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Scientific notation

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

The Earth weighs about 5 970 000 000 000 000 000 000 000 kg – a number with 25 digits. In short: \( 5.97 \cdot 10^{24} \) kg. This is how to write very big and very small numbers.
📄 Open worksheet (PDF, German)

The video

Watch the colours: the digits in front are blue, the exponent of the power of ten is orange.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:23 The order of magnitude
  3. 0:44 Powers of ten
  4. 1:07 Scientific notation
  5. 1:39 Big numbers
  6. 2:13 Small numbers
  7. 2:47 Physics: the electron
  8. 3:11 Chemistry: the Avogadro number
  9. 3:33 Calculating and the calculator
  10. 4:03 Careful, mistake!
  11. 4:25 Try it yourself

The most important points

The mass of the Earth written out: 25 digits.
The mass of the Earth written out: 25 digits.

Often not every digit matters, but the order of magnitude (die Größenordnung). For this we need the powers of ten (die Zehnerpotenzen): \( 10^1 = 10 \), \( 10^2 = 100 \), \( 10^6 = 1\,000\,000 \). Times 10 means: add a zero. The exponent counts the zeros.

Scientific notation (die wissenschaftliche Schreibweise)

A number from 1 to below 10, times a power of ten: \( 5.97 \cdot 10^{24} \). Put the decimal point after the first digit that is not zero and count the places. Moved to the left: positive exponent. Moved to the right: negative exponent.

The decimal point moves 24 places to the left: 5.97 · 10^{24} kg.
The decimal point moves 24 places to the left: \( 5.97 \cdot 10^{24} \) kg.

Examples: \( 45\,000 = 4.5 \cdot 10^4 \), light \( \approx 3 \cdot 10^8 \) m/s, \( 0.0005 = 5 \cdot 10^{-4} \). If the number is smaller than 1, the exponent is negative.

Mass of an electron: 30 zeros after the decimal point, in short 9.11 · 10^{-31} kg.
Mass of an electron: 30 zeros after the decimal point, in short \( 9.11 \cdot 10^{-31} \) kg.

In about 12 g of carbon there are about \( 6.02 \cdot 10^{23} \) atoms (the Avogadro number). The calculator often shows such numbers with E: 5.97E24. Careful: \( 5.97 \cdot 10^{24} \) is not \( 5.97^{24} \). Note: in German, the decimal point is written as a comma: 5,97.

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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The worksheet – Page 5
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Show the worksheet solutions
The worksheet – Page 6
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Try it yourself

Task

Write in scientific notation: a) \( 45\,000\,000 \) b) \( 0.003 \) c) \( 380\,000 \) d) \( 0.00072 \)

Show solution

a) \( 4.5 \cdot 10^7 \) b) \( 3 \cdot 10^{-3} \) c) \( 3.8 \cdot 10^5 \) d) \( 7.2 \cdot 10^{-4} \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: scientific notation (for example 5.97 · 10 to the 24 or 3 · 10 to the minus 3). 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.