# What is this in scientific notation?

Converts a number to scientific notation (a × 10ⁿ with 1 ≤ |a| < 10), E notation, engineering notation, and plain decimals, with its significant figures and optional rounding.

- Page: https://www.acalculator.org/math/scientific-notation-calculator
- JSON spec: https://www.acalculator.org/math/scientific-notation-calculator.json
- Version: 2fb3ad50b4be

## Default answer

Example with the default inputs (Number 123,400): 123,400 in scientific notation is 1.234 × 10⁵.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| n | Number | The number: a decimal such as 0.00045 or 123,400, E notation such as 4.5e-4, or 4.5 × 10^-4. |
| sf | Round to significant figures | Optional: how many significant figures to keep, from 1 to 30. Leave empty to keep them all. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| scientific | Scientific notation | The number as a × 10ⁿ, with one non-zero digit before the point of a. |
| e | E notation | The same number as calculators and spreadsheets write it: a e n means a × 10ⁿ. |
| engineering | Engineering notation | The number with a power of ten that is a multiple of 3, matching the SI prefixes. |
| decimal | Decimal notation | The number written out in full, for powers from 10⁻³⁰ to 10³⁰. |
| coefficient | Coefficient a | The number before × 10ⁿ. |
| exponent | Exponent n | The power of ten. |
| figures | Significant figures | How many significant figures the result has. |

## Method

Write the number as a × 10ⁿ with 1 ≤ |a| < 10: n is the number of places the decimal point moves (left is positive), and a keeps the significant digits.

## Assumptions

- Leading zeros are never significant. Trailing zeros are significant after a decimal point (0.0010 has 2), and not in a whole number without one (1200 has 2; type 1200. to keep 4).
- Rounding to significant figures drops digits: more than half rounds up, less rounds down, and exactly half rounds to the even digit (2.45 → 2.4, 2.35 → 2.4).
- Asking for more significant figures than the number has adds zeros (1.2 to 4 figures is 1.200).
- Digits are exact as typed: nothing is rounded through binary floating point.
- Zero is written as 0 in every form. An empty box counts as 0.

## Worked examples

1. n = 123,400 gives scientific = 1.234 × 10⁵, e = 1.234e5, engineering = 123.4 × 10³, decimal = 123,400, exponent = 5, sf = 4. Source: hand calculation in content.mdx: move the point 5 places left; trailing zeros of a whole number do not count.
2. n = 37,000 gives scientific = 3.7 × 10⁴. Source: OpenStax Prealgebra 2e, section 10.5: 37,000 = 3.7 × 10⁴.
3. n = 0.00045 gives scientific = 4.5 × 10⁻⁴, e = 4.5e-4, engineering = 450 × 10⁻⁶, decimal = 0.00045, sf = 2. Source: hand calculation in content.mdx: move the point 4 places right; leading zeros do not count.
4. n = 299792458, sf = 3 gives scientific = 3.00 × 10⁸, e = 3.00e8, sf = 3. Source: NIST CODATA: the speed of light is exactly 299 792 458 m/s; hand calculation in content.mdx: 2.99|79… rounds up to 3.00.
5. n = 6.02214076e23 gives scientific = 6.02214076 × 10²³, engineering = 602.214076 × 10²¹, decimal = 602,214,076,000,000,000,000,000, sf = 9. Source: NIST CODATA 2022: the Avogadro constant is exactly 6.022 140 76 × 10²³ per mole.
6. n = 9.1093837139 × 10^-31, sf = 4 gives scientific = 9.109 × 10⁻³¹, e = 9.109e-31, exponent = -31. Source: NIST CODATA 2022: the electron mass is 9.109 383 7139 × 10⁻³¹ kg; rounded to 4 figures by hand.
7. n = 2.45, sf = 2 gives scientific = 2.4 × 10⁰. Source: OpenStax Chemistry 2e, section 1.5: a dropped digit of exactly 5 rounds to the even digit, so 2.45 → 2.4.
8. n = -0.0010 gives scientific = −1.0 × 10⁻³, e = -1.0e-3, decimal = −0.0010, sf = 2. Source: OpenStax Chemistry 2e, section 1.5: a trailing zero after the decimal point is significant.

## FAQ

### How do I write a number in scientific notation?

Move the decimal point until one non-zero digit is in front of it, and count the places you moved it. Moving left makes the power positive, moving right makes it negative. 37,000 becomes 3.7 × 10⁴ (4 places left), and 0.00045 becomes 4.5 × 10⁻⁴ (4 places right).

### What does E mean on a calculator?

E (or e) means "times ten to the power". 4.5E-4 is 4.5 × 10⁻⁴ = 0.00045, and 1.234E5 is 1.234 × 10⁵ = 123,400. Calculators and spreadsheets use it when a number is too long for the screen.

### How do I convert scientific notation back to a normal number?

Move the decimal point by the power: right for a positive power, left for a negative one, filling gaps with zeros. 6.02214076 × 10²³ is 602,214,076,000,000,000,000,000. Type the scientific form and read the decimal notation.

### What is engineering notation?

Scientific notation where the power of ten is a multiple of 3, so the number before it is from 1 to 999. The powers match the SI prefixes: 450 × 10⁻⁶ m is 450 micrometres, and 123.4 × 10³ W is 123.4 kilowatts.

### How many significant figures does a number have?

Count every digit from the first non-zero digit, except trailing zeros of a whole number written without a decimal point. 0.00045 has 2, 0.0010 has 2, 123,400 has 4, and 1,200. (with the point) has 4. Scientific notation removes the doubt: 1.2 × 10³ has 2 and 1.200 × 10³ has 4.

### How is a number rounded to significant figures?

Keep that many digits and look at what is dropped: more than half rounds up, less than half rounds down, and exactly half rounds to the even digit. 299,792,458 to 3 figures is 3.00 × 10⁸; 2.45 to 2 figures is 2.4, and 2.35 is also 2.4.

### Is scientific notation the same as standard form?

In the UK, "standard form" means scientific notation, a × 10ⁿ. In the US, "standard form" usually means an ordinary decimal, such as 123,400.

## Sources

- OpenStax, Prealgebra 2e, §10.5 Integer Exponents and Scientific Notation (a × 10ⁿ with 1 ≤ a < 10 and n an integer; 37,000 = 3.7 × 10⁴). https://openstax.org/books/prealgebra-2e/pages/10-5-integer-exponents-and-scientific-notation
- OpenStax, Chemistry 2e, §1.5 Measurement Uncertainty, Accuracy, and Precision (significant figures; rounding a dropped 5 to the even digit). https://openstax.org/books/chemistry-2e/pages/1-5-measurement-uncertainty-accuracy-and-precision
- NIST, CODATA 2022 values of the fundamental physical constants: speed of light 299 792 458 m/s (exact), Avogadro constant 6.022 140 76 × 10²³ mol⁻¹ (exact), electron mass 9.109 383 7139 × 10⁻³¹ kg. https://physics.nist.gov/cuu/Constants/
- NIST, Metric (SI) Prefixes (kilo 10³, mega 10⁶, milli 10⁻³, micro 10⁻⁶). https://www.nist.gov/pml/owm/metric-si-prefixes