Convert numbers to and from scientific notation (US) / standard form (UK). Multiply, divide, add, and subtract numbers in scientific notation. Shows significant figures and engineering notation.
Scientific notation is the internationally accepted method for expressing very large and very small numbers compactly and precisely. In the United States, it is called scientific notation. In the United Kingdom and most Commonwealth countries, the same concept is called standard form. This free calculator converts any number to and from scientific notation, performs arithmetic on numbers in scientific notation, and shows significant figures and engineering notation.
Scientific notation expresses any number as a product of a coefficient and a power of 10:
N = a × 10n
where:
| Ordinary Number | Scientific Notation (US) | Standard Form (UK) |
|---|---|---|
| 5,000,000 | 5 × 10⁶ | 5 × 10⁶ |
| 0.00042 | 4.2 × 10⁻⁴ | 4.2 × 10⁻⁴ |
| 299,792,458 (speed of light, m/s) | 2.998 × 10⁸ | 2.998 × 10⁸ |
| 0.000000000529 (Bohr radius, m) | 5.29 × 10⁻¹¹ | 5.29 × 10⁻¹¹ |
Multiply the coefficients and add the exponents: (a × 10^m) × (b × 10^n) = (a × b) × 10^(m+n). Then adjust so the coefficient is in the range 1–10 if needed.
Example: (3 × 10⁴) × (2 × 10³) = 6 × 10⁷
Divide the coefficients and subtract the exponents: (a × 10^m) ÷ (b × 10^n) = (a/b) × 10^(m−n).
Example: (9 × 10⁶) ÷ (3 × 10²) = 3 × 10⁴
First convert both numbers to the same power of 10, then add or subtract the coefficients.
Example: (3 × 10⁵) + (2 × 10⁴) = (3 × 10⁵) + (0.2 × 10⁵) = 3.2 × 10⁵
Significant figures (sig figs) are the meaningful digits in a measurement. Scientific notation makes counting sig figs unambiguous because only the coefficient digits count:
In calculations, the result should have no more sig figs than the least precise measurement used. In multiplication/division, use the smallest number of sig figs. In addition/subtraction, use the smallest number of decimal places.
Engineering notation is a variant where the exponent is always a multiple of 3 (corresponding to SI prefixes). For example, 47,000 becomes 47 × 10³ (not 4.7 × 10⁴). This directly aligns with unit prefixes like kilo (10³), mega (10⁶), giga (10⁹), milli (10⁻³), and micro (10⁻⁶).
| Prefix | Symbol | Power | Example |
|---|---|---|---|
| Tera | T | 10¹² | 1 TB = 10¹² bytes |
| Giga | G | 10⁹ | 1 GHz = 10⁹ Hz |
| Mega | M | 10⁶ | 1 MW = 10⁶ W |
| Kilo | k | 10³ | 1 km = 10³ m |
| Milli | m | 10⁻³ | 1 mm = 10⁻³ m |
| Micro | μ | 10⁻⁶ | 1 μm = 10⁻⁶ m |
| Nano | n | 10⁻⁹ | 1 nm = 10⁻⁹ m |
| Pico | p | 10⁻¹² | 1 pF = 10⁻¹² F |
Scientific notation is essential in physics because the relevant scales span over 40 orders of magnitude:
The mathematics is identical — only the terminology differs. In England, Wales, and Northern Ireland, GCSE Mathematics (Higher Tier) includes a unit on Standard Form. Students must:
In the US, scientific notation is introduced in Middle School Math (Grade 8, CCSS 8.EE.A.3–4) and developed further in Algebra 2, Chemistry, and Physics courses. The CCSS specifically requires students to multiply and divide numbers in scientific notation and interpret results in context.
Chemistry uses scientific notation extensively. Avogadro's number (6.022 × 10²³) is perhaps the most famous example — it represents the number of particles in one mole of substance. Equilibrium constants, reaction rates, atomic masses, and concentrations all routinely involve numbers that span many orders of magnitude, making standard form / scientific notation indispensable in chemistry laboratories and examinations.
They are the same mathematical concept expressed in a × 10^n form where 1 ≤ |a| < 10. In the USA it is called scientific notation; in the UK and Commonwealth it is called standard form. Both require the coefficient to have exactly one non-zero digit before the decimal point.
Move the decimal point until you have a number between 1 and 10 (the coefficient). Count how many places you moved it — that is the exponent. If you moved left (for large numbers), the exponent is positive. If you moved right (for small numbers), the exponent is negative. Example: 0.0000045 = 4.5 × 10⁻⁶.
Multiply the coefficients and add the exponents: (a × 10^m) × (b × 10^n) = (a×b) × 10^(m+n). If the product coefficient is ≥ 10 or < 1, adjust: for example 20 × 10³ = 2 × 10⁴. Example: (3.0 × 10⁴) × (2.5 × 10⁻²) = 7.5 × 10².
Engineering notation is a form where the exponent is always a multiple of 3. This aligns with SI prefixes: 10³ (kilo), 10⁶ (mega), 10⁹ (giga), 10⁻³ (milli), 10⁻⁶ (micro), etc. For example, 47,000 Hz = 47 × 10³ Hz = 47 kHz in engineering notation, rather than 4.7 × 10⁴ Hz in scientific notation.
Significant figures are the meaningful digits in a measured or calculated value. Scientific notation makes counting them unambiguous: only the digits in the coefficient count. 3.0 × 10⁵ has 2 sig figs; 3.00 × 10⁵ has 3; 3 × 10⁵ has 1. In calculations, round the result to match the least precise input.
Avogadro's number is 6.022 × 10²³ mol⁻¹ (to 4 sig figs). It represents the number of atoms, molecules, or formula units in one mole of a substance. Written out in full it would be 602,200,000,000,000,000,000,000 — clearly demonstrating why scientific notation is essential in chemistry.
Yes. Standard form is a required topic in GCSE Mathematics Higher Tier in England, Wales, and Northern Ireland. AQA, OCR, and Edexcel specifications all include converting to/from standard form and performing arithmetic operations. Students are expected to work with standard form both with and without a calculator.
The speed of light in a vacuum is exactly 299,792,458 m/s, which is 2.998 × 10⁸ m/s in scientific notation (to 4 sig figs). It is commonly approximated as 3 × 10⁸ m/s (1 sig fig). In miles per hour, the speed of light is approximately 6.71 × 10⁸ mph.
Results are calculated using JavaScript floating-point arithmetic. Extreme numbers may lose precision due to IEEE 754 double-precision limits (approximately 15–17 significant decimal digits). For scientific research, use dedicated precision software.