Scientific notation
Scientific notation (called standard form in the UK) is a short way to write very large or very small numbers using a power of ten. A number is written as
a × 10ⁿ,
where the coefficient a is at least 1 and less than 10, and the exponent n is a whole number.
Why we use it
The distance to the Sun is about 150 000 000 km. Writing all those zeros is slow and easy to get wrong. In scientific notation it is just 1.5 × 10⁸ km — much shorter and clearer.
From scientific notation to standard form
Multiplying by 10ⁿ moves the decimal point n places to the right. For 3.4 × 10⁴:
- start with 3.4
- move the point 4 places right: 3.4 → 34 000
So 3.4 × 10⁴ = 34 000. Fill any empty places with zeros.
From standard form to scientific notation
Put the decimal point after the first non-zero digit so the coefficient is between 1 and 10, then count how many places the point moved — that count is the exponent. For 52 000:
- coefficient: 5.2
- the point moved 4 places, so the exponent is 4
So 52 000 = 5.2 × 10⁴.
A common mistake
Writing 34 × 10³ instead of 3.4 × 10⁴ is not proper scientific notation, because the coefficient 34 is not between 1 and 10. Always check that 1 ≤ a < 10.
Three rules that always help
- The coefficient a must satisfy 1 ≤ a < 10 — exactly one non-zero digit before the decimal point.
- A large number has a positive exponent equal to the number of places the decimal point moves left to form the coefficient.
- To expand, move the decimal point right by the exponent and add zeros as needed.