Scientific Notation Guide: Converting to and from Standard Form
Scientific notation is the standard way to write very large or very small numbers in science and engineering. Instead of writing 0.000000000000000000000000000000911 kg, physicists write 9.11 × 10⁻³¹ kg — compact, precise, and easy to work with.
How Scientific Notation Works
Scientific notation expresses any number as a × 10ⁿ, where 1 ≤ a < 10 (the coefficient) and n is an integer exponent. The exponent tells you how many places to shift the decimal point.
Key Rules & Examples
3.0 × 10⁴ + 1.5 × 10³ → convert: 0.15 × 10⁴ + 3.0 × 10⁴ = 3.15 × 10⁴
Real Scientific Measurements in Scientific Notation
| Measurement | Scientific Notation |
|---|---|
| Speed of light | 2.998 × 10⁸ m/s |
| Mass of electron | 9.11 × 10⁻³¹ kg |
| Earth radius (equatorial) | 6.371 × 10⁶ m |
| Avogadro's number | 6.022 × 10²³ mol⁻¹ |
| Planck constant | 6.626 × 10⁻³⁴ J·s |
| Distance to Sun (mean) | 1.496 × 10¹¹ m |
| Elementary charge (e) | 1.602 × 10⁻¹⁹ C |
| Mass of Earth | 5.972 × 10²⁴ kg |
Conversion Practice: Standard Form ↔ Scientific Notation
| Standard Form | Direction | Scientific Notation | How |
|---|---|---|---|
| 93,000,000 | → sci | 9.3 × 10⁷ | Moved decimal 7 places left |
| 0.00047 | → sci | 4.7 × 10⁻⁴ | Moved decimal 4 places right |
| 1,234,567 | → sci | 1.234567 × 10⁶ | Moved decimal 6 places left |
| 0.000000082 | → sci | 8.2 × 10⁻⁸ | Moved decimal 8 places right |
| 50,000 | ← standard | 5.0 × 10⁴ | Move decimal 4 places right |
| 0.00314 | ← standard | 3.14 × 10⁻³ | Move decimal 3 places left |
| 7,200,000,000 | ← standard | 7.2 × 10⁹ | Move decimal 9 places right |
| 0.000001 | ← standard | 1.0 × 10⁻⁶ | Move decimal 6 places left |
Frequently Asked Questions
How do I write a number in scientific notation?
To write a number in scientific notation (a × 10ⁿ): 1) Move the decimal point until exactly one non-zero digit remains to its left. 2) Count how many places you moved it — this is the exponent n. 3) If you moved the decimal left (large number), n is positive. If you moved it right (small number), n is negative. Example: 6,370,000 → move decimal 6 places left → 6.37 × 10⁶.
What is the difference between scientific and engineering notation?
Scientific notation uses any integer exponent (e.g., 2.6 × 10⁵). Engineering notation restricts exponents to multiples of 3 (…10⁻⁶, 10⁻³, 10⁰, 10³, 10⁶, 10⁹…) so they align with SI metric prefixes. For example, 2,600 W is 2.6 × 10³ W in engineering notation — which directly maps to 2.6 kW (kilo = 10³).
How do I multiply numbers in scientific notation?
To multiply two numbers in scientific notation: multiply the coefficients and add the exponents. Example: (3 × 10⁴) × (2 × 10³) = (3 × 2) × 10⁴⁺³ = 6 × 10⁷. If the resulting coefficient is not between 1 and 10, adjust: 15 × 10⁵ = 1.5 × 10⁶. To divide, divide the coefficients and subtract the exponents.