acalculator

How many sig figs are in a number?

Count the significant figures in a number, round a number to a set number of them, or calculate with measured numbers and get the answer to the right precision.

Your numbers

What to do
Write 1.20 × 10^3 as 1.20e3 or 1.20 × 10^3.
Significant figures
4

0.004050 has 4 significant figures.

Decimal places
6
Scientific notation
4.050 × 10^-3
Significant digits
4050
In short
0.004050 has 4 significant figures.

Significant figures: 4. 0.004050 has 4 significant figures.

How to calculate

Counts the significant figures (sig figs) in a number, rounds a number to any number of them, and adds, subtracts, multiplies, or divides measured numbers under the sig fig rules.

Example with the default inputs (What to do Count, Number 0.004050): 0.004050 has 4 significant figures.

Method: Count: every non-zero digit, zeros between them, and trailing zeros after a decimal point. × and ÷: keep the fewest significant figures of the numbers. + and −: round to the last significant place of the less precise number. Halves round away from zero.

  • Leading zeros never count. Trailing zeros count only when the number has a decimal point (1200 has 2 significant figures, 1200. has 4, 1.200 × 10^3 has 4).
  • Every number is a measurement; the rules do not treat exact counts or defined constants differently.
  • Rounding works on the exact value; a value exactly halfway rounds away from zero (2.25 to 2 figures is 2.3), not to the even digit that some chemistry texts use.
  • The answer is written so that typing it back gives the same significant figures: 2.50 × 10^3 where 2500 would read as 2 figures.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. What to do Count, Number 0.004050 gives Significant figures 4, Decimal places 6, Scientific notation 4.050 × 10^-3, Significant digits 4050.Source: Rules from OpenStax Chemistry 2e, section 1.5: https://openstax.org/books/chemistry-2e/pages/1-5-measurement-uncertainty-accuracy-and-precision
  2. What to do Count, Number 1200 gives Significant figures 2, Scientific notation 1.2 × 10^3, Note The trailing zeros in 1200 are not significant, because it has no decimal point..
  3. What to do Count, Number 1200. gives Significant figures 4, Decimal places 0.
  4. What to do Round, Number 0.0034567, Significant figures 3 gives Answer 0.00346, Decimal places 5, Scientific notation 3.46 × 10^-3.
  5. What to do Round, Number 31.57, Significant figures 2 gives Answer 32.
  6. What to do Round, Number 8.1649, Significant figures 3 gives Answer 8.16.
  7. What to do Round, Number 0.051065, Significant figures 4 gives Answer 0.05107.
  8. What to do Round, Number 2468, Significant figures 2 gives Answer 2500, Scientific notation 2.5 × 10^3.
  9. What to do Round, Number 2496, Significant figures 3 gives Answer 2.50 × 10^3.
  10. What to do Round, Number 2499.7, Significant figures 4 gives Answer 2.500 × 10^3.
  11. What to do Calculate, Number 3.24, Operation ×, Second number 2.5 gives Answer 8.1, Significant figures 2, Unrounded result 8.1, Rule used Multiplication: the answer keeps as many significant figures as the number with the fewest, 2..
  12. What to do Calculate, Number 1.0023, Operation +, Second number 4.383 gives Answer 5.385, Significant figures 4, Decimal places 3, Unrounded result 5.3853.
  13. What to do Calculate, Number 486, Operation −, Second number 421.23 gives Answer 65, Significant figures 2, Unrounded result 64.77.
  14. What to do Calculate, Number 0.6238, Operation ×, Second number 6.6 gives Answer 4.1, Significant figures 2, Unrounded result 4.11708.
  15. What to do Calculate, Number 421.23, Operation ÷, Second number 486 gives Answer 0.867, Significant figures 3.
  16. What to do Calculate, Number 10, Operation ÷, Second number 3.0 gives Answer 3, Significant figures 1, Unrounded result ≈ 3.33333333333333.
  17. What to do Calculate, Number 5.0, Operation −, Second number 5.0 gives Answer 0.0, Decimal places 1.

