Light travels about 300,000,000 meters each second. A small bacterium can be about 0.000002 meter long. Numbers like these are easy to misread, because you have to count zeros. Scientific notation writes them as a short number times a power of 10: $3 \times 10^{8}$ and $2 \times 10^{-6}$.
It rests on multiplying and dividing by powers of 10: each factor of 10 moves every digit one place.
Key idea
A number is in scientific notation when it is written as $a \times 10^{n}$, where $1 \le a < 10$ and $n$ is an integer. Big numbers get a positive $n$. Numbers between 0 and 1 get a negative $n$.
Why it works
$3.08 \times 10^{6}$ multiplies 3.08 by 10 six times, which moves its digits six places to the left: 3,080,000. A negative exponent divides instead: $10^{-4} = \frac{1}{10{,}000}$, so $5.06 \times 10^{-4}$ moves the digits four places to the right: 0.000506.
Worked examples
Worked example: A big number
Write $3{,}080{,}000$ in scientific notation.
- Put the decimal point after the first nonzero digit: $3.08$, which is between 1 and 10.
- From 3.08 to 3,080,000 the point moves 6 places to the right.
- So $3{,}080{,}000 = 3.08 \times 10^{6}$.
Answer: $3.08 \times 10^{6}$
Worked example: A small number
Write $0.000506$ in scientific notation.
- The first factor is $5.06$.
- To get from 5.06 back to 0.000506, the point moves 4 places to the left, which is dividing by $10^{4}$.
- Dividing by $10^{4}$ is multiplying by $10^{-4}$.
Answer: $5.06 \times 10^{-4}$
Worked example: Back to standard form
Write $6.1 \times 10^{-3}$ as an ordinary decimal.
- A negative exponent: the number is less than 1.
- Move the point 3 places left, filling with zeros.
Answer: $0.0061$
Common mistake
Two common slips. (1) Using a positive exponent for a small number: $0.0045 = 4.5 \times 10^{3}$ is wrong, because $4.5 \times 10^{3}$ is 4,500. A number less than 1 needs a negative exponent: $4.5 \times 10^{-3}$. (2) Stopping at $45 \times 10^{3}$. It equals 45,000, but 45 is not less than 10. Move one more place: $4.5 \times 10^{4}$.