Scientific notation writes a number as a value between 1 and 10 times a power of 10. It keeps very large and very small numbers readable. The distance from Earth to the Sun is about 150,000,000 km, or 1.5 × 10⁸ km. A red blood cell is about 0.000008 m across, or 8 × 10⁻⁶ m.
The form
a × 10ⁿ, where 1 ≤ a < 10 and n is an integer
- 4.2 × 10⁵ is in scientific notation.
- 42 × 10⁴ is not, because 42 is 10 or more.
- 0.42 × 10⁶ is not, because 0.42 is less than 1.
All three are the same number, 420,000.
Writing a number in scientific notation
Move the decimal point until exactly one nonzero digit is left of it. Count the moves.
- If the original number is 10 or more, n is positive.
- If it is less than 1, n is negative.
Example Write 6,370,000 in scientific notation
- Move the decimal to get 6.37.
- It moved 6 places left.
- 6.37 × 10⁶.
Example Write 0.00052 in scientific notation
- Move the decimal to get 5.2.
- It moved 4 places right.
- 5.2 × 10⁻⁴.
Converting back to standard form
Move the decimal n places: right if n is positive, left if n is negative. Fill in zeros as needed.
| Scientific | Standard |
|---|---|
| 3.08 × 10⁴ | 30,800 |
| 7 × 10⁰ | 7 |
| 1.9 × 10⁻³ | 0.0019 |
| 6.02 × 10²³ | 602,000,000,000,000,000,000,000 |
Negative exponent, negative number
5 × 10⁻³ is 0.005, a small positive number. A negative exponent makes the number small. Only a minus sign in front, like −5 × 10³, makes it negative.
Multiplying
Multiply the front numbers and add the exponents, using the exponent rule 10ᵃ · 10ᵇ = 10ᵃ⁺ᵇ. Then fix the front number if it isn’t between 1 and 10.
Example (4 × 10⁵)(3 × 10²)
- 4 × 3 = 12, and 10⁵ × 10² = 10⁷.
- 12 × 10⁷ isn’t in proper form.
- 12 = 1.2 × 10¹, so the answer is 1.2 × 10⁸.
Dividing
Divide the front numbers and subtract the exponents.
Example (6 × 10³) ÷ (8 × 10⁻²)
- 6 ÷ 8 = 0.75, and 10³ ÷ 10⁻² = 10⁵.
- 0.75 × 10⁵ = 7.5 × 10⁴.
Subtracting a negative exponent
3 − (−2) = 5. Writing 10³⁻² = 10¹ is the most common slip in division problems.
Adding and subtracting
You can only add the front numbers when the powers of 10 match. Rewrite one number first.
Example 5.2 × 10⁴ + 3.1 × 10³
- Rewrite 3.1 × 10³ as 0.31 × 10⁴.
- Add: 5.51 × 10⁴.
Calculator E notation
Calculators and spreadsheets show 6.02 × 10²³ as 6.02E23 or 6.02e23. The E means “times 10 to the”. To type 3 × 10⁻⁵, use the EE or EXP key: 3 EE −5. Typing 3 × 10 ^ −5 also works, but forgetting parentheses in a division like 1/(2 × 10⁶) is a common error.
Significant figures
The front number shows how precise a measurement is. 3.0 × 10⁴ has two significant figures, 3.00 × 10⁴ has three. When you multiply or divide measurements, round the answer to the fewest significant figures among the numbers you started with.
Try a few
Write 93,000,000 in scientific notation.
Show answer
9.3 × 10⁷.
Write 0.0000071 in scientific notation.
Show answer
7.1 × 10⁻⁶.
Write 2.45 × 10⁻² in standard form.
Show answer
0.0245.
Compute (2.5 × 10⁴)(6 × 10³).
Show answer
1.5 × 10⁸. 15 × 10⁷ adjusted.
Compute (9 × 10⁻³) ÷ (3 × 10⁻⁵).
Show answer
3 × 10².
Compute 4.6 × 10⁵ − 2 × 10⁴.
Show answer
4.4 × 10⁵. 4.6 − 0.2 = 4.4.