Scientific Notation Converter
Convert between decimal, scientific, E- and engineering notation, count significant figures, and multiply or divide in scientific notation.
- Scientific
- 4.5600 × 10-4
- E-notation
- 4.5600E-4
- Engineering
- 456.00 × 10-6 (SI prefix: micro (µ))
- Decimal
- 0.00045600
Significant figures as typed: 5
Updated Formula covered by automated tests
Scientific notation writes any number as a value between 1 and 10 multiplied by a power of ten, so 0.00045600 becomes 4.5600 × 10^-4 and 602,214,076,000,000,000,000,000 becomes 6.02214076 × 10^23. Type a number in ordinary decimal form, in E-notation such as 1.6E-19, or as 3.0 × 10^8, and the converter shows the same value in scientific notation, E-notation, engineering notation with its SI prefix, and as a full decimal.
It also counts significant figures, rounds to the number of figures you choose, and multiplies or divides two numbers while reporting the answer to the correct precision. Chemistry and physics homework, lab reports, spreadsheet exports and calculator displays all use these forms, and moving between them by hand is where misplaced decimal points creep in.
Ordinary floating-point arithmetic keeps only about 16 significant digits, which quietly corrupts very long numbers. This tool works on the digits as text and exact integers instead, so a 30-digit input keeps all 30 digits and a value like 1E-300 is expanded without turning into zero.
The formula
- m
- Mantissa (also called the coefficient or significand): the significant digits with one nonzero digit before the decimal point
- n
- Exponent: how many places the decimal point moved; positive for large numbers, negative for numbers smaller than 1
- k
- Integer chosen so the engineering exponent 3k matches an SI prefix such as kilo (10^3) or micro (10^-6)
- a, b, p, q
- Mantissas and exponents of the two numbers being multiplied or divided
To convert a decimal, move the decimal point until exactly one nonzero digit sits to its left, and count the moves: moves to the left give a positive exponent and moves to the right a negative one. E-notation is the same thing with the "× 10^" replaced by the letter E, a convention from early computers and printers that could not show superscripts. Engineering notation restricts the exponent to multiples of three so that the mantissa reads directly with a metric prefix: 4.7 × 10^4 ohms becomes 47 × 10^3, or 47 kilohms.
Significant figures follow the usual rules: nonzero digits count, zeros between them count, leading zeros never count, and trailing zeros count only when a decimal point is written. A whole number such as 1500 is ambiguous, which is one reason scientists prefer 1.5 × 10^3 or 1.500 × 10^3, where the precision is explicit. When multiplying or dividing measured values, the answer keeps as many significant figures as the least precise input.
Worked example
Aisha is estimating how far light travels in a year for a physics assignment. She uses the speed of light rounded to 2.998 × 10^8 m/s and a year of 3.156 × 10^7 seconds, both with four significant figures.
- 1 Multiply the mantissas: 2.998 × 3.156 = 9.461688.
- 2 Add the exponents: 8 + 7 = 15, giving 9.461688 × 10^15 m.
- 3 Both inputs have four significant figures, so round the mantissa to four: 9.461688 becomes 9.462.
- 4 Write the forms: 9.462 × 10^15 m, or 9.462E+15 in E-notation; 15 is already a multiple of three, so in engineering form it is 9.462 petametres.
- 5 As a plain decimal that is 9,462,000,000,000,000 m, where the trailing zeros are placeholders, not measured digits.
Result: Aisha reports a light-year as about 9.462 × 10^15 m, which agrees with the accepted 9.4607 × 10^15 m to within the precision of her rounded year length.
Before you rely on the result
- The significant-figure rule applied to products and quotients is the classroom convention; formal uncertainty analysis in a lab may call for a different number of digits.
- Division results that never terminate (such as 1 ÷ 3) are computed to about 40 digits and then rounded, so the full-precision display is accurate but not infinitely long.
- Plain decimal output is limited to exponents between -400 and 400 to keep the page readable; beyond that, only scientific and E-notation are shown.
Frequently asked questions
Why does my calculator show 6.02E23 instead of 6.02 × 10^23?
They are the same number. Calculators, spreadsheets and programming languages print E (or e) as shorthand for "times ten to the power", because a screen line cannot easily show a raised exponent. 6.02E23, 6.02e+23 and 6.02 × 10^23 are interchangeable, and this tool accepts any of them as input.
Is 1500 two significant figures or four?
Written that way it is ambiguous. Most textbooks treat the trailing zeros as not significant (two figures) unless a decimal point is added, as in 1500., or the number is written as 1.500 × 10^3. The converter flags such inputs so you can decide which reading your teacher or lab manual expects.
How is engineering notation different from scientific notation?
Scientific notation keeps one digit before the decimal point and allows any exponent. Engineering notation allows one to three digits before the point so the exponent is always a multiple of three, matching SI prefixes. Electronics and electrical work use it because 330 × 10^-9 F reads straight off as 330 nanofarads.
Why not just type huge numbers into a normal calculator?
Standard double-precision numbers carry roughly 15 to 17 significant digits, so a 25-digit integer loses its last digits and anything below about 1E-308 rounds to zero. Here the digits are handled as text and exact integers, so long inputs and extreme exponents keep every digit you typed.
Sources
- NIST: Metric (SI) Prefixes (nist.gov)
- NIST CODATA: speed of light in vacuum (physics.nist.gov)