Scientific Notation Calculator

A way of writing any number as a coefficient times a power of ten, where the coefficient is at least 1 and less than 10. The exponent counts how many places the decimal point has moved: positive for a large number, negative for a small one. So 250,000 is 2.5 × 10^5 and 0.00015 is 1.5 × 10^-4. It exists because the alternative is unreadable — the mass of an electron written out is a decimal point followed by thirty zeros — and because it makes the precision of a measurement explicit, since every digit in the coefficient is a significant figure and there is nowhere for a placeholder zero to hide.

Multiply the coefficients, add the exponents, then move the decimal point if the coefficient has landed outside 1 to 10. For 6.02 × 10^23 times 3.11 × 10^8: 6.02 × 3.11 = 18.7222 and 23 + 8 = 31, giving 18.7222 × 10^31, which is 1.87222 × 10^32 once the point moves one place left and the exponent takes the place instead. Division is the same in reverse — divide the coefficients and subtract the exponents. The exponents are added as ordinary integers, which is why this stays exact even when the answer is far too large for a computer to store as a number.

You cannot add the coefficients until both numbers are on the same power of ten, because 6 apples and 3 oranges is not 9 of anything. Rewrite the smaller one to match the larger exponent, add the coefficients, then put the result back into standard form. So 1.23 × 10^4 plus 4.5 × 10^3 becomes 1.23 × 10^4 plus 0.45 × 10^4, which is 1.68 × 10^4. Subtraction works identically. This is also where the precision rule changes: addition keeps the coarser decimal place rather than the smaller number of significant figures.

Because the significant-figure rules say it should, and reporting the extra digits would claim a precision the inputs never had. Multiplication and division keep the count: the answer gets as many significant figures as whichever input had the fewest, so a factor known to one figure caps the whole product at one. Addition and subtraction keep the place instead: the answer stops where the coarser input stopped, so 6.02 × 10^23 plus 3 × 10^8 is still 6.02 × 10^23, because the second number is far below the place the first one stops making claims about. The working on this page shows the unrounded answer as well, so you can see exactly what was given up.

The same idea with the exponent restricted to a multiple of three, so that it lines up with the SI prefixes. 6.02 × 10^23 in engineering notation is 602 × 10^21, which reads as 602 zetta-something; 1.5 × 10^-7 is 150 × 10^-9, which is 150 nano-something. The coefficient then sits between 1 and 1000 instead of between 1 and 10. Electronics and instrumentation use it because the answer arrives already in the unit people speak in — kilohms, microfarads, nanoseconds — rather than needing a further conversion in your head.

Because the coefficient boxes take numbers, and a stored number has no memory of the zeros it was written with. 3.0 and 3 are the same value in a computer, so this page reads either as one significant figure and says so in a note whenever it matters. The rule is applied to what it can actually see. If your 3.0 was measured to two figures, count the figures yourself and read the working against that count — or use the significant figures calculator, which takes the number as text and keeps every trailing zero you type.

Yes, and that is the main reason the coefficient and the exponent are separate fields rather than one box. A double-precision number stops at about 1.8 × 10^308, so evaluating 6.02 × 10^300 times 3 × 10^300 the ordinary way gives Infinity and loses every digit of a perfectly well-defined answer. Here the exponents are added as integers — exact at any size — and the coefficients are multiplied on their own, where they are small and well behaved, so the answer comes back as 2 × 10^601. The same holds underneath, where results below about 5 × 10^-324 would otherwise collapse to zero.

You can lose every significant figure you had, and this calculator reports that rather than hiding it. 1.5 × 10^5 minus 1.49 × 10^5 is 1000 as plain arithmetic, but 1.5 × 10^5 says nothing at all about the hundreds place — it stops at the ten-thousands — so no digit of that 1000 is significant. This is called catastrophic cancellation: the leading digits agree and cancel, leaving only the digits that were uncertain to begin with. It is why measurements intended to be subtracted are taken to more figures than the difference itself appears to need.