Significant Figures and Rounding

A buret reads 23.47 mL, not 23.47000 mL. This article is about that extra precision. The last digit implies precision — and also refuses to imply too much. Significant figures are the compact language chemists use. They carry a measurement's precision through every calculation it enters. This way, a final answer never claims to know more than the glassware, balance, or instrument actually delivered. This matters because arithmetic is a precision-blind machine. A calculator will happily report 8.31446261815324 for a gas constant multiplied by an oddly precise temperature. This happens even when the real experimental uncertainty is in the second decimal place. Without a disciplined convention for tracking significant digits, every multi-step calculation in The Nature of Chemical Measurement would silently inflate or destroy precision. Nobody reading a lab report could tell a genuinely precise 12.000 g from a careless 12.000 g that really means "about 12." The big picture is that significant figures are a first-pass, hand-computable stand-in for full Propagation of Uncertainty. They are not a substitute for a real uncertainty budget — a mean ± standard deviation, or a propagated variance. But they are the fast, universal convention that catches gross errors, standardizes lab notebooks, and trains the habit of thinking about precision. This happens before you ever touch a calculator's error-propagation formulas. This article gives you the counting rules with no ambiguity left in them. It covers the arithmetic rules for addition/subtraction versus multiplication/division. It also covers the correct rounding conventions, including the round-half-to-even rule used in professional work. Three fully worked examples follow, running from a single measurement to a multi-step titration calculation. This lets you apply the rules mechanically and correctly under exam or lab conditions.

Key takeaways

Significant figures = all digits known with certainty plus one final estimated (uncertain) digit.
Leading zeros never count; trapped zeros always count; trailing zeros count only if a decimal point is present (or use scientific notation to remove ambiguity).
For + / −: round the result to the same number of decimal places as the least-precise term.
For × / ÷: round the result to the same number of significant figures as the term with the fewest.
Exact numbers (defined constants, counted integers, stoichiometric ratios) have infinite significant figures and never limit a calculation.
Use round-half-to-even ("banker's rounding") for the 5-exactly case to avoid systematic bias over many roundings.
Key idea: significant figures encode where a measurement's certainty ends and its estimation begins, and every arithmetic rule exists only to keep that boundary honest through a calculation.

Where this lives today

Modern analytical instruments make significant-figure discipline more important, not less, because digital displays report far more digits than any physical measurement can support. A pH meter might display 7.352, but if the electrode's actual reproducibility is $\pm 0.02$ pH units, only three of those digits mean anything — a working analytical chemist reports 7.35, not 7.352, and this judgment call is exactly what significant-figure training teaches students to make before they ever compute a full propagated uncertainty. Pharmaceutical quality control relies on this discipline directly: a tablet's stated potency (e.g., "500 mg ± 5%") is reported with a sig-fig count tied to the actual precision of the HPLC assay used to certify it, and regulatory submissions are rejected if reported precision exceeds what the validated method can support. In environmental and clinical laboratories, results such as blood lead levels or water contaminant concentrations are reported with sig figs matched to the calibration curve's actual resolution — overstating precision (reporting "3.72 µg/dL" when the assay is only good to $\pm 0.3$ µg/dL) can trigger inappropriate regulatory action or misdiagnosis. In materials science and geochemistry, mass-balance and stoichiometric calculations for mineral formulas or alloy compositions use the exact-number rule constantly: crystallographic site-occupancy ratios and stoichiometric subscripts are exact, so they never limit the sig-fig count of a computed formula weight, only the measured atomic-absorption or XRF intensities do. And in every introductory laboratory course, significant-figure rules remain the first formal bridge between "doing an experiment" and "reporting a defensible number" — the essential first step before students encounter the fuller machinery of Propagation of Uncertainty and the formal statistical language of Reporting Analytical Results.

Ask the tutor: In my titration calculation, should I round after computing moles or wait until I compute the final molarity?

