What scientific notation actually does
Scientific notation exists because ordinary digits stop being useful once a number gets extreme enough. Avogadro's number has 24 digits before the decimal point; the mass of an electron needs 30 zeros after it before a meaningful digit turns up. Nobody can compare, multiply or even read numbers like that reliably in plain digit form, so scientists rewrite them a different way: a single digit from 1 to 9, followed by whatever comes after the decimal point, multiplied by a power of ten that carries the actual size.
The mantissa carries the precise digits; the exponent carries the scale. Splitting a number this way means you can compare the size of two numbers just by glancing at their exponents, and multiply huge or tiny quantities together by doing simple arithmetic on the exponents instead of chasing zeros across a page.
The same number, four ways
This tool always shows a converted number in four forms, because different tools and different audiences expect different ones.
- Standard (plain digits) is the everyday form, fine for anything short enough to read comfortably, unwieldy once the digit count climbs into double figures.
- Scientific notation is the mantissa-and-exponent form above, always exactly one digit before the decimal point.
- E notation is what a calculator or spreadsheet prints when its display cannot show a proper superscript power of ten: the "×10" is simply replaced with the letter E, so 3.45 × 10⁵ becomes 3.45E5. It means exactly the same thing; nothing about the value has changed, only the typography.
- Engineering notation is scientific notation with a rule added: the exponent must always be a multiple of three. That keeps the mantissa between 1 and 1000 rather than 1 and 10, and it lines the number up neatly with the metric unit prefixes, kilo, mega, milli, micro and the rest, which is why engineers favour it when a value is about to be quoted with a physical unit attached.
How the conversion works: moving the point
Converting to scientific notation is really just counting how many places the decimal point has to move to leave a single digit in front of it, and in which direction. Moving the point to the left makes the exponent positive, because the point started further right than it ends up, in a genuinely large number. Moving it to the right makes the exponent negative, for a number smaller than one.
Nothing here is rounded. The tool moves the point the same way you would on paper, by counting digits, which is also why it can take a 30-digit number and convert it without a single digit drifting, something that trips up tools that quietly convert your number to a binary floating-point value first.
Engineering notation and the SI prefixes
Because engineering notation locks the exponent to a multiple of three, every value lines up with a metric prefix used on everything from a food label to a data sheet: 3 is kilo, 6 is mega, −3 is milli, −6 is micro. A resistor marked 4.7 kΩ is 4.7 × 103 ohms; a capacitor marked 100 nF is 100 × 10−9 farads. Reading either figure straight off an engineering-notation result is the entire point of using it.
| Exponent | Prefix | Symbol | Example |
|---|---|---|---|
| 10²⁴ | yotta | Y | 1 YB ≈ every byte of data on Earth, several times over |
| 10⁹ | giga | G | 1 GB, a fairly ordinary file |
| 10³ | kilo | k | 1 km = 1,000 m |
| 10⁰ | (none, the base unit) | — | 1 m |
| 10⁻³ | milli | m | 1 mm = 0.001 m |
| 10⁻⁶ | micro | µ | 1 µm, roughly a bacterium |
| 10⁻⁹ | nano | n | 1 nm, a few atoms wide |
| 10⁻¹² | pico | p | 1 pF, a tiny capacitor |
The full run from smallest to largest in common use is yocto, zepto, atto, femto, pico, nano, micro, milli, (none), kilo, mega, giga, tera, peta, exa, zetta, yotta, at exponents −24 up to 24 in steps of three; the results panel above names whichever one applies to your number, when one exists.
Why scientific notation is honest about precision
Write a number as 1200 and you have not actually said how precise it is. Those trailing zeros might be measured digits, or they might just be holding the decimal point in place while meaning "somewhere around 1200". Plain digits cannot tell the two apart. Scientific notation can, because the mantissa only ever contains digits you chose to write: 1.200 × 103 states four significant figures outright, while 1.2 × 103 states two, and both are unambiguous the moment you see them.
This converter follows that convention by default: a bare whole number's trailing zeros are treated as placeholders unless you add a decimal point, so 345,000 converts to 3.45 × 105, not 3.45000 × 105. Typing 345000. or 3.45000e5 instead keeps every zero, because a decimal point is a deliberate statement that those digits were actually measured. If you are not sure how many figures a rounded result should really carry, our Rounding Calculator covers significant figures and the school-versus-banker's rounding conventions in more depth.
