Significant Figures Calculator

What Is the Significant Figures Calculator?

How to use the significant figures calculator

1. Choose the unknown. The Solve for menu opens on Relative precision, and the other two choices are the half-unit in the last place and the number of significant figures a target precision needs. The box a mode does not read is hidden while that mode is showing.
2. Type the quoted value. It opens on 2500 and it is the one box every mode reads. No unit is asked for, because the percentage is the same whatever you measured in — the same digits on a length and on a pressure give the identical answer.
3. Type the figure count, or the precision you want. The count opens on 4 and has to be a whole number of at least one. In the third mode you give a target precision instead, in per cent or in parts per million.
4. Read the headline. Four significant figures with trailing zeros dropped by the engine. At the opening boxes it reads 0.02 %, which the working and the readings print as 0.02000 %.
5. Read the half-unit, the written value and the interval. They are 0.5000, 2.500e+3 and 2.500e+3 ± 0.5000 at the opening boxes. The interval is a worst-case bound rather than a standard deviation, and it is labelled that way.
6. Read what one figure either way is worth. The last two readings are 0.002000 % and 0.2000 % at the opening boxes. They differ from the answer by exactly a factor of ten in each direction, which is the whole reason the count is a coarse instrument.
7. Switch per cent to parts per million. The relative-precision box carries a two-entry unit menu and the engine converts for you, so 0.02 in per cent and 200 in ppm are the same entry.
8. Switch the menu to the figure count. Choose Significant figures and the answer comes back continuous — 3.602 for a target of 0.05 % on 2.50 — and you round it up, to four. Read the first reading rather than the headline: the headline shows the raw count to four figures, so a raw 2.00043 prints there as 2 when the answer is three.
9. Open Show working. The decade, the place of the last digit, the half-unit and the percentage appear line by line, then the closed form that skips the decade. Every line is produced from the full-precision figures and formatted once.

This page is the arithmetic, and the guide to significant figures in physics is where the method is set out with eight worked problems, the counting rules for all four zero cases, and the two propagation rules that are not the same rule. The digits you type usually arrive from somewhere else: the SI units guide is where a reading gets its unit in the first place, and the SI prefix converter moves one between prefixes without changing how many figures it carries.

Three mistakes account for most wrong answers here. The first is reading the count as a precision: 1.00 and 9.99 carry the same three figures and differ in precision by a factor of 9.99, which is why this page reports the percentage beside the count. The second is dropping a trailing zero that was written on purpose — 0.04070 is four figures and 0.0407 is three, and the difference is a factor of ten in the claim.

The third is rounding on the way through instead of once at the end. The unit conversions guide sets that rule out properly for a chain of conversions, and the unit conversions lab is where you can watch a chain run. Nothing on this page converts anything: it prices the digits you already have.

Worked example: change one thing at a time

The first table starts at the opening boxes and then changes one thing at a time: the same value read at three counts, the same count on two different mantissas, the four zero cases, a negative value, and finally the entries the calculator declines. Every answer and reading cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool.

Rows 1 to 3 are one value read three ways, and they are the clearest thing on the page. The digits never change; only the claim does. Each figure dropped multiplies the half-unit by ten and the percentage with it, from 0.02000 % to 0.2000 % to 2.000 %.

Notice what rows 1 to 3 print in the interval column: 2.500e+3, 2.50e+3 and 2.5e+3. Written plainly, all three would read 2500, which counts anywhere from two figures to four, so the plain form would hide the very thing the row is about. The e form is the only way to write a trailing zero that means something.

Rows 4 and 5 are this page's own result. Both quotes carry three figures, both claim a half-unit of 0.005000, and they are 0.5000 % and 0.05005 % precise — a factor of 9.99 apart for the same number of digits. A reader who only counts digits cannot see that, and it is the difference between a measurement that settles a question and one that does not.

Rows 6 to 9 are the four zero cases. A trailing zero after a decimal point counts, so 0.04070 is four figures; drop it and row 7 is three figures and ten times coarser. A captive zero counts too, so 10.053 is five; leading zeros never do, so 0.053 is two.

Rows 10 and 11 are two resolutions written out. 20 as written does not say how many of its digits are being claimed, so the only honest way to use it is to state the count you meant — and every count you add or drop moves the answer by a factor of ten. Resolved to two figures it is 2.500 % precise; resolved to three, 0.2500 %. The interval column is where the resolution becomes visible: row 10 writes back 2.0e+1 and row 11 writes back 20.0, which is the string a three-figure claim needs. The physics formulas reference is where that habit matters most, because a formula fed an ambiguous figure returns an ambiguous answer.

Row 12 is the coarsest count there is, one figure, at 5.005 %. Row 13 shows that a sign is not a digit: -0.04070 carries the same four figures and the same 0.01229 % as the positive reading.

