Significant Figures Calculator

How it works

Significant figures are the digits of a number that carry measured information: every non-zero digit, every zero between non-zero digits, and every zero after the decimal point that follows a non-zero digit. Leading zeros only locate the decimal point and never count, so 0.00123 has three significant figures, the same as 1.23 or 123,000 written as 1.23 × 10^5.

To round to n significant figures the calculator takes the first n digits, looks at the next one, and rounds half up on the decimal digits you typed. Where the result ends in zeros to the left of the decimal point, standard notation cannot show how many of them are significant; the scientific-notation form always can, which is why both are given.

The count shown for the value as entered is the count of digits the number field can carry. A numeric field drops trailing zeros after the decimal point, so 1.500 arrives as 1.5 and counts as 2, and trailing zeros of a whole number such as 1,500 are treated as place-holders (2 significant figures), following the usual convention that they are not significant unless a decimal point or bar marks them.

Formula

e = ⌊log10 |x|⌋                      (exponent of the leading digit)
scaled = |x| × 10^(n − 1 − e)        (first n digits left of the point)
r = ⌊scaled + 0.5⌋                   (round half up)
rounded = sign(x) × r × 10^(e − n + 1)
scientific: d.ddd × 10^e with n digits

Example

1,234.5678 to 3 significant figures: the leading digit is in the thousands (e = 3), so scale by 10^(3 − 1 − 3) = 10^−1 to get 123.45678, round to 123, and scale back to 1230. In scientific notation that is 1.23 × 10^3 (E notation 1.23e3); the trailing zero of 1230 is a place-holder, not a significant digit. The value as entered has 8 significant figures.

31.57 to 2 significant figures is 32 (3.2 × 10^1), and 8.1649 to 3 significant figures is 8.16 (8.16 × 10^0). 0.0012345 to 3 is 0.00123 (1.23 × 10^−3). 9.99 to 2 carries over to 10 (1.0 × 10^1).

Assumptions and limitations

• Ties are rounded half up (away from zero on the magnitude): 0.051065 to 4 significant figures gives 0.05107. Some textbooks and IEEE 754 round ties to even, which would give 0.05106; the two rules differ only when the dropped part is exactly 5.
• The number field cannot carry trailing zeros after a decimal point or a trailing decimal point, so the count of significant figures in the value as entered is a lower bound: 1.500 is received as 1.5 (2), and 1,500 counts as 2 under the convention that bare trailing zeros are not significant.
• Rounding is done on the decimal digits you typed, so 1.005 to 3 significant figures is 1.01, not the 1.00 that rounding its binary value produces.
• Zero is not accepted: it has no leading digit, so its number of significant figures is a matter of convention rather than calculation.
• Up to 15 significant figures can be kept, the precision a double-precision number reliably carries.

Frequently asked questions