Scientific Notation Converter
With number 0.00042, significant figures 3, scientific notation converter comes to 4.200000 — mantissa. It is reached in 6 steps, the last of which is 0.00042 / pow(10, -4), and each one is printed on the page with its numbers filled in. The formula is the one published by NIST Special Publication 811, not an approximation fitted to it.
A number in scientific, engineering and plain notation at once, with its order of magnitude and significant figures.
Formula and sources checked · How we check
Number 0, Significant figures 3
4.200000
Mantissa for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Order of magnitude
floor(log10(abs(0.00042)))-4- Mantissa, between 1 and 10
0.00042 / pow(10, -4)4.2- Engineering exponent
floor(-4 / 3) * 3-6- Engineering mantissa, 1 to 1000
0.00042 / pow(10, -6)420- Rounded to your significant figures
round(4.2 * pow(10, 3 - 1)) / pow(10, 3 - 1) * pow(10, -4)0- Decimal places that implies
max(0, 3 - 1 - -4)6
Worked example
0.00042 is 4.2 × 10⁻⁴ in scientific notation and 420 × 10⁻⁶ in engineering notation — the second being the one an electronics datasheet writes, because 10⁻⁶ is "micro" and 10⁻⁴ is nothing.
How to work it out yourself
- 1.Enter the number as you have it. The mantissa comes back between 1 and 10 and the exponent is the count of places the point moved.
- 2.Use engineering notation when the answer needs an SI prefix: exponents that are multiples of three map onto kilo, mega, milli and micro.
- 3.Set the significant figures to what your measurement actually supports. Writing 4.20 × 10⁻⁴ claims three figures of precision; writing 4.2 × 10⁻⁴ claims two.
The formula
- Order of magnitude
floor(log10(abs(0.00042))) - Mantissa, between 1 and 10
0.00042 / pow(10, -4) - Engineering exponent
floor(-4 / 3) * 3 - Engineering mantissa, 1 to 1000
0.00042 / pow(10, -6) - Rounded to your significant figures
round(4.2 * pow(10, 3 - 1)) / pow(10, 3 - 1) * pow(10, -4) - Decimal places that implies
max(0, 3 - 1 - -4)
Source: NIST Special Publication 811 — Guide for the Use of the International System of Units
Questions people actually ask
- What is the difference between scientific and engineering notation?
- The exponent. Scientific notation keeps the mantissa between 1 and 10, so any exponent is allowed. Engineering notation forces the exponent to a multiple of three and lets the mantissa run from 1 to 1000, which lines it up with the SI prefixes — 420 × 10⁻⁶ is 420 microunits, while 4.2 × 10⁻⁴ has no prefix at all.
- How many significant figures should I keep?
- As many as the measurement supports and no more. A ruler reading to the millimeter gives three figures on a 12 cm object, so writing 120.00 mm claims a precision the ruler never had. Trailing zeros in a plain decimal are ambiguous about this, which is one of the reasons scientific notation exists.
- What does the E in 4.2E-4 mean?
- "Times ten to the power of" — it is how calculators and programming languages write an exponent when superscripts are unavailable. 4.2E-4 and 4.2 × 10⁻⁴ are the same number. The letter is not Euler’s e, which is a different constant entirely and a real source of confusion.
- How do I multiply numbers in scientific notation?
- Multiply the mantissas and add the exponents, then renormalise if the mantissa leaves the 1-to-10 range. (3 × 10⁴)(4 × 10³) is 12 × 10⁷, which becomes 1.2 × 10⁸. Division subtracts the exponents. That is the whole reason the notation was invented — it turns awkward multiplications into additions.
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