Significant Figures Simulator

What Is the Significant Figures Simulator?

How to use the significant figures physics simulator

1. Read the four cards before you touch anything. They are at the top of the panel, above the sliders, and they are already answering for two readings quoted 5.00 and 4.00: The answer 20.0, Figures you may keep 3 significant figures, Precision of the answer 0.2250 % and Honest to the last digit? 0.90 — every digit supported.
2. Read the sub-line under the second card, because it names the rule that fired. It says fewer figures wins while an operation of × or ÷ is pressed and coarser decimal place wins for + or -. Those two are different rules, not two wordings of one.
3. Drag Figures in B from 3 down to 2 and watch the first card change form. It stops printing a plain decimal and prints 2.0e+1, with the sub-line the e form, because the plain 20 would read as 1 to 2 figures and not 2. The precision card moves with it, from 0.2250 % to 1.350 %.
4. Press + and read the same two cards again. The rule name changes to coarser decimal place wins, the answer becomes 9.0 at 2 significant figures, and the fourth card leaves the supported band for the first time: 1.10 — last digit marginal.
5. Now raise Figures in B back to 3 and watch a sum gain a figure. With Reading B at 4.900 the answer goes from 9.9 to 9.90 and the count from 2 significant figures to 3 — while the verdict gets worse, from 1.10 — last digit marginal to 2.00 — last digit not supported.
6. Use Decade of B, the fifth slider, to watch a perfectly good reading stop counting. It reads 10^0 on arrival and runs from 10^-3 to 10^3. At 10^-3 a B dialled to 9.900 is quoted 0.00990, and adding it to A changes no digit of the answer at all.
7. Read the wrapping strip under the drawing, which carries six more readings. At the opening state it gives A, as quoted 5.00, A’s precision 0.1000 %, B, as quoted 4.00, B’s precision 0.1250 %, Answer bound ± 0.04500 and Answer range 19.96 to 20.05.
8. Read the sentence strip above it, then press Reset. At the opening state it reads: 5.00 x 4.00 = 20.0000 before any rounding, and the rounded answer written unambiguously is 2.00e+1. Multiply and divide keep the FEWER of the two figure counts, 3 against 3, so the answer keeps 3 significant figures and stops at 10^-1. Reset puts all five sliders and the operation back to that state.

Dragging is the right way to find out which readings a control is allowed to move, and a poor way to hit an exact figure — the reading sliders step a thousandth at a time and the figure counts a whole figure. When you already have a value in front of you and want the half-unit, the percentage, or the figure count a target precision would need, the significant figures calculator takes typed input and runs the question in either direction. The written guide to significant figures in physics is where the counting rules themselves are set out, with the zero cases and the eight worked problems.

One thing this panel will not decide for you is how many figures your own instrument entitles you to claim, and that choice is upstream of every number on the screen. The sliders take the claim you make and price it; they cannot tell you whether the claim was reasonable. Settle that first, then come back and see what the arithmetic does to it.

Worked example: change one thing at a time

Nothing in the two tables below was worked out here. Each row is a setting the lab was put into through its own sliders and then read back from, cell by cell, so every string in them is the panel’s own. The first table moves one control per row; the second is the eight preset buttons in the order they appear above.

One: six rows, and only one control moves between any two of them

Reading A never moves: it sits at 5.000 quoted to three figures in all six rows. Everything that happens below is done by Figures in B, Reading B and the operation buttons, one change at a time.

Read the fifth column down, because it is the one a figure count hides. It runs 0.2250 %, 1.350 %, 0.6111 %, 0.5556 %, 0.1010 %, 10.00 %. The last step is one button press and it costs two decades of precision; nothing else in the table comes close to it.

Now read the third column against the sixth. Row five keeps more figures than row four — 3 significant figures against 2 — and is less honest for it, moving from 1.10 — last digit marginal to 2.00 — last digit not supported. The count went up and the claim behind it got worse.

The fourth column is the point of having both rules on one panel. It says fewer figures wins for the first two rows and coarser decimal place wins for the last four, and the switch happens at a button press rather than at a change of reading. The rule that applies is a property of the operation, not of the numbers. The habit of rounding once at the end rather than at every step belongs to the guide to unit conversions in physics, which owns that half of it.

Two: the eight presets, and what each one is for

These are the buttons above with the readings they produce. The readings column gives the written forms from the strip under the drawing rather than the slider positions, because the written form is what the figure count applies to.

Rows one and two differ by one figure in one reading and nothing else. The answer changes form rather than value, from 20.0 to 2.0e+1, because a plain 20 would not read back as exactly two figures. Both rows are in the supported band, at 0.90 and 0.54.

Row three is the first overreach, and it is a plain division. The rule keeps 3 significant figures and prints 4.21, while the bound behind it supports two: 2.15 — last digit not supported. Nothing exotic was dialled in to get there.

Rows five and six are the two additive surprises, and only the second of them needs B’s decade slider. In row five, with B still in A’s own decade, a reading quoted to two figures plus one quoted to four gives 11.0 at 3 significant figures, which is more than one input carries. In row six a three-figure reading of 0.00990 changes no digit of the sum at all, and the answer is the same 1.0 that A gives alone.

