Sig Fig Calculator

123 → 3 significant figures; 1002 → 4; 0.005 → 1; 0.004560 → 4; 1200 → 2 but 1200. → 4; round 0.004560 to 2 figures = 0.0046 = 4.6 × 10⁻³; round 9.99 to 2 figures = 10 = 1.0 × 10¹

0.004560

The number being counted, in ordinary decimal or e notation. It can be a measurement of any size, positive or negative; the sign takes no part in either the count or the rounding, so it is the digits alone that are read

0.005 → 1

Why leading zeros do not count. The three zeros in front of the 5 do nothing but hold the decimal point out where it belongs; the only digit carrying information is the 5, so a reading quoted as 0.005 has one significant figure, and rewriting it as 5 × 10⁻³ changes nothing about that

1002 → 4

Why zeros in the middle do count. The two zeros in 1002 sit between the 1 and the 2 and are holding the hundreds and tens places open, which is information: without them the number would be 12 or 102. Every digit of 1002 is therefore significant

1200 → 2, 1200. → 4

The rule that depends on the spelling rather than the value. In a whole number written without a decimal point, the trailing zeros may be nothing more than place holders, so they are not counted: 1200 has two significant figures. Adding the point says they were written deliberately, and 1200. has four

0.004560 → 4

Why trailing zeros after a decimal point count. The zero at the end of 0.004560 sits after the decimal point, so it was written on purpose and it says the measurement reached the millionths place. It is the fourth significant figure, which is why 0.004560 and 0.00456 are not the same claim even though they are the same value

0.004560 → 0.0046 = 4.6 × 10⁻³

Rounding to a chosen number of figures. Half-up means a following 5 rounds away from zero, and no padding means a request for more figures than the number has is returned unchanged. The scientific form shows both the figures that survive and the size that does not change

Significant figures are how a measured number states its own precision, and most of their use is in writing results down. A balance that reads to four decimal places, a ruler graduated in millimetres, and a timer that resolves hundredths all produce numbers whose last digit is the last one the instrument could actually see; recording 0.004560 rather than 0.00456 is what preserves that, and the count of significant figures is the shorthand for it. When measured quantities are combined the answer cannot be more precise than the weakest input, so the standard practice in physics, chemistry and engineering is to work out how many significant figures the result deserves and then round it to that many. The usual working rule is that a product or quotient keeps as many significant figures as the least precise factor, and a sum or difference keeps as many decimal places as the least precise term — which is why rounding to significant figures and rounding to decimal places are two different operations and this page offers only the first. Chemistry and physics also use the count as a checking habit: an answer with more significant figures than its inputs is a sign that something was not measured but computed, and it is the fastest way to spot a result that has been copied out of a display without thinking. In a classroom this is an examinable skill in its own right, since the count of figures in 1200 as against 1200. is a standard question and the answer turns on the written form rather than on the value. Rounding to a number of decimal places instead is the rounding calculator's job, and writing a value as a coefficient and a power of ten, with no judgement about precision at all, is the standard form page.

1. Counting the figures in 0.004560

   1. The leading zeros, 0.00, only position the decimal point and do not count
   2. The digits 4, 5 and 6 all count, and so does the zero after them, because it sits after the decimal point and was written deliberately
   3. That is four significant figures in all
   4. Rounding to four figures asks for exactly the figures the number already has, so the value is returned unchanged
   5. In scientific notation the coefficient shows the same four figures and the exponent carries the size: 4.560 × 10⁻³

   The default, and the case that shows the difference between the value of a number and the claim it makes. 0.00456 and 0.004560 are equal as values and different as measurements, and the trailing zero is the entire difference. The scientific form of the answer is the same statement written where the zeros cannot be mistaken for layout: a coefficient of 4.560 has four figures and says so.

2. Rounding 0.004560 to one figure

   1. The number has four significant figures, and one is being asked for
   2. Keep the first figure, which is the 4 in the thousandths place, and look at the digit after it
   3. That digit is a 6, so the 4 rounds up to 5
   4. Everything to the right is dropped, and the leading zeros stay where they were: 0.005
   5. Written in scientific notation it is 5 × 10⁻³, with a coefficient of one figure

   Rounding to one figure is the crudest case and the one where the size is easiest to lose, since the answer 0.005 no longer shows the digit that was kept except as a single 5. The scientific form makes the trade explicit: one figure of the answer survived and the exponent records the size it applies to.