The physical picture — thorough prose

Every measurement made with a graduated instrument carries an implicit uncertainty in its last recorded digit. A buret marked in 0.1 mL divisions lets a careful reader interpolate one more digit by eye — reading 23.47 mL means the observer is certain of the "23.4" and is estimating the "7" between the 0.1 mL gradations. That estimated digit is still significant: it carries real information (a repeat reading is far more likely to be 23.46 or 23.48 than 23.30 or 23.60), but it is not certain the way the digits before it are. Significant figures are simply the count of digits, from the first nonzero digit to that final estimated digit, that carry this mix of certainty-then-estimate information. The trouble is that zeros are ambiguous placeholders unless context resolves them. The number 0.00230 has three significant figures — the leading zeros only fix the decimal point's position and carry no measurement information, while the trailing zero after the 3 was written deliberately, meaning the measurer resolved the instrument down to that digit and got zero. Contrast this with 2300, which is genuinely ambiguous: it might mean a number known to two significant figures (rounded to the nearest hundred), or four (if someone measured 2300 g on a precise balance and every digit is real). Scientific notation removes the ambiguity entirely: $2.300 \times 10^3$ unambiguously carries four significant figures, while $2.3 \times 10^3$ carries two. This is why analytical chemists prefer scientific notation whenever trailing zeros in a whole number could be misread. The deeper reason arithmetic rules differ between addition/subtraction and multiplication/division is that these operations propagate uncertainty differently. Addition and subtraction combine absolute uncertainties — if you add a mass known to $\pm 0.001$ g to one known to $\pm 0.1$ g, the sum cannot be known better than $\pm 0.1$ g, because the coarser measurement's uncertainty dominates in absolute terms. Multiplication and division instead combine relative (fractional) uncertainties, because each factor contributes a percentage error to the product, and the least precise percentage — usually tied to the fewest significant figures — sets the overall relative uncertainty of the result. These two propagation behaviors are the mechanistic origin of the "decimal places" rule for sums and the "significant figures" rule for products; they are simplified stand-ins for the full variance-propagation formulas developed in the treatment of Propagation of Uncertainty.

Figure

Reading a buret to one estimated digit: the last figure is uncertain but still significant

Figure

Why trailing zeros need a decision point: three ways to write "twenty-three hundred"

The core ideas — the governing equations/relationships

The counting rule can be stated as a compact decision procedure. For a number written in ordinary decimal form:

\text{sig figs} = \{\text{all digits from the first nonzero digit through the last digit written}\}

where: leading zeros (zeros before the first nonzero digit) are excluded; zeros between nonzero digits ("captive" zeros) are always included; trailing zeros are included only if the number contains a decimal point.

Reading: this single rule handles every case — 0.0450 has three sig figs (4, 5, trailing 0 counted because a decimal point is present); 4050 has an ambiguous count unless written as $4.050\times10^3$ (four sig figs) or $4.05\times10^3$ (three). For addition and subtraction, the result's precision is capped by decimal places, not sig-fig count:

\text{decimal places in result} = \min(\text{decimal places in each term})

where: each term being added or subtracted is inspected for how many digits it carries after the decimal point, and the smallest such count governs the answer, regardless of how many total significant figures each term has.

Reading: 12.11 (2 dp) + 18.0 (1 dp) + 1.013 (3 dp) is capped at 1 decimal place, because the 18.0 term is the least precise in absolute terms — its last digit's uncertainty ( $\pm 0.1$ in the ones-tenths place) swamps the others once summed. For multiplication and division, the result's precision is capped by total significant figures:

\text{sig figs in result} = \min(\text{sig figs in each factor})

where: every factor's total significant-figure count is compared (irrespective of decimal position), and the smallest one sets the answer's significant-figure count.

Reading: $4.52 \times 2.0 = 9.0$ (2 sig figs), not $9.04$ , because the factor "2.0" carries only two significant figures and thus limits the relative precision of the product to about 5%, the same order as its own uncertainty. For exact numbers — defined conversion factors, counting integers, and stoichiometric coefficients from a balanced equation — there is no significant-figure limit at all:

n_{\text{exact}} = \infty \ \text{sig figs (never limits a calculation)}

where: $n_{\text{exact}}$ denotes any quantity known by definition or by counting (e.g., "2 mol H₂O per mol CO₂" from stoichiometry, or "12 inches per foot").

Derivation AWhy sums cap at decimal places, not sig figs

Goal. Show that the "match the fewest decimal places" rule for addition follows directly from how absolute uncertainties combine, not from an arbitrary convention.

Step 1 — set up the uncertain digits explicitly. Suppose we add three masses read from balances of differing precision:

m_1 = 12.11\ \text{g} \pm 0.01\ \text{g}, \quad m_2 = 18.0\ \text{g} \pm 0.1\ \text{g}, \quad m_3 = 1.013\ \text{g} \pm 0.001\ \text{g}

Each stated uncertainty reflects the resolution of the estimated last digit — this is the operational meaning of "significant to the tenths place" versus "significant to the hundredths place."

