What Is Scientific Notation
Scientific notation writes any number as a coefficient between 1 and 10 multiplied by a power of 10. The format is a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. It compresses very large and very small numbers into a readable form that makes arithmetic easier and comparisons instant.
To convert a number to scientific notation: move the decimal point until you have a single non-zero digit to its left. The number of places you moved is the exponent. Moved left (large number) = positive exponent. Moved right (small number) = negative exponent. 300,000,000 becomes 3 × 10⁸ (moved 8 places left). 0.000042 becomes 4.2 × 10⁻⁵ (moved 5 places right).
To convert back to a decimal: move the decimal point by the exponent. Positive exponent moves right (bigger number). Negative exponent moves left (smaller number). 6.022 × 10²³ = 602,200,000,000,000,000,000,000.
Powers of 10 Reference Table
| Power | Value | Name | Prefix |
|---|---|---|---|
| 10¹² | 1,000,000,000,000 | Trillion | Tera (T) |
| 10⁹ | 1,000,000,000 | Billion | Giga (G) |
| 10⁶ | 1,000,000 | Million | Mega (M) |
| 10³ | 1,000 | Thousand | Kilo (k) |
| 10⁰ | 1 | One | — |
| 10⁻³ | 0.001 | Thousandth | Milli (m) |
| 10⁻⁶ | 0.000001 | Millionth | Micro (μ) |
| 10⁻⁹ | 0.000000001 | Billionth | Nano (n) |
| 10⁻¹² | 0.000000000001 | Trillionth | Pico (p) |
Engineering Notation and SI Prefixes
Engineering notation restricts the exponent to multiples of 3. This maps directly to SI prefixes that engineers, scientists, and technicians use daily. Instead of writing 4.7 × 10⁴, you write 47 × 10³ — which reads as “47 kilo.” A 2.2 GHz processor runs at 2.2 × 10⁹ Hz. A 470 nF capacitor stores 470 × 10⁻⁹ farads.
The most common SI prefixes in everyday life: kilo (k) = 10³ for kilobytes, kilometers, kilograms. mega (M) = 10⁶ for megabytes, megapixels, megawatts. giga (G) = 10⁹ for gigabytes, gigahertz, gigawatts. tera (T) = 10¹² for terabytes. On the small side: milli (m) = 10⁻³ for milliseconds, millimeters. micro (μ) = 10⁻⁶ for microseconds, micrometers. nano (n) = 10⁻⁹ for nanometers, nanoseconds.
Real-World Numbers in Scientific Notation
| Quantity | Value | Scientific Notation |
|---|---|---|
| Speed of light | 299,792,458 m/s | 2.998 × 10⁸ |
| Avogadro's number | 602,214,076,000,000,000,000,000 | 6.022 × 10²³ |
| Electron mass | 0.000000000000000000000000000000911 kg | 9.11 × 10⁻³¹ |
| Distance to Sun | 149,597,870,700 m | 1.496 × 10¹¹ |
| Hydrogen atom radius | 0.0000000000529 m | 5.29 × 10⁻¹¹ |
| US national debt (2025) | $36,000,000,000,000 | 3.6 × 10¹³ |
| Planck's constant | 0.000000000000000000000000000000000663 J·s | 6.63 × 10⁻³⁴ |
Multiplying and Dividing in Scientific Notation
Multiplication: Multiply the coefficients and add the exponents. (3 × 10⁴) × (2 × 10³) = 6 × 10⁷. If the resulting coefficient is 10 or greater, normalize: (5 × 10³) × (4 × 10²) = 20 × 10⁵ = 2 × 10⁶.
Division: Divide the coefficients and subtract the exponents. (8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴. If the resulting coefficient is less than 1, normalize: (3 × 10⁵) ÷ (6 × 10²) = 0.5 × 10³ = 5 × 10².
This is why scientific notation exists. Try multiplying 602,214,076,000,000,000,000,000 by 0.000000000000000000000000000000000663 by hand. In scientific notation: (6.022 × 10²³) × (6.63 × 10⁻³⁴) = 39.93 × 10⁻¹¹ = 3.993 × 10⁻¹⁰. That took seconds.
For powers and roots, the exponent calculator handles xⁿ and nth roots. For quick percentage conversions of any number, use the percentage calculator.