What scientific notation is
Scientific notation writes a number as a number from 1 to just under 10, times a power of ten: 93,000,000 is 9.3 × 10⁷ and 0.000456 is 4.56 × 10⁻⁴. The exponent says how many places the decimal point moved. A positive exponent means a large number, a negative exponent a small one, and 10⁰ is 1. It turns long strings of zeros into something you can read, compare and type without miscounting.
In the GCSE mathematics curriculum for England this same form is called standard form, written A × 10ⁿ with 1 ≤ A < 10 and n an integer.
How to convert, both ways
To scientific notation: move the decimal point until one nonzero digit stands in front of it. Count the places. Moving the point to the left, as in 93,000,000 → 9.3, gives a positive exponent, here 7. Moving it to the right, as in 0.000456 → 4.56, gives a negative exponent, here −4.
Back to a decimal: do the reverse. For a positive exponent, move the point that many places to the right and fill with zeros: 6.02 × 10⁵ = 602,000. For a negative exponent, move it to the left: 3.2 × 10⁻⁵ = 0.000032.
Here are some conversions done by the converter itself:
| Typed | Scientific | E notation | Engineering | Sig figs |
|---|---|---|---|---|
| 93,000,000 | 9.3 × 10⁷ | 9.3E7 | 93 × 10⁶ | 2 |
| 0.000456 | 4.56 × 10⁻⁴ | 4.56E-4 | 456 × 10⁻⁶ | 3 |
| 6.0221e23 | 6.0221 × 10²³ | 6.0221E23 | 602.21 × 10²¹ | 5 |
| −0.0050 | −5.0 × 10⁻³ | -5.0E-3 | −5.0 × 10⁻³ | 2 |
| 1500. | 1.500 × 10³ | 1.500E3 | 1.500 × 10³ | 4 |
E notation
E notation puts the power of ten after the letter E so the number fits on one line: 4.56E-4 is 4.56 × 10⁻⁴ and 6.0221E23 is 6.0221 × 10²³. Lowercase e means the same, and JavaScript, for example, accepts both 6.02e23 and 6.02E23 as numbers. The converter reads both, as well as × 10^ with an x, a star or the real times sign.
Engineering notation and metric prefixes
Engineering notation only uses exponents that are multiples of 3, so the number in front runs from 1 to under 1,000. 45,600 is 45.6 × 10³ and 0.000456 is 456 × 10⁻⁶. Each multiple of 3 has a metric prefix: 10³ is kilo (k), 10⁶ mega (M), 10⁻³ milli (m), 10⁻⁶ micro (μ) and 10⁻⁹ nano (n). So 456 × 10⁻⁶ meters is 456 micrometers. The converter names the prefix for every exponent from 10⁻³⁰ (quecto) to 10³⁰ (quetta), the range of prefixes listed by NIST.
Keeping the significant figures
The digits in front of the power of ten are exactly the significant figures of the number. 4.50 × 10³ has 3, and the final zero says the value is known to the tens. That makes scientific notation the clearest way to show precision, and it is why this converter works on the digits you typed instead of a rounded computer number. 0.0004560 stays 4.560 × 10⁻⁴, with its final zero.
Whole numbers with zeros at the end are the exception: 93,000,000 does not say whether the zeros were measured. The usual rule treats them as placeholders, so the converter gives 9.3 × 10⁷. If they were measured, choose the number of significant figures, or type a decimal point at the end, like 1500., to keep them. When a decimal result has to end in placeholder zeros, the converter puts a bar over the last significant digit, a marking described in the significant figures calculator.
To round to fewer digits, choose the number of significant figures. 6.0221 × 10²³ to 3 figures is 6.02 × 10²³. The converter rounds the exact digits, with 5 or more rounding up.
Exact for huge and tiny numbers
If a converter first turns your number into a 64-bit binary floating point number, only about 15 to 17 significant digits survive, most decimals are stored slightly off, and 1e-500 becomes 0. This converter keeps your digits as text and only moves the decimal point, so 123456789012345678901234567891 keeps all 30 digits and 1e-500, far below what such a number can hold, converts without trouble. It works for powers of ten from 10⁻¹⁰⁰⁰ to 10¹⁰⁰⁰ and up to 200 digits; very long decimals are shown in scientific notation only.
How to use the scientific notation converter
- 1Type or paste a number in any form: a plain decimal, a number with thousands separators, E notation, or a × 10^n.
- 2Keep "As typed" to preserve your significant figures, or pick how many figures to round to.
- 3Read the result in scientific notation, E notation, engineering notation with its metric prefix, and as a decimal, and follow the steps.
- 4Copy any single form with its own button, copy all of them at once, or copy a link to the conversion.
Sources
- GCSE mathematics: subject content and assessment objectives, Department for Education, 2013: standard form A × 10ⁿ
- Metric (SI) prefixes, NIST: prefix names and symbols from quecto to quetta
Frequently asked questions
How do you convert a number to scientific notation?
Move the decimal point until exactly one nonzero digit stands in front of it, then count how many places you moved it. That count is the exponent: positive if you moved the point to the left, as for big numbers, and negative if you moved it to the right, as for small ones. 93,000,000 becomes 9.3 × 10⁷ and 0.000456 becomes 4.56 × 10⁻⁴.
What is E notation?
E notation writes the power of ten after the letter E, so it fits on one line without superscripts: 4.56E-4 means 4.56 × 10⁻⁴ and 6.02E23 means 6.02 × 10²³. JavaScript, for one, accepts 6.02e23 as a number. The E has nothing to do with the number e; read it as times ten to the power of.
What is the difference between scientific and engineering notation?
Scientific notation keeps one digit in front of the decimal point, so the number in front is from 1 to under 10. Engineering notation only allows exponents that are multiples of 3, so the number in front is from 1 to under 1,000: 45,600 is 4.56 × 10⁴ in scientific notation and 45.6 × 10³ in engineering notation. Multiples of 3 match the metric prefixes like kilo and milli.
How many significant figures does a number in scientific notation have?
Every digit in front of the power of ten counts, and the power of ten does not. 4.50 × 10³ has 3 significant figures, because the trailing zero after the decimal point was written on purpose. That is why scientific notation is the clearest way to show precision: 4500 alone could have 2, 3 or 4.
How do I write a negative exponent as a decimal?
A negative exponent means the number is small: move the decimal point that many places to the left and fill with zeros. 3.2 × 10⁻⁵ is 0.000032. A negative sign in front of the whole number, as in −3.2 × 10⁵, is something else: it makes the value negative, here −320,000.
Is scientific notation the same as standard form?
In the GCSE mathematics curriculum for England, standard form means exactly this: a number written as A × 10ⁿ with 1 ≤ A < 10 and n a whole number.