From an electron to the whole world
A handful of constants show just how much ground scientific notation has to cover in ordinary scientific work.
| Quantity | Value |
|---|---|
| Mass of an electron | about 9.109 × 10−31 kg |
| Typical transistor feature size on a modern chip | around 5 × 10−9 m (roughly, node names are marketing terms rather than exact measurements) |
| Avogadro constant | 6.02214076 × 1023 per mole (exact, by definition) |
| World population | around 8.2 × 109 people (and still climbing) |
| Average distance from the Earth to the Sun | 1.496 × 108 km |
Set side by side like that, the electron and the world population sit 40 orders of magnitude apart, a gap that plain digits would need a line long enough to fill this page just to write out.
Multiplying, dividing, adding and subtracting in scientific notation
Once two numbers are already in mantissa-and-exponent form, the Operate tab above works through all four operations at once, because each one follows its own short rule.
Multiplying multiplies the mantissas and adds the exponents. (3 × 104) × (2 × 105) is (3 × 2) × 104+5, which is 6 × 109. The mantissa product will not always land back between 1 and 10, though: (8 × 104) × (5 × 103) gives a mantissa of 8 × 5 = 40 first, which has to be renormalised, moving the point one place and adding one to the exponent, to arrive at 4 × 108.
Dividing divides the mantissas and subtracts the exponents, the mirror image of multiplying.
Adding and subtracting cannot touch the exponents at all until they match. 3 × 104 and 2 × 103 are not directly comparable while one counts in ten-thousands and the other in thousands, so the smaller exponent becomes the common one first: 3 × 104 is rewritten as 30 × 103, which can now be added to 2 × 103 directly, giving 32 × 103, renormalised to 3.2 × 104. Skipping that rewrite and adding the mantissas as they stood, 3 and 2, would silently throw away an entire factor of ten.
Common mistakes
- Reading "e5" as "times 5". The e in E notation stands for exponent, not a multiplication by 5; 3.4e5 means 3.4 × 105 (340,000), not 3.4 × 5.
- Treating a negative exponent as a negative number. 4 × 10−4 is 0.0004, a small positive number, not a negative one. The minus sign belongs to the exponent and says "small", not to the value itself.
- Being surprised by E notation on a calculator display. Most calculators and spreadsheets switch to E notation automatically once a result gets too long for the screen; it is a display convenience, not a different kind of number, and converts straight back to the same value in plain digits.
Questions people ask
What is scientific notation and why is it used?
It writes any number as a single digit followed by a decimal fraction, multiplied by a power of ten, such as 3.45 × 105 for 345,000. It is used because it makes very large and very small numbers short to write, easy to compare by exponent, and unambiguous about how many figures are actually known.
What is the difference between scientific notation and standard form?
They are the same thing under two different names, mantissa times a power of ten with exactly one digit before the decimal point. "Standard form" is the more common term in UK schools; "scientific notation" is used more widely elsewhere, including in this tool's other three notations for comparison.
What is E notation, and why does my calculator show it?
E notation writes the power of ten after the letter E instead of as "×10 to the power of", because most calculator and spreadsheet displays cannot show a proper superscript. 3.45E5 means exactly 3.45 × 105; it usually appears automatically once a result is too long to fit the screen in plain digits.
What is engineering notation, and how is it different from scientific notation?
Engineering notation is scientific notation with the exponent forced to a multiple of three, which keeps the mantissa between 1 and 1000 instead of 1 and 10. That lines every value up with a metric prefix (kilo, mega, milli, micro and so on), which is why it is the preferred form in engineering and electronics.
Why does 345,000 become 3.45 × 10⁵ and not 3.45000 × 10⁵?
Written as a bare whole number with no decimal point, the trailing zeros in 345,000 are ambiguous, they could be measured digits or just placeholders, so this calculator treats them as placeholders by default and keeps only the figures that are clearly significant. Typing the number with an explicit decimal point, such as 345000. or 3.45000e5, keeps every zero instead.
Does a negative exponent mean the number is negative?
No. A negative exponent means small, not negative: 4 × 10−4 equals 0.0004, a positive number less than one. Whether the number itself is negative depends only on the sign in front of the mantissa.
How do you add two numbers in scientific notation?
Match the exponents first by rewriting the number with the larger exponent in terms of the smaller one, then add the mantissas directly, then renormalise if needed. 3 × 104 plus 2 × 103 becomes 30 × 103 plus 2 × 103, which is 32 × 103, or 3.2 × 104 renormalised.
The Convert tab reads your number as text and moves the decimal point with exact string and integer arithmetic, never a floating-point approximation, so long numbers keep every digit. The Operate tab uses the same exact arithmetic for multiplying, adding and subtracting; only the quotient, which can be a genuinely never-ending decimal, is rounded for display.