Scientific notation writes a number as a × 10ⁿ, where a is at least 1 and less than 10, and n is a whole number. It makes huge and tiny numbers easy to read and compare.
Try it with your own problem ↓
The steps
- Move the decimal point so there is exactly one nonzero digit to its left. That gives a number between 1 and 10.
- Count how many places you moved the point. That count is n.
- If you moved it left (the original number was big), n is positive. If you moved it right (the number was small), n is negative.
- Write a × 10ⁿ. Drop any extra zeros at the end of a.
- To go back, move the decimal point n places right for positive n, or left for negative n.
A worked example
Write 45,000,000 in scientific notation.
- The decimal point is at the end: 45,000,000.
- Move it left until one digit is before it: 4.5000000. That is 7 places.
- Moving left for a big number gives a positive exponent: n = 7.
- Drop the extra zeros: a = 4.5.
- So the number is 4.5 × 10⁷. Check: 4.5 × 10,000,000 = 45,000,000.
45,000,000 = 4.5 × 10⁷. For a small number, 0.00032 = 3.2 × 10⁻⁴.
Common mistakes
- Counting places wrong. Count the moves, not the zeros.
- Wrong sign on the exponent. Big numbers have positive exponents and numbers below 1 have negative ones.
- Leaving a not between 1 and 10. 45 × 10⁶ is not scientific notation. Write 4.5 × 10⁷.
- Forgetting to adjust after multiplying. (3 × 10⁴)(2 × 10⁵) = 6 × 10⁹, but 6 × 10⁴ × 7 × 10⁵ = 42 × 10⁹ = 4.2 × 10¹⁰.
Questions people ask
How do I multiply numbers in scientific notation?
Multiply the front numbers, add the exponents, and adjust so the front number is between 1 and 10.
How do I know if the exponent is negative?
If the number is less than 1 (like 0.00032), the exponent is negative. The point moved right.
Why use scientific notation?
It's shorter, it shows size at a glance, and it avoids miscounting zeros. Scientists use it for things like the mass of an atom or the distance to a star.
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