How it works

Reading a number. A number is typed as a sign, digits (commas are allowed only between groups of three digits before the point), an optional decimal point with digits, and an optional power of ten written e3, E-3, × 10^3, x10^3, *10^3 or × 10^(-3) (spaces are ignored; the power is from −1000 to 1000). The value is exact: 0.004050 is 4050 ÷ 10^6.

Counting significant figures. Take the digits of the number (before the power of ten), without the sign and the point.

  1. Leading zeros (before the first non-zero digit) never count.
  2. Every digit from the first non-zero digit on counts, including zeros between non-zero digits,
  3. except trailing zeros when the number has no decimal point: in 1200 and in 120e3 they do not count; in 1200. and 1.20e3 they do.

The place of the last significant digit is the power of ten it stands for: in 0.004050 it is 10^-6 (millionths); in 1200 it is 10^2 (hundreds); in 1.20e3 it is 10^1 (tens). A number with no non-zero digit (0, 0.00) has no significant figures; its place is that of its last typed digit (ones for 0, hundredths for 0.00, thousands for 0e3), which is what adding and subtracting use.

Rounding always works on the exact value and rounds a value exactly halfway away from zero (half up on the size of the number): 2.25 to 2 figures is 2.3, and −2.25 is −2.3.

  • Round to n figures (1 to 30): find k with 10^k ≤ |x| < 10^(k+1), and round |x| to the place 10^(k − n + 1). A number with no significant figures (0, 0.00) cannot be rounded to significant figures, so it has no answer in Round mode too. When the rounding carries to one more digit (9.96 to 2 figures), the answer is 1.0 × 10^1, still with n figures.
  • Multiply or divide: the exact result is rounded to as many figures as the number with the fewest. A number with no significant figures (0) cannot be used, and dividing by 0 has no answer.
  • Add or subtract: the exact result is rounded to the place of the less precise number: the higher of the two places of their last significant digits (for 486 and 421.23, the ones place). The result may be 0.

How answers are written. Let the rounded answer have the digits q (its significant digits, starting with a non-zero digit) and the last place 10^p, and let k be the power of ten of its first digit.

  • Scientific notation: the first digit of q, then (when q has more than one digit) a point and the other digits, then × 10^ and k: 2.50 × 10^3, 4.050 × 10^-3, 5 × 10^0. A negative number starts with a plain hyphen.
  • Answer: in plain notation when that shows the same figures, else in scientific notation. Plain notation is used when k is from −6 to 20 and either p is negative (0.00346, 1.00: zeros after a point count) or the last digit of q is not 0 (2500 for the digits 25, 65). Otherwise the answer is in scientific notation (2.50 × 10^3 for 2496 to 3 figures, 1.0 × 10^1 for 9.96 to 2 figures). A zero result of adding or subtracting is written 0 when p is 0 or more, and with p decimal places otherwise (0.0). There are no thousands separators.

The calculator shows:

  • Count mode: Significant figures (the count), Decimal places (the larger of 0 and −p), Scientific notation (the typed number with exactly its significant figures), and Significant digits (q as typed, such as 4050).
  • Round mode: Answer, Significant figures (n), Decimal places (the larger of 0 and −p of the answer), and Scientific notation.
  • Calculate mode: Answer, Significant figures of the answer (left out when the answer is 0), Decimal places, Scientific notation (left out for 0), the Rule used, and the Unrounded result. The rule reads Multiplication: the answer keeps as many significant figures as the number with the fewest, 2. (or Division), or Addition: the answer is rounded to the last significant place of the less precise number, the tenths place. (or Subtraction). Place names are ones, tens, hundreds, thousands, tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths and millionths; other places are written the 10^-9 place.
  • Unrounded result: the exact value with no trailing zeros, in plain notation when its first digit is between 10^-6 and 10^20 and in scientific notation otherwise. When it has more than 15 significant digits (or never ends, as 1 ÷ 7), it is rounded to 15 significant figures, trailing zeros dropped, and starts with ≈ and a space.
  • Note (any mode), for each typed number that lost trailing zeros under rule 3: The trailing zeros in 1200 are not significant, because it has no decimal point. (with the number as typed).
  • In short: 0.004050 has 4 significant figures., 0.0034567 rounded to 3 significant figures is 0.00346., or 3.24 × 2.5 = 8.1 to 2 significant figures. (the numbers as typed, + − × ÷ for the operation, a negative second number in brackets, and no figure count when the answer is 0). "1 significant figure" is singular.