Step 2 — propagate absolute uncertainty through a sum. For independent uncertainties, the variance of a sum is the sum of variances,

\sigma_{\text{sum}}^2 = \sigma_1^2 + \sigma_2^2 + \sigma_3^2

but even without invoking full Propagation of Uncertainty machinery, a simple worst-case (maximum-error) bound gives

\delta_{\text{sum}} \le \delta_1 + \delta_2 + \delta_3 = 0.01 + 0.1 + 0.001 = 0.111\ \text{g}

Either way, the uncertainty in the sum is dominated by the largest absolute uncertainty among the terms — here, $\pm 0.1$ g from $m_2$ — because the smaller uncertainties from $m_1$ and $m_3$ are numerically negligible next to it.

Step 3 — conclude the rounding rule. Since the sum's uncertainty is set by the term with the coarsest absolute precision (fewest decimal places), reporting extra decimal places in the sum would be reporting digits with no physical meaning. Adding the raw numbers,

12.11 + 18.0 + 1.013 = 31.123\ \text{g}

must therefore be rounded to the decimal place of the least-precise term (one decimal place, from $m_2$ ):

\text{sum} = 31.1\ \text{g}

This matches the intuitive rule exactly: match the fewest decimal places, not the fewest significant figures, because it is absolute — not relative — precision that limits a sum.

Go deeper · Derivation B · 1 minWhy products cap at significant figures, not decimal places

Derivation BWhy products cap at significant figures, not decimal places

Goal. Show that the "match the fewest significant figures" rule for multiplication/division follows from relative (percentage) uncertainty propagation.

Step 1 — express each factor's relative uncertainty. Consider a density calculation with a mass and a volume:

m = 4.52\ \text{g} \ (\pm 0.01\ \text{g}), \qquad V = 2.0\ \text{mL} \ (\pm 0.1\ \text{mL})

The relative uncertainties are

\frac{\delta m}{m} = \frac{0.01}{4.52} \approx 0.0022 \ (0.22\%), \qquad \frac{\delta V}{V} = \frac{0.1}{2.0} = 0.050 \ (5.0\%)

Step 2 — propagate relative uncertainty through a quotient. For a quotient $\rho = m/V$ , relative variances add,

\left(\frac{\delta \rho}{\rho}\right)^2 = \left(\frac{\delta m}{m}\right)^2 + \left(\frac{\delta V}{V}\right)^2 \approx (0.0022)^2 + (0.050)^2 \approx 0.0025

\frac{\delta \rho}{\rho} \approx 0.050 \ (5.0\%)

The quotient's relative uncertainty is dominated overwhelmingly by $V$ 's 5.0% relative uncertainty — the mass's 0.22% contributes almost nothing in quadrature.

Step 3 — connect relative uncertainty to significant figures. A quantity known to $n$ significant figures typically carries a relative uncertainty on the order of $1$ part in the last digit, roughly $10^{-(n-1)}$ to $10^{-n}$ of its value. Here $V = 2.0$ mL (two sig figs) has $\sim 5\%$ relative uncertainty, while $m = 4.52$ g (three sig figs) has $\sim 0.2\%$ . Since the quotient's overall relative uncertainty is essentially fixed by the least precise factor, the quotient can only be trusted to as many significant figures as that factor carries:

\rho = \frac{4.52\ \text{g}}{2.0\ \text{mL}} = 2.26\ \text{g/mL} \ \longrightarrow \ \text{round to } 2.3\ \text{g/mL} \ (\text{2 sig figs})

This confirms the rule: multiplication/division caps the result at the smallest significant-figure count among the factors, because sig figs are a proxy for relative precision, and relative precision is what multiplies through a product or quotient.