3. A carry that changes the exponent: 9.99 to two figures

   1. The number has three significant figures, all of them non-zero
   2. Keep the first two, giving 9.9, and look at the next digit
   3. That digit is a 9, so 9.9 rounds up — and the carry runs all the way through, turning 9.9 into 10
   4. The rounded value is now a power of ten larger, so the exponent rises rather than the digit count
   5. In scientific notation that is 1.0 × 10¹, which shows two figures where the plain 10 could be read as one

   Why the scientific line earns its place. A carry that runs off the end of the number changes its size, and the plain form cannot say what happened: 10 could be the number ten, or it could be a two-figure result whose figures are 1 and 0. The scientific form settles it — a coefficient of 1.0 has two figures and an exponent of 1 records the size. The same thing happens one step further up with 999 rounded to two figures, which is 1000.

4. Asking for more figures than the number has: 123 to five

   1. The number has three significant figures, and none of them can be removed
   2. Five figures are asked for, which is more than the number carries
   3. Rounding never pads, so the value is returned exactly as it came in
   4. The count line still reports three, because the number did not gain two figures by being asked for them

   The case where the calculator refuses to help. Padding 123 to 123.00 would turn three measured digits into five, and the two extra zeros would be read as precision by anyone who took the result at face value — which is exactly the error the whole subject exists to prevent. The rounded-scientific line shows why the request cannot be honoured: 1.23 × 10² has three figures in it and there is nowhere for two more to come from.

The count follows the school rules and one of them depends on how the number is spelled. 1200 has two significant figures and 1200. has four; there is no third answer, because the calculator has only the written form to go on and does not know whether the zeros were measured. A whole number whose trailing zeros are significant but whose decimal point was omitted will therefore be undercounted — write 1200. if the zeros are meant. Rounding is half-up and away from zero, so a following 5 always rounds up; bank-style round-half-even is not available here. Rounding never pads, so asking for more figures than the number has returns the number unchanged, and the count line reports the figures the number actually carries. Leading zeros are not preserved in any input, so 007 is read as 7. The plain rounded output cannot show trailing zeros that are significant, which is why a second line gives the same value in scientific notation; for 1002.0 rounded to two figures the plain line reads 1000 and only the scientific line, 1.0 × 10³, reveals that the figure count is two. Inputs are limited to fifteen significant digits and to exponents within ±99, and the figure count must be a whole number from 1 to 15. Written forms such as 4.5 × 10⁴ and thousands separators are refused, as they are throughout this site. Zero has one significant figure. Finally, this page rounds to significant figures only. Rounding to a number of decimal places is the rounding calculator, and the two do not agree in general — 0.004560 to two decimal places is 0.00 and to two significant figures is 0.0046.

How do I count significant figures?

Every non-zero digit counts, and so does a zero between two non-zero digits, as in 1002, which has four. Leading zeros never count, because they only position the decimal point, so 0.005 has one. A zero after a decimal point counts, so 0.004560 has four. And a zero at the end of a whole number written without a decimal point does not count, so 1200 has two while 1200. has four.

Why does 1200 have only two significant figures?

Because the trailing zeros of a whole number written without a decimal point may be nothing more than place holders — they could be saying that the number is in the thousands rather than that anyone measured down to the ones. The written form cannot distinguish those two cases, so the conservative reading is taken and only the 1 and the 2 count. Write 1200. if the zeros are meant to be significant.

What is the difference between significant figures and decimal places?

Decimal places count from the decimal point outward; significant figures count from the first non-zero digit. For numbers near 1 the two roughly agree, and for numbers far from 1 they do not: 0.004560 rounded to two decimal places is 0.00, while rounded to two significant figures it is 0.0046. Rounding to decimal places is the rounding calculator's job; this page counts and rounds significant figures.

Why does asking for more figures than the number has not add zeros?

Because adding them would invent precision. 123 has three significant figures; printing it as 123.00 would show five, and a reader would take the last two as measured when nothing measured them. The calculator returns the number unchanged and the count line still reports three, which is its way of saying that the request could not be honoured.

Why is the rounded value also given in scientific notation?

Because the plain form cannot show significant trailing zeros. Round 1002.0 to two figures and the answer is 1000, where the zeros that hold the size open look exactly like zeros that were measured. The scientific line writes it as 1.0 × 10³, where the coefficient holds the two figures and the exponent holds the size, and the ambiguity is gone.

How many significant figures does zero have?

One. Zero is a digit that was written down, and unlike the leading zeros in 0.005 it is the whole of what is known about the quantity rather than a marker positioning a decimal point. Rounding it to any number of figures leaves it at 0, since there is nothing beyond the first digit to keep or drop.

• Significant digits — the definition, the rules for counting them and the conventions for rounding a result to a given number of them — Wolfram MathWorld (United States)
• GB/T 8170-2008《数值修约规则与极限数值的表示和判定》——The bibliographic page for this national standard on rounding (significant digits, rounding interval and rounding rules); the definitions of the rounding interval and of significant digits are governed by the text of the standard — 国家标准馆·国家数字标准馆
• 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); a first acquaintance with decimal approximations and significant figures is part of the Number and Algebra strand of these standards, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部