What Is Scientific Notation and How Is It Calculated?
Scientific notation is a way of writing very large or very small numbers as a coefficient between 1 and 10 multiplied by a power of ten, written a × 10n. For example, 300,000 is written 3 × 105 and 0.00042 is written 4.2 × 10−4.
Arithmetic follows two rules. To multiply, (a × 10m) × (b × 10n) = (a × b) × 10m+n; to divide, (a × 10m) ÷ (b × 10n) = (a ÷ b) × 10m−n. Here a and b are the coefficients, and m and n are the integer exponents. To add or subtract, first rewrite both numbers with the same exponent, then add or subtract the coefficients. Finally, normalize the answer so the coefficient is between 1 and 10.
The International System of Units (SI), maintained by the BIPM, uses powers of ten in steps of three for its prefixes, from kilo (103) to quetta (1030), which is why engineering notation restricts exponents to multiples of 3. The prefixes ronna (1027) and quetta (1030) were added in 2022.
How to Use This Scientific Notation Calculator
Type two numbers, choose an operation, and the result updates instantly in four formats:
- First and second number: enter plain decimals (0.00045), E notation (4.5e-4), or the written form (4.5 x 10^-4).
- Operation: choose add, subtract, multiply, or divide.
- Results: scientific notation (a × 10ⁿ), E notation (aen, as used by calculators and programming languages), engineering notation (exponent a multiple of 3), and the full decimal form.
Scientific notation is used worldwide, but in some countries a comma is the decimal separator. This calculator expects a period (3.2e5) and shows exponents the same way in every locale.
Scientific, E, and Engineering Notation Compared
The same value can be written three ways. The table shows the number 48,700,000 in each format.
| Format | Example | Rule |
|---|---|---|
| Scientific | 4.87 × 10⁷ | Coefficient from 1 to under 10 |
| E notation | 4.87e7 | "e" replaces × 10^ |
| Engineering | 48.7 × 10⁶ | Exponent is a multiple of 3 |
According to NIST, the speed of light in a vacuum is exactly 299,792,458 m/s, or about 2.998 × 10⁸ m/s, and Avogadro's constant is 6.02214076 × 10²³ per mole. Both are routinely written in scientific notation because the plain decimals are impractical.
Frequently Asked Questions
How do you multiply numbers in scientific notation?
Multiply the coefficients and add the exponents. For example, (3 × 10⁴) × (2 × 10³) = 6 × 10⁷. If the new coefficient is 10 or more, shift the decimal point and raise the exponent, so (5 × 10⁴) × (4 × 10³) = 20 × 10⁷ = 2 × 10⁸.
How do you add and subtract numbers in scientific notation?
First give both numbers the same power of ten, then add or subtract the coefficients. For example, 3.2 × 10⁵ + 4 × 10⁴ becomes 3.2 × 10⁵ + 0.4 × 10⁵ = 3.6 × 10⁵. Adding the exponents, as you would for multiplication, is a common mistake.
What does the E mean on a calculator?
The E stands for "exponent" and means "times ten to the power of". So 6.5E4 means 6.5 × 10⁴ = 65,000, and 2E-3 means 2 × 10⁻³ = 0.002. Spreadsheets and programming languages use the same convention.
What is the difference between scientific and engineering notation?
Scientific notation always has one non-zero digit before the decimal point, so any exponent is allowed. Engineering notation only allows exponents that are multiples of 3, so the coefficient can be from 1 to 999. This matches SI prefixes: 4.7 × 10³ ohms is 4.7 kΩ and 2.2 × 10⁻⁶ farads is 2.2 µF.