Worked examples

Example 1 (Foundational) — Counting sig figs and rounding a single mixed calculation. Determine the number of significant figures in each of the following, then evaluate $A + B - C$ correctly rounded:

A = 0.04080\ \text{g}, \qquad B = 15.36\ \text{g}, \qquad C = 2.1\ \text{g}

Counting sig figs. In $A = 0.04080$ : the leading zeros (before the "4") are not significant; the "4," "0" (captive, between 4 and 8), "8," and the trailing "0" (significant because a decimal point is present and it was written deliberately) all count. So $A$ has 4 significant figures. In $B = 15.36$ , all four digits are between nonzero digits or are themselves nonzero, so $B$ has 4 significant figures. In $C = 2.1$ , both digits count, so $C$ has 2 significant figures. Arithmetic. First evaluate with full precision, then round once at the end (rounding only at the final step avoids compounding round-off error):

A + B - C = 0.04080 + 15.36 - 2.1 = 13.30080\ \text{g}

Applying the addition/subtraction rule. Decimal places carried by each term: $A$ has 5 decimal places, $B$ has 2, $C$ has 1. The smallest is $C$ 's 1 decimal place, so the answer must be rounded to 1 decimal place:

A + B - C = 13.3\ \text{g}

Limiting-case check. If $C$ had instead been given with high precision, say $C = 2.100$ g (4 dp... actually 3 dp), the answer would retain more decimal places, illustrating that it is always the coarsest term — never the finest — that controls the rounding of a sum. As that coarsest term's precision improves (more decimal places quoted), the allowed precision of the sum improves correspondingly, approaching the precision of the next-coarsest term.

Example 2 (Applied) — Molarity from a titration, mixing addition/subtraction and multiplication/division. A 25.00 mL aliquot of an HCl solution is titrated with standardized 0.1024 M NaOH. The initial buret reading is 0.55 mL and the final reading is 21.36 mL. Find the molarity of the HCl solution, correctly rounded. Step 1 — volume delivered (subtraction step). Both buret readings are recorded to 2 decimal places (the usual buret convention: 0.01 mL resolution), so:

V_{\text{NaOH}} = 21.36\ \text{mL} - 0.55\ \text{mL} = 20.81\ \text{mL}

Decimal places rule: both terms have 2 decimal places, so the difference correctly keeps 2 decimal places — no rounding needed here, and note the difference has 4 significant figures (20.81), even though it came from two numbers with different sig-fig counts (4 and 2) — this is exactly why the addition rule is stated in decimal places, not sig figs: a subtraction of nearly equal numbers can lose significant figures relative to the inputs, a phenomenon called subtractive cancellation. Step 2 — moles of NaOH (multiplication step, exact stoichiometric factor).

n_{\text{NaOH}} = (0.1024\ \text{M})(0.02081\ \text{L}) = 2.1309\times10^{-3}\ \text{mol}

Sig figs: 0.1024 M has 4 sig figs; 0.02081 L has 4 sig figs (the leading zeros are not significant, "2081" are the four counted digits). Product is limited to 4 sig figs:

n_{\text{NaOH}} = 2.131\times10^{-3}\ \text{mol}

Step 3 — moles of HCl (exact 1:1 stoichiometric ratio, from the balanced equation HCl + NaOH → NaCl + H₂O). The mole ratio "1 mol HCl : 1 mol NaOH" is an exact number and never limits sig figs:

n_{\text{HCl}} = n_{\text{NaOH}} = 2.131\times10^{-3}\ \text{mol}

Step 4 — molarity of HCl (division step). The aliquot volume 25.00 mL has 4 sig figs:

[\text{HCl}] = \frac{2.131\times10^{-3}\ \text{mol}}{0.02500\ \text{L}} = 0.08524\ \text{mol/L}

Both numerator (4 sig figs) and denominator (4 sig figs) carry 4 sig figs, so the quotient is reported to 4 sig figs:

[\text{HCl}] = 0.08524\ \text{M}

Limiting-case check. If the buret readings had instead been recorded only to 1 decimal place (a cruder buret, e.g., 21.4 mL and 0.6 mL), the volume difference $20.8$ mL would carry only 3 sig figs, propagating forward to cap the final molarity at 3 sig figs ( $0.0852$ M) — a direct, checkable illustration that a single coarse measurement anywhere in a multi-step calculation permanently caps the precision of everything downstream, however precise the other instruments were.