Rows 14 to 17 sit outside the domain and are declined there rather than answered. Row 14 is the interesting one: a written zero has no significant figures, so there is no last place to take half a unit of, and the answer is undefined rather than 0. Writing 0.0 does assert a resolution and still counts as no figures — a real wrinkle in the rule, not a limitation of the tool.

Rows 15 and 16 refuse a count below one and a fractional count, because figures are whole things and interpolating between two counts would interpolate between precisions a factor of ten apart. Row 17 is refused by the arithmetic rather than by the rule: 400 figures is a legal whole number, and then the half-unit underflows a double to zero, so the page declines instead of printing a precision of 0 that is simply false.

The second table is the menu's third position, which is the half of this page that answers the question a reader usually arrives with: not what my digits are worth, but how many I need. The count that comes back is continuous and you round it up, and the two columns beside it show what that count delivers and what one fewer would have delivered.

Read the Rounded up column, not the headline. The headline shows the raw count to four significant figures with trailing zeros dropped, so a raw count of 2.00043 prints there as 2 figures while the answer is three. The working prints the raw count to six figures beside it for exactly that reason, and the first reading beside the answer carries the rounded-up count on its own.

Row 2 is the row to read twice. A target of 0.05 % on a value starting 2.50 computes as 3.602 figures, so you write four. Four deliver 0.02000 % and three deliver 0.2000 %, so three miss the target by a factor of four while four beat it by a factor of 2.5 — and there is nothing in between to settle for.

Rows 3 and 4 are the sharpest illustration of that on the page. 0.5 % on a value starting 9.99 computes as 2.00043 figures, which rounds up to three, because two figures there deliver 0.5005 % and that is a miss. Relax the target by one hundredth of a percentage point, to 0.51 %, and the raw count falls to 1.992, so the same value needs only two.

Row 5 closes the round trip on the decade effect from the other side. The same 0.5 % target on a value starting 1.00 comes out at exactly three figures, where the value starting 9.99 needed three as well — but there the three buy 0.05005 %, a factor of 9.99 more precision than you asked for. The count is a decade-wide dial and the mantissa decides where inside the decade you land.

Row 6 types the same target as row 1 in parts per million instead of per cent and gets the same answer, because the engine converts the entry rather than the physics. Row 7 is a target so coarse that one figure already beats it: 99.5 % is accepted, because the refusal boundary is a strict 100 per cent, and the last reading says in words that one figure is the floor rather than reporting that you need less than one.

Rows 8 to 11 are refused. A target of zero or less is not a target; a target at or above 100 per cent cannot be met by any count, because the relative precision of a real quote never reaches 100 per cent; and a quoted zero is refused here for the same reason it is refused in the first table.

Formula and symbol reference

One relation does all the work. A quote of n significant figures stops at the place p = E - n + 1, where E = floor(log10 abs(x)) is the decade of the value, and it admits half a unit of that place: u = 0.5 × 10^(E - n + 1). Divide that by the value and you have the relative precision, u_rel = 100 × u / abs(x).

Written as a closed form the decade drops out entirely: u_rel = 50 × 10^(1 - n) / m, where m = abs(x) / 10^E is the mantissa between 1 and 10. That is the most useful line on the page. The count sets the decade of the answer and the mantissa sets everything else, which is why the same count is worth up to ten times more on a value starting 9 than on one starting 1.

Invert it and you have the third mode: n = 1 + log10(50 / (m × u_rel)). The count that comes out is continuous and you round it up, and the decimal-place count is just dp = -p = n - 1 - E, which can legitimately be negative when a quote stops short of the units column.

Two neighbouring quantities look like this one and are not. The precision of a published constant is the physics constants guide's material, and this page prints no constant's value or uncertainty; the physics constants lab is where that precision is explored. The quantum limit on what can be measured at all belongs to the Heisenberg uncertainty principle guide, and a quote's resolution is a statement about how a number was written rather than about nature.

The physics: why a figure count is a coarse instrument

The count can only move in whole steps, and each step is worth a factor of ten. So the precisions a figure count can express are not spread evenly: they come in decades, and inside each decade the mantissa stretches the answer across a further factor of almost ten. Three figures buys anything from 0.5000 % at the bottom of a decade to 0.05005 % at the top, and the count alone cannot tell you which you have.

That is also why each count's range ends exactly where the next begins. Three figures at the bottom of a decade — a value starting 1.00 — claims 0.5000 %, and two figures at the top of one claims 0.5005 % on 9.99. The closer the leading digits run to ten, the closer that gets to the same 0.5 %, so the curves tile the line with no gaps and no overlaps.

One consequence is worth stating because it is easy to get backwards. The relative precision of a real quote never reaches 100 per cent, however few figures you write: one figure on a value starting 1 is a 50 per cent claim, and that is the coarsest there is. So a quoted interval can never straddle zero, which is why the calculator refuses a target precision of 100 per cent rather than reporting an enormous figure count.

