Very big and very small numbers
Scientists deal with numbers like the distance to the Sun (about 150,000,000 km) and the mass of a bacterium (about 0.000000000001 g). Writing all those zeros is slow and error-prone. Scientific notation is a shortcut.
The form
a × 10ⁿ
- a is the coefficient: 1 ≤ a < 10
- n is an integer exponent (the power of 10)
| Standard form | Scientific notation |
|---|---|
| 3,500 | 3.5 × 10³ |
| 150,000,000 | 1.5 × 10⁸ |
| 0.0042 | 4.2 × 10⁻³ |
| 0.000001 | 1 × 10⁻⁶ |
| 7.8 | 7.8 × 10⁰ |
Converting to scientific notation
- Move the decimal point so there’s one non-zero digit to its left.
- Count how many places you moved it.
- Big numbers (≥ 10): positive exponent. Small numbers (< 1): negative exponent.
Example: 62,000 → 6.2 (moved 4 left) → 6.2 × 10⁴
Example: 0.00087 → 8.7 (moved 4 right) → 8.7 × 10⁻⁴
Converting back to standard form
- Positive exponent: move the decimal point right.
- Negative exponent: move it left.
Example: 3.04 × 10⁵ = 304,000
Example: 5.1 × 10⁻³ = 0.0051
Multiplying and dividing
Work with coefficients and powers separately.
Multiply: (3 × 10⁴)(2 × 10⁵) = 6 × 10⁹
Divide: (8 × 10⁷) ÷ (2 × 10³) = 4 × 10⁴
If the coefficient leaves the range, adjust: (5 × 10³)(4 × 10²) = 20 × 10⁵ = 2.0 × 10⁶.
Comparing
Compare exponents first. 3 × 10⁸ > 9 × 10⁷ because 10⁸ > 10⁷.
On a calculator
Calculators show 2.5E8 or 2.5 × 10^8. Use the EXP or EE key to enter numbers like 6.02 × 10²³ (Avogadro’s number).
Real examples
| Quantity | Value |
|---|---|
| Speed of light | about 3.0 × 10⁸ m/s |
| Earth’s population (2020s) | about 8 × 10⁹ |
| Diameter of a hydrogen atom | about 1 × 10⁻¹⁰ m |
| Width of a human hair | about 7 × 10⁻⁵ m |
The coefficient shows the significant digits: 3.00 × 10⁸ is more precise than 3 × 10⁸.
Worked examples
Example 1: Write 3,500 in scientific notation. 3.5 × 10³.
Example 2: Write 0.0602 in scientific notation. 6.02 × 10⁻².
Example 3: (6 × 10⁴)(7 × 10⁻²) = 42 × 10² = 4.2 × 10³.
Common mistakes
- Writing 35 × 10² and calling it scientific notation. The coefficient must be less than 10.
- Using the wrong sign on the exponent for small numbers.
- Adding exponents when dividing (subtract them).
Try it yourself
- 47,000,000 in scientific notation
- 0.00309 in scientific notation
- 9.2 × 10⁻⁴ in standard form
- (2.5 × 10⁶)(4 × 10³)
- Which is larger: 5 × 10⁻³ or 4 × 10⁻²?
Answers: 1) 4.7 × 10⁷ 2) 3.09 × 10⁻³ 3) 0.00092 4) 1.0 × 10¹⁰ 5) 4 × 10⁻²
Quick recap
- Scientific notation: a × 10ⁿ with 1 ≤ a < 10.
- Positive exponent = big number; negative = small number.
- Multiply/divide coefficients, add/subtract exponents.