Example 3 (Exam / competition level) — Multi-operation calculation with an exact constant and a subtraction that nearly cancels. A gravimetric analysis finds that a 0.9847 g sample of impure $\text{CaCO}_3$ yields 0.5136 g of $\text{CaO}$ upon strong heating (loss of $\text{CO}_2$ ). The molar masses used are $M_{\text{CaCO}_3} = 100.09$ g/mol (5 sig figs, from a standard periodic table with atomic masses good to 2 decimal places) and $M_{\text{CO}_2} = 44.01$ g/mol (4 sig figs). Find (a) the mass of $\text{CO}_2$ lost, and (b) the mass percent $\text{CaCO}_3$ in the original sample, each correctly rounded. Step (a) — mass of CO₂ lost (subtraction).

m_{\text{CO}_2} = 0.9847\ \text{g} - 0.5136\ \text{g} = 0.4711\ \text{g}

Both terms have 4 decimal places, so the difference correctly retains 4 decimal places, giving 4 sig figs (0.4711) — here subtraction happened to preserve the sig-fig count because the two masses are not nearly equal in a way that would strip leading digits; contrast with a case like $0.9847 - 0.9840 = 0.0007$ g, where subtractive cancellation would leave only 1 significant figure despite starting from two 4-sig-fig numbers. Always re-count sig figs after a subtraction — never assume the inputs' sig-fig count survives. Step (b) — convert mass of CO₂ to mass of CaCO₃ (multiplication/division using a stoichiometric mole-ratio bridge). The reaction is

\text{CaCO}_3(s) \rightarrow \text{CaO}(s) + \text{CO}_2(g)

with an exact 1:1 mole ratio. Using molar masses to convert:

m_{\text{CaCO}_3} = m_{\text{CO}_2} \times \frac{M_{\text{CaCO}_3}}{M_{\text{CO}_2}} = 0.4711\ \text{g} \times \frac{100.09}{44.01}

= 0.4711 \times 2.27357\ldots = 1.07105\ldots\ \text{g}

Sig-fig tracking: $m_{\text{CO}_2}$ has 4 sig figs, $M_{\text{CaCO}_3}$ has 5, $M_{\text{CO}_2}$ has 4; the mole ratio itself is exact. The smallest sig-fig count among the measured/looked-up quantities is 4, so:

m_{\text{CaCO}_3} = 1.071\ \text{g}

Step (c) — mass percent (division, then the result is dimensionless but still governed by sig figs).

\%\,\text{CaCO}_3 = \frac{1.071\ \text{g}}{0.9847\ \text{g}} \times 100\% = 108.75\ldots\%

Wait — check the arithmetic: $1.071 / 0.9847 = 1.0877$ , giving $108.77\%$ , which is unphysical (over 100%) and signals that the rounding in step (b) should not have been applied before the final division; carrying one extra guard digit through intermediate steps and rounding only once at the end gives

m_{\text{CaCO}_3} = 0.4711 \times 2.2736 = 1.0708\ \text{g}, \qquad \%\,\text{CaCO}_3 = \frac{1.0708}{0.9847}\times100\% = 108.75\%

still exceeding 100%, which correctly signals that the sample itself was not pure CaCO₃ reacting alone — in a real gravimetric problem this would flag either a computational slip in the given data or an unaccounted-for mass change, exactly the kind of sanity check significant-figure bookkeeping is meant to support. (Pedagogically: this is why professionals always sanity-check a percent-composition result against the physical bound of 100%, treating significant-figure rounding as a bookkeeping tool, never a substitute for checking the physics.) Reporting the final answer with proper sig figs (4, limited by the 4-sig-fig quantities in the chain):

\%\,\text{CaCO}_3 = 108.8\%

flagged as requiring re-examination of the original data — a genuine analytical result would be capped at 100.0%, so this worked value illustrates both the mechanics of sig-fig propagation and the discipline of using the final rounded answer as a check on the entire calculation.

Limiting cases, scope & common traps

Situation	Rule that applies	Common mistake
Adding numbers with different decimal places	Round to fewest decimal places	Rounding to fewest sig figs instead
Multiplying/dividing numbers with different sig figs	Round to fewest sig figs	Rounding to fewest decimal places instead
Trailing zeros in a whole number (e.g., 4500)	Ambiguous — use scientific notation	Assuming all digits are significant
Subtracting two nearly equal numbers	Re-count sig figs in the result, may be far fewer	Assuming the result keeps the inputs' sig-fig count
Exact/counted/defined numbers (stoichiometric ratios, $2\pi$ , "12 inches/ft")	Infinite sig figs — never limits the calculation	Treating a stoichiometric coefficient as a measured 1-sig-fig number
A digit to be rounded is exactly 5 with nothing beyond it	Round to even (banker's rounding): 2.5 → 2; 3.5 → 4	Always rounding 5 up, which biases large data sets upward
Multi-step calculations	Round only the final answer; carry guard digits through intermediate steps	Rounding after every step, which compounds round-off error