The other consequence is the one this page exists for. If the precision you need falls between what two counts deliver, the count is not the dial to turn — you need a better reading, or you need to say the precision outright instead of implying it through digits. Published constants are quoted with an explicit uncertainty for exactly that reason, and they differ from one another in precision by many orders of magnitude; their figures live in the physics constants guide rather than here.

The counting convention behind all of this — which digits of a written number count, and why a bare integer ending in zeros is ambiguous about how many of them do — is the standard one, and you can read it at OpenStax, University Physics Volume 1, section 1.6. The closed form and its inversion are this page’s own working from that definition, as the references list records.

Where the significant figures calculator breaks down

The arithmetic here is exact to the last digit a double can hold, and there is no measured constant in it to be uncertain about. What goes wrong is everything around the count: whether the digits on the page are the digits that were measured, whether the rule is the right tool for the question, and whether a resolution was ever the thing worth knowing.

It cannot tell you how many figures your reading really carries

The count is something you supply, and getting it right means reading the number as it was written rather than as a calculator stored it. A bare integer ending in zero is the common trap: 20 could be one figure or two and nothing in the digits settles it. If you type a count the writing does not support, every reading on the page is confidently wrong.

A resolution is not a standard deviation

Every interval here is a worst-case bound: the quote admits anything inside the last digit's own column, with no distribution behind it. That means these bounds do not combine the way statistical uncertainties do, and nothing on this page adds them in quadrature. If your figures come from repeated measurements with a spread, that is a different calculation and this is not the tool for it.

It prices one quote, not a calculation

Multiplying or adding two quoted numbers has its own rules, and they are not the same rule as each other: a product keeps the smaller figure count while a sum keeps the coarser decimal place. This page deliberately stops at one value. The significant figures guide is where both propagation rules are worked through, and where you can see a sum end up with more figures than an input carries.

Rounding the written digits and rounding the stored double can disagree

The rule is taught on the digits you wrote, and a binary double is not those digits. Rounding 0.045 to one figure gives 0.05 on the digits, because the tie breaks away from zero, and 0.04 through a calculator's own rounding, because the stored value sits just below 0.045. This page rounds on the written decimal digits in integer arithmetic for exactly that reason, so your marking scheme and your calculator can honestly disagree on an apparent tie.

A very large figure count runs out of double

Somewhere past three hundred figures the half-unit underflows a double to zero, and the page declines rather than printing a precision of 0 that is simply false. On the opening value of 2500 the last count it will answer is 325 and the first it declines is 326; the exact boundary moves with the decade of the value. That is a limit of the arithmetic rather than of the rule, and no real measurement comes anywhere near it.

It says nothing about whether the measurement was any good

Resolution and accuracy are different things, and this page only knows about the first. A reading written to six figures can be six figures of a systematically wrong number, and the calculator has no way to see that. Checking an answer against an independent route is the habit that catches it, and that is the guide to solving physics problems's territory rather than this page's.

Where significant figures are actually used

Deciding how many digits to write in an answer

This is the question a figure count exists for, and the calculator answers it from the direction that matters: type the precision the reading supports and read off the count. Writing more digits than that claims a resolution you do not have; writing fewer throws away precision you paid for.

Comparing two measurements of different sizes

A half-unit of 0.5 means nothing until you know what it is half a unit of. The percentage reading is what makes two quotes comparable, and it is why a 2500 read to four figures and a 0.04070 read to four are not equally precise at all: 0.02000 % against 0.01229 %.

Choosing the resolution a measurement needs

Run the third mode before you measure rather than after. If you need 0.05 % on a reading that will start with a 2, you need four figures, and that tells you what the reading has to resolve. The answer depends on the leading digits, which is why the mode asks for a value rather than only a target.

Working in prefixes without changing the claim

Moving a value between SI prefixes shifts the decade and leaves the figure count alone, so the relative precision does not move at all. The SI prefix converter does that shift, and the SI units lab is where the prefixes themselves are laid out; this page is what tells you the claim survived the move.

Scaling a quantity by a power law

When a result depends on an input raised to a power, the input's relative precision is multiplied by that power on its way through. The dimensional scaling calculator does the scaling itself, and the percentage reading here is the figure you carry into it.

Reading a figure count out of a published quote

Tables of measured values are written to a deliberate number of figures, and the number chosen is a claim you can price. Constants are the extreme case and are quoted with explicit uncertainties instead; the physics constants lab is where those are explored, and none of their figures appears here.

Where to go next

For the method rather than the arithmetic — eight worked problems, the counting rules for all four zero cases, the two propagation rules and the subtraction that destroys figures — read the guide to significant figures in physics, which this tool is the calculating half of. The SI units guide and the unit conversions guide are the two pages this one hands off to in both directions.

Three more guides pick up what this page leaves out. The physics constants guide owns the precision of published constants and their uncertainties, the Heisenberg uncertainty principle guide owns the quantum limit that a quote's resolution is not, and the guide to solving physics problems is where checking an answer three ways belongs.