaker note: Teacher: ask how 1/7's digits differ from √3's. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Irrational

**Which number is irrational?**

- √10
- 3/11
- 0.121212…
- √25

* Answer: √10

<!-- Speaker note: 10 is not a perfect square, so √10 is irrational. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Rational

**Which number is rational?**

- √36
- √2
- √7
- π

* Answer: √36

<!-- Speaker note: √36 = 6. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Never repeats

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

- Recurring
- Irrational
- Rational
- A whole number

* Answer: Irrational

<!-- Speaker note: Never stopping and never repeating means irrational. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: square roots

**Review: what is √49?**

* Answer: 7

<!-- Speaker note: 7 × 7 = 49. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: π

Then explain why 3.1416 is not exactly π.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/real-numbers/irrational-numbers/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Finding π to 4 places with only 'too big, too small' feedback applies the zooming method to a number with no squares to help. Hints, in order: π is a little more than 3. / Its first two decimal digits are 14. -->

<!-- Speaker note: Teacher: ask what exact means here. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

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

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

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This starts AC9M8N01. Where it leads: placing irrational numbers on the number line (next lesson), Pythagoras' exact answers, and surds (Year 10). -->