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Heron's method

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

Square roots without a calculator: step by step a rectangle becomes a square. The formula: \( x_{\text{new}} = \left(x + \frac{a}{x}\right) : 2 \).
📄 Open worksheet (PDF, German)

The video

Watch the colours: every new rectangle has its own colour. The target square is dashed green.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:14 The idea
  3. 0:31 The first step
  4. 0:54 The second step
  5. 1:18 The formula
  6. 1:37 An example
  7. 1:59 History
  8. 2:16 Try it yourself

The most important points

We look for a square with area \( 2 \). Start: rectangle \( 2 \cdot 1 \). New side: the mean \( (2 + 1) : 2 = 1.5 \), other side \( 2 : 1.5 \approx 1.333 \).

After three steps: 1.414216 – almost exactly \sqrt{2} = 1.414213\ldots
After three steps: \( 1.414216 \) – almost exactly \( \sqrt{2} = 1.414213\ldots \)

Heron's method

\( x_{\text{new}} = \left(x + \frac{a}{x}\right) : 2 \). Start with a number that roughly fits. Example \( \sqrt{10} \): start \( 3 \), after two steps \( 3.16228 \).

The formula: the mean of x and a/x .
The formula: the mean of \( x \) and \( \frac{a}{x} \).

History (from Wikipedia): Heron of Alexandria described the method around 100 AD. In Mesopotamia it was already known at the time of Hammurabi (around 1750 BC).

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
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The worksheet – Page 3
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The worksheet – Page 4
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Show the worksheet solutions
The worksheet – Page 5
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Try it yourself

Task

Do one step: a) \( \sqrt{5} \), Start \( x_0 = 2 \) b) \( \sqrt{17} \), Start \( x_0 = 4 \) c) \( \sqrt{3} \), Start \( x_0 = 2 \)

Show solution

a) \( (2 + 2{,}5) : 2 = 2{,}25 \) b) \( (4 + 4{,}25) : 2 = 4{,}125 \) c) \( (2 + 1{,}5) : 2 = 1{,}75 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Heron's method (approximating square roots with x new = (x + a divided by x) divided by 2). Give me five short tasks, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Write in simple German and put an English translation in brackets after each sentence.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.