Rows seven and eight are the two the rest of the page is about. The subtraction takes two readings each about a tenth of a per cent precise — 0.1000 % and 0.1002 % — and returns one that is 100.0 % precise. The leading 1 keeps three figures and earns 10.99 — last digit not supported, which is the largest ratio any of the eight reaches.

Formula and symbol reference

The lab prints its own definitions in a fixed line under the drawing. Written out the same way: the half-unit of a quote is half a unit in the column it stops at; the relative precision is that half-unit over the reading as a percentage; multiply and divide keep the fewer figure count while add and subtract keep the coarser decimal place; the worst-case half-width comes from the corners or from a sum of half-units; and the honesty ratio is that half-width against the half-unit the rule claims.

Eleven rows follow and only five of them are something you drag. The rest are cards, cells of the strip under the drawing, or quantities the panel names without giving you a control for. For the sliders the last column gives the end stops and the step; for everything else it gives the strings the lab printed and the span it printed them over.

One row is worth reading before the table. Decade of B is the only control that moves a quantity between decades, and it is there instead of a second decade slider for A because a reading can only be too small to change a digit of the sum when it sits a decade or more below the reading it is added to. The powers of ten it steps through are handled as named prefixes on a real unit by the SI prefix converter, which is the other way to meet the same ladder.

If you read one row twice, make it u_rel. It is the only quantity on the panel that tells you what a figure count is worth, and it is the reason the lab prints percentages at all rather than just counting digits for you.

The physics: the rule counts digits, the bound prices them

The first claim the panel lets you test is that a figure count is not a precision. Set both readings to three figures and the strip under the drawing prints two different percentages for them, because the half-unit is fixed by the column the quote stops at while the reading it is divided by is not. Inside one decade that proxy is wrong by up to a factor of ten, which is the whole of the leading-1 preset.

The second claim is that the two propagation rules really are two rules. Multiplication and division compare figure counts and keep the fewer; addition and subtraction compare decimal places and keep the coarser. Inside one decade both rules pick the same place on the two readings, and the counts still part company as soon as the sum carries into the decade above — which is the 11.0 row of the preset table, reached with B at 10^0. What the fifth slider buys is a different case: move B three decades down and a perfectly good three-figure reading stops affecting the answer at all.

The third claim is the awkward one, and the fourth card is arranged around it. The rule is not conservative: it hands you a last digit that the inputs may not support, and the ratio on that card is how far out it is. Two of the eight presets are conservative and three sit above 1.5, so neither verdict can be mistaken for the normal case.

The rules the panel applies are not this site’s invention. Read them at the source: OpenStax, University Physics Volume 1, section 1.6 — Significant Figures. The references list below says which parts of this page are that source’s and which are the page’s own.

None of this is the quantum limit on what can be measured, and the lab does not borrow its authority. What the bound here describes is the resolution the person quoting the reading claimed, which is an arithmetic consequence of how they wrote it down; the guide to the Heisenberg uncertainty principle is where the physical limit is set out, and nothing on this page computes or implies it.

Where the significant figures simulator breaks down

Every slider position this lab can reach is answered, and the arithmetic is not where it can let you down. What it cannot do is show you a refusal, and what it chooses not to do is give reading A a decade. Nine limits sit under the readings on this page; three are decisions the panel makes on purpose and six are bounds on what a panel of this shape can show at all.

Most of the refusals this topic has are unreachable here, and the lab does not pretend otherwise

The five sliders cannot produce a reading of zero, a figure count below one, a fractional figure count or a divisor of zero, and a validly quoted interval can never straddle zero, so even the division guard has no reachable input. Those refusals are real and they matter — a written zero has no significant figures at all, which is a genuine asymmetry in the rule — but they belong to the significant figures calculator, where you type the figures yourself and it says why it will not answer. The guards exist in this lab too and were driven directly in testing; they simply have no input you can reach.

Reading A has no decade, so the lab cannot show you a scale effect that is not there

A always lies in one decade. That costs nothing, because a figure count on a given leading mantissa buys the same relative precision in any decade, and it buys the fifth slider: only a difference of decades between the two readings can leave a correctly quoted reading changing no digit of the sum at all. What it does mean is that you cannot use this panel to explore a quantity whose size is the question. A later reader who wants A to roam should know the omission is deliberate.

A figure count above the digits the slider supplies is allowed, not clamped

Set reading A to 5.000 and Figures in A to 5 and the strip quotes it 5.0000, with a half-unit of 0.00005 and a relative precision of 0.001000 %. That is not the lab inventing precision: quoting a measurement to a figure count is an assertion about resolution, and the panel takes the assertion at face value and then prices it. If the claim is unreasonable, the fourth card is where you will see it.

The exact tie is reachable, it is not a preset, and software will disagree with it

Put both readings at 5.000 with one figure each and multiply. The product is an exact tie, the lab rounds it the way the taught rule does and prints 3e+1 with a ratio of exactly 1.00 — every digit supported, and the range runs 20.25 to 30.25. A tool that rounds the stored double instead can give 2e+1 for the same question, because the two routes genuinely differ on a tie. It is left off the preset row for that reason: it needs the paragraph, and the written guide is where the argument belongs.

