Maths
What is scientific notation?
Quick answer
Scientific notation writes very large or very small numbers as a number from 1 to 10 times a power of 10. For example, 3,500,000 is 3.5 × 10⁶. In the UK it is called standard form.
In one sentence
Scientific notation is a short way to write very big or very small numbers as a number between 1 and 10 multiplied by a power of ten.
Explained
Some numbers are so big or so small that writing every zero is tiring. The distance from Earth to the Sun is about 150,000,000 kilometres. A hydrogen atom weighs about 0.00000000000000000000000000167 kilograms. Scientific notation fixes this.
The format is
a × 10ⁿ, where a is at least 1 but less than 10, and n is a whole number.
The power of ten tells you how many places the decimal point moves. See powers and indices.
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Large numbers. Move the decimal point to the left until you have a number between 1 and 10. Count the moves: that is the power.
3,500,000 → 3.5 (moved 6 places) → 3.5 × 10⁶.
Small numbers. Move the decimal point to the right. The power is negative.
0.00042 → 4.2 (moved 4 places) → 4.2 × 10⁻⁴.
To go back, do the opposite. 6 × 10⁵ means multiply 6 by 100,000, giving 600,000. And 5 × 10⁻³ is 5 ÷ 1,000 = 0.005.
Scientific notation also makes comparing easy. A number with a bigger power is bigger: 2 × 10⁷ is larger than 9 × 10⁶. The speed of light is about 3 × 10⁸ metres per second. On many calculators, a result like 3.5E6 means 3.5 × 10⁶. Because it relies on place value, it is easy once you know how each place is ten times the one before. The same idea runs through decimals.
Examples
| Number | Scientific notation |
|---|---|
| 4,200 | 4.2 × 10³ |
| 3,500,000 | 3.5 × 10⁶ |
| 150,000,000 | 1.5 × 10⁸ |
| 0.005 | 5 × 10⁻³ |
| 0.00042 | 4.2 × 10⁻⁴ |
Common mistakes
- First number outside 1 to 10. 35 × 10⁴ is correct in value but should be 3.5 × 10⁵.
- Wrong sign. Small numbers (below 1) have a negative power; big ones have a positive power.
- Counting the moves wrongly. Count from where the decimal point started to where it ends.
Quick quiz
Scientific notation
Check what you know with 4 quick questions.
Question 1 of 4
Write 4,200 in scientific notation.
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For grown-ups
Related topics
Powers and indices
A power (also called an index or exponent) is a small raised number showing how many times to multiply a number by itself. So 2⁴ means 2 × 2 × 2 × 2 = 16.
Place value
Place value means a digit's value depends on its position in a number. In 352, the 3 is worth 300, the 5 is worth 50 and the 2 is worth 2.
Decimals
Decimals are numbers with a decimal point. Digits after the point show tenths, hundredths and so on, so 2.5 means two and five tenths.
Highest common factor and lowest common multiple
The HCF is the largest number that divides exactly into two numbers. The LCM is the smallest number that both divide into. For 12 and 18, the HCF is 6 and the LCM is 36.
Prime factorisation
Prime factorisation means writing a number as a product of prime numbers. For example, 60 = 2 × 2 × 3 × 5, which can be written as 2² × 3 × 5.
Square roots
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 25 is 5, because 5 × 5 = 25. The sign is √.