Assumptions

  • Every number is a measured value. The rules do not treat exact counts (12 eggs) or defined constants differently.
  • A value exactly halfway rounds away from zero, not to the even digit.

Worked examples by hand

Count 0.004050. The zeros before 4 are leading zeros and do not count. 4, the 0 between 4 and 5, 5, and the trailing 0 after the decimal point count: 4 significant figures. The last one is in the millionths place, so 6 decimal places, and the number is 4.050 × 10^-3.

Count 1200. No decimal point, so the trailing zeros do not count: 2 significant figures (1.2 × 10^3). Written 1200. with a point, it has 4.

Round 0.0034567 to 3 figures. The first non-zero digit is 3 (thousandths). Keep 3, 4, 5; the next digit is 6, so round up: 0.00346.

Round 2468 to 2 figures gives 2500, which reads as 2 figures: 2500. Round 2496 to 3 figures also gives 2500, but that would read as 2 figures, so the answer is 2.50 × 10^3. Round 2499.7 to 4 figures: 2.500 × 10^3.

Round 31.57 to 2 figures: 32. Round 8.1649 to 3 figures: 8.16. Round 0.051065 to 4 figures: the dropped part is exactly 5, which rounds up: 0.05107.

3.24 × 2.5. 3.24 × 2.5 = 8.100. 2.5 has 2 significant figures, so the answer is 8.1.

1.0023 + 4.383. 5.3853 exactly. 4.383 stops at the thousandths place, so the answer is 5.385.

486 − 421.23. 64.77 exactly. 486 stops at the ones place, so the answer is 65.

0.6238 × 6.6. 4.11708 exactly; 6.6 has 2 figures, so 4.1. 421.23 ÷ 486 = 0.86673…; 486 has 3 figures, so 0.867.

10 ÷ 3.0. 10 has 1 significant figure (its trailing zero does not count), so 3.333… becomes 3. The unrounded result is shown as ≈ 3.33333333333333.

5.0 − 5.0 = 0, rounded to the tenths place: 0.0.

Other questions people ask

How do I count significant figures?

Count every non-zero digit, every zero between non-zero digits, and every zero at the end of a number that has a decimal point. Leading zeros never count. 0.004050 has 4 significant figures (4, 0, 5, 0), and 1.20 × 10³ has 3.

Are trailing zeros significant?

After a decimal point, yes: 3.10 has 3 significant figures. In a whole number with no decimal point, such as 1200, trailing zeros are ambiguous, and this calculator does not count them (1200 has 2). To show that they were measured, write 1200. with a decimal point or use scientific notation, 1.200 × 10³.

How do significant figures work when multiplying or dividing?

The answer keeps as many significant figures as the number with the fewest. 0.6238 × 6.6 = 4.11708, and 6.6 has 2 significant figures, so the answer is 4.1.

How do significant figures work when adding or subtracting?

Round the answer to the last decimal place that both numbers know. 1.0023 + 4.383 = 5.3853; 4.383 stops at the thousandths place, so the answer is 5.385. 486 − 421.23 = 64.77 rounds to 65, because 486 stops at the ones place.

How do I round to a number of significant figures?

Find the first non-zero digit, count that many digits to the right, and look at the next digit: 5 or more rounds up, less than 5 rounds down. 0.0034567 to 3 significant figures is 0.00346.

Why is the answer sometimes in scientific notation?

Because the plain number would hide the significant figures. 2496 rounded to 3 significant figures is 2500, but 2500 reads as 2 significant figures, so the calculator writes 2.50 × 10³. You can copy any answer back into the calculator and it counts the same number of figures.

What happens when the dropped digit is exactly 5?

The calculator rounds halves up (away from zero): 0.051065 to 4 figures is 0.05107. Some chemistry texts round a final 5 to the even digit instead and get 0.05106, so check which rule your class uses.