Two states can print the same ratio and mean different things

The verdict is decided on the unrounded ratio and printed next to the ratio rounded to two decimal places. The exact tie just above is a multiplication whose ratio is exactly one, and it prints 1.00 — every digit supported; the sixth preset, the reading too small to matter, is an addition whose ratio is a shade over one, and it prints 1.00 — last digit marginal beside the same two digits. Read the words rather than the digits, and never quote the boundary as being "at 1.00 as displayed".

One state says nothing, and it says so rather than printing a number

Subtract two identical readings and the answer is exactly zero. The precision card then reads undefined rather than a figure, an infinity or an error, the figure count reads 0 significant figures, and the answer strip is drawn with no gold cell and no tick under it. The bound is still real, which is the honest part: the answer is zero to within a stated half-width, and that half-width is still in the strip.

The bound is a worst case, so it is wider than a statistical treatment would give

Everything here combines by corners and by addition, because a figure count asserts a resolution rather than a distribution. A root-sum-of-squares combination is the statistical question and is deliberately not this lab’s subject, so no interval on this page carries a probability and nothing on it is a standard deviation. If your readings are repeated measurements with scatter, this is the wrong tool for them and the bound it prints will be pessimistic.

No digit cell is a measurement, and no pixel count means anything

The scene is drawn to whatever width the canvas has, so the same reading occupies a different number of pixels on a phone and on a desktop, and the drawing promises no fixed width per digit. What it does guarantee is the correspondence: the gold cells in a strip always number the figures the captions claim, and the tick always sits under the last of them. The exact figures are in the four cards and the strip underneath, which is where to read them from.

Nothing here is a constant, and nothing here is a quantum limit

Every reading on the panel is a slider position chosen to divide cleanly, not a published value, and no manufacturer, model or instrument specification is named anywhere in this cluster. Constants do differ from one another in how precisely they are known, which matters when one of them ends up in your arithmetic, and the guide to physics constants is where those figures live; none is printed or implied here. The limit a quantum measurement carries is a different subject again and is not this bound.

Where significant figures are actually used

Deciding how many digits of an answer to write down

This is the everyday use and the panel is the fastest way to feel it. Dial your two readings and their figure counts, press the operation, and the first two cards give the answer and the count the rule allows. Then look at the fourth card before you write it: if it reads last digit not supported, the digit you were about to publish is not one your readings paid for.

Finding out which of two readings is holding the answer back

Put both readings on the panel and compare the two precision cells in the strip. On the leading-1 preset they read 0.5000 % and 0.05005 % at an identical three figures — 9.99 to one, not the round ten it looks like — and the answer inherits the worse of them. Improving the better reading buys you nothing; it is the coarser one that sets the answer, and the axis in the drawing puts the three markers on one scale so you can see which.

Spotting a subtraction that has destroyed your precision

Dial 5.000 against 4.990, each quoted to three figures and each about a tenth of a per cent precise, then press -. The answer reads 0.01 at 1 significant figure and 100.0 %, and the range starts at 0, which means the two readings may have been identical. Any difference of two nearly equal measurements has this problem, and the panel is the quickest demonstration that the rule cannot rescue it.

Checking that a quoted precision and a quoted figure count agree

If somebody hands you both a figure count and a percentage, the two have to be consistent. Dial the reading and the count here and compare the precision cell with what you were told: a percentage better than the count can buy means a digit has been dropped somewhere, and one much worse means the count was never the limiting thing. The panel will not tell you which, but it will tell you the two cannot both be right.

Keeping a guard digit through a conversion instead of rounding twice

Most readings get converted before anybody argues about their digits, and rounding at each step throws away more than rounding once at the end. Carry the extra digit, do the arithmetic, and use this panel on the final pair to decide where to stop. The unit conversions lab is where the conversion itself is drawn, and this one is where its answer gets its figure count.

Showing a class why the rule is a convention and not a theorem

The habit of trusting the digit count is very hard to shift by being told. Press the leading-1 preset, read 3 significant figures on one card and 10.99 — last digit not supported on another, then press the first preset and watch the same rule behave perfectly. Three minutes of that does more than a paragraph, which is why the ratio is on the panel at all.

Where to go next

A panel can show you what a figure count is worth, and that is the end of what a panel is for. Significant figures in physics is where the subject itself is written out: what counts as a significant figure, the four zero cases, why a bare integer is ambiguous, and eight worked problems with every step shown. The calculator that takes typed figures heads the related tools below.

Three guides sit either side of this one. The guide to SI units in physics and the guide to unit conversions are the two pages a reading passes through before its digits are counted, and the guide to physics constants is where a quantity arrives with a precision already attached to it.

Two more are about what you do with the answer. The physics formulas reference is where the arithmetic this panel prices usually comes from, and the guide to solving physics problems is where deciding what to write down sits in a longer method.

On the tool side there are the SI units lab, the unit conversions lab, the physics constants lab and the dimensional analysis lab, with the dimensional scaling calculator for the typed version of the last of those. Beyond them there is the simulation library, an index of calculators, and the blog for everything written out at length.