Scientific Notation Calculator
Convert any number to scientific notation, expand it back to a decimal, or perform arithmetic (add, subtract, multiply, divide, exponent, square root) directly in scientific notation. Results appear in scientific, E-notation and engineering notation simultaneously, with a full SI prefix label.
Formula
Worked example
Take 149,597,870,700 (one astronomical unit in metres). The decimal point moves 11 places left to leave 1.49597870700, so b = 11 and the result is 1.496 × 10^11 to four significant figures, or 1.496e+11 in E-notation. In engineering notation the nearest multiple-of-3 exponent is 9, giving 149.6 × 10^9 (giga).
How scientific notation works
Scientific notation writes any real number as a coefficient multiplied by a power of ten: a × 10^b. The coefficient a is normalized so its absolute value is at least 1 but less than 10, leaving exactly one non-zero digit before the decimal point. The exponent b counts how many places the decimal point must move to reach that normalized form: positive for large numbers, negative for fractions smaller than one. This compact form keeps very large and very small quantities readable and makes their relative scale obvious at a glance. For example, the mass of the Earth (5,972,000,000,000,000,000,000,000 kg) becomes 5.972 × 10^24 kg.
Converting in both directions
To convert a decimal to scientific notation, shift the decimal point until one non-zero digit remains on the left, then use the number of shifts as the exponent (positive for left-shifts, negative for right-shifts). To reverse, multiply the coefficient by 10 raised to the exponent, which moves the decimal point that many places. The coefficient you enter does not need to be normalized first: this calculator re-normalizes automatically, so 14.96 × 10^10 becomes 1.496 × 10^11.
Arithmetic in scientific notation
Multiplication and division are straightforward: multiply or divide the coefficients, then add or subtract the exponents, then normalize. For example, (3 × 10^4) × (2 × 10^5) = 6 × 10^9. Addition and subtraction first require both numbers to share the same exponent: rewrite the smaller-exponent number with the larger exponent, then add or subtract the coefficients. Exponentiation applies the power rule: (a × 10^b)^n = a^n × 10^(b×n). Square root applies the standard rule, adjusting the exponent so it remains even before taking the root. The arithmetic mode in this calculator handles all of these steps automatically and shows the working.
E-notation and engineering notation
E-notation (6.022e+23) is how most calculators, spreadsheets and programming languages display scientific notation because superscripts are hard to type. The letter "e" or "E" stands for "times ten to the power of". Engineering notation restricts the exponent to multiples of three (0, 3, 6, 9, 12...) so it aligns perfectly with SI metric prefixes: kilo (10^3), mega (10^6), giga (10^9), tera (10^12), milli (10^-3), micro (10^-6), and so on. The coefficient in engineering notation can therefore range from 1 to 999 rather than 1 to 9, which is the trade-off for prefix alignment.
Significant figures
Significant figures control how many meaningful digits the coefficient retains, which is how scientists express the precision of a measurement. Rounding 6.022140857 × 10^23 to four significant figures gives 6.022 × 10^23 without changing the order of magnitude. When multiplying or dividing in scientific notation, the result should be rounded to the same number of significant figures as the least-precise input. When adding or subtracting, the result should be rounded to the same number of decimal places as the least-precise input after aligning exponents. The significant figures setting in this calculator applies uniformly to the coefficient of the output.
Scientific and engineering notation reference
| Quantity | Decimal | Scientific notation | Engineering notation | SI prefix |
|---|---|---|---|---|
| Proton charge | 0.000000000000000000160217 | 1.602 × 10^-19 | 160.2 × 10^-21 | zepto (z) |
| Nanometre | 0.000000001 | 1 × 10^-9 | 1 × 10^-9 | nano (n) |
| Millimetre | 0.001 | 1 × 10^-3 | 1 × 10^-3 | milli (m) |
| One thousand | 1,000 | 1 × 10^3 | 1 × 10^3 | kilo (k) |
| Speed of light (m/s) | 299,792,458 | 2.998 × 10^8 | 299.8 × 10^6 | mega (M) |
| Avogadro's number | 602,214,076,000,000,000,000,000 | 6.022 × 10^23 | 602.2 × 10^21 | zetta (Z) |
| Gravitational const. | 0.0000000000667 | 6.674 × 10^-11 | 66.74 × 10^-12 | pico (p) |
Common quantities shown in scientific, E-notation and engineering notation with SI prefix.
Frequently asked questions
What is the difference between scientific notation and E-notation?
They represent the same value. Scientific notation uses a superscript power of ten (6.022 × 10^23), while E-notation writes it as 6.022e+23 because calculators and spreadsheets cannot easily display a superscript. The number after the "e" is the exponent. Both are mathematically identical.
How do I add or subtract numbers in scientific notation?
First rewrite both numbers with the same exponent (use the larger one). Then add or subtract the coefficients and keep the shared exponent. Finally normalize the result so the coefficient stays between 1 and 10. For example, (3.2 × 10^5) + (4.8 × 10^4) = (3.2 × 10^5) + (0.48 × 10^5) = 3.68 × 10^5. The arithmetic mode in this calculator does all of that automatically.
How do I multiply or divide numbers in scientific notation?
For multiplication, multiply the coefficients and add the exponents: (3 × 10^4) × (2 × 10^5) = 6 × 10^9. For division, divide the coefficients and subtract the exponents: (6 × 10^9) / (2 × 10^5) = 3 × 10^4. Normalize if the resulting coefficient falls outside 1 to 10.
What is engineering notation and how is it different from scientific notation?
Engineering notation is a variant where the exponent is always a multiple of three (0, 3, 6, 9...). This keeps the exponent aligned with SI prefixes like kilo (10^3), mega (10^6) and giga (10^9), making it easier to read in engineering and electronics contexts. The coefficient in engineering notation can range from 1 to 999, unlike scientific notation where it must be between 1 and 10.
How do I write a small number like 0.00042 in scientific notation?
Move the decimal point right until one non-zero digit is to the left of it. Here, 0.00042 becomes 4.2 after moving four places right. Moving right means the exponent is negative, so the result is 4.2 × 10^-4. In E-notation that is 4.2e-4.
Does the coefficient have to be between 1 and 10?
In standard normalized scientific notation, yes. If you enter 14.96 × 10^10 this calculator re-normalizes it to 1.496 × 10^11 automatically. Engineering notation is the main exception: its coefficient can reach up to 999 so the exponent stays a multiple of three.