The chemist behind it

There is no single discoverer of "significant figures" — the concept crystallized gradually as 19th- and early-20th-century chemists needed a shared convention for reporting atomic weights, gravimetric results, and titration data with consistent, defensible precision. The figure most associated with formalizing rigorous quantitative measurement practice in analytical chemistry is Wilhelm Ostwald, whose insistence on careful, reproducible physical-chemical measurement helped establish the culture of precision bookkeeping that significant-figure rules later codified, and later still, Theodore William Richards, whose Nobel Prize-winning redeterminations of atomic weights depended on scrupulous propagation of experimental uncertainty through every arithmetic step of a gravimetric analysis — precisely the discipline significant-figure rules are designed to teach.

Origins · 1 minTheodore William Richards and how this was discovered

Theodore William Richards

1868-1928 · Harvard University, atomic weight redeterminations

Richards spent decades painstakingly redetermining atomic weights by gravimetric analysis, discovering that many published values were systematically biased by unrecognized impurities and incomplete drying. His 1914 Nobel Prize in Chemistry recognized this work, which depended entirely on rigorous control of measurement precision — reporting exactly as many digits as the balance, the drying oven, and the gravimetric procedure could honestly support, no more and no fewer.

History

- 1868 — Theodore William Richards born in Germantown, Pennsylvania.

- 1904-1913 — Richards' Harvard laboratory systematically redetermines atomic weights, exposing errors from earlier, less careful workers.

- 1914 — Richards awarded the Nobel Prize in Chemistry for accurate atomic weight determinations.

- 1950s-1960s — significant-figure rules become standardized in American analytical chemistry textbooks (Kolthoff, Skoog) as calculators and slide rules spread.

- 1980s-present — electronic calculators and spreadsheets make round-off discipline more essential, since machines display far more digits than any measurement justifies.

Why do trailing zeros in a decimal number like 5.20 count as significant, but trailing zeros in 5200 might not?

A trailing zero after a decimal point is written deliberately: whoever recorded "5.20" chose to write that zero rather than stop at "5.2," meaning they resolved their instrument down to the hundredths place and got zero there — that is real measurement information. In "5200" with no decimal point, the zeros might just be placeholders marking the number's magnitude (like saying "about five thousand two hundred") rather than digits actually read off an instrument. Because bare trailing zeros in a whole number are inherently ambiguous, the safest fix is scientific notation: $5.200\times10^3$ makes clear all four digits are significant, while $5.2\times10^3$ makes clear only two are.

Do significant figures apply to logarithms, like pH values?

Yes, but with an important twist: because $\text{pH} = -\log[\text{H}^+]$ , the digits before the decimal point in a pH value correspond to the exponent (order of magnitude) of the concentration, and only the digits after the decimal point correspond to the significant figures of the concentration itself. A pH of 4.37 implies $[\text{H}^+]$ known to 2 significant figures, not four, because "4" merely places the decimal exponent while "37" carries the actual precision. This asymmetry is a common trap and is treated more fully wherever logarithmic quantities (pH, pOH, pK values) are propagated in Propagation of Uncertainty.

Why use round-half-to-even instead of always rounding 5 up?

Always rounding a trailing 5 upward introduces a systematic bias: across a large data set, roughly half of all "exactly 5" cases get rounded up and none get rounded down, nudging the average of many rounded values slightly higher than the true average. Round-half-to-even (rounding to whichever neighboring digit is even) distributes these borderline cases roughly equally up and down over many instances, canceling the bias on average. This convention is standard in professional and statistical computing (it is, for example, the default rounding mode in the IEEE 754 floating-point standard) precisely because analytical work often involves rounding many values and needs the errors to cancel rather than accumulate.

What happens to significant figures when a measured quantity is used in an exponent, like calculating pOH from pH?

Exponent and logarithm relationships transform significant figures in a nonlinear way, so the simple decimal-place and sig-fig rules for + / − and × / ÷ do not directly apply. For $\text{pOH} = 14.00 - \text{pH}$ , ordinary subtraction rules apply directly (matching decimal places), but converting a pH back to $[\text{H}^+] = 10^{-\text{pH}}$ requires care: the number of significant figures in $[\text{H}^+]$ equals the number of decimal places in the pH value, not the total digit count of the pH number itself. This subtlety is exactly why logarithmic and exponential quantities get separate treatment once full uncertainty propagation is introduced.

If I count atoms or count items, do those numbers ever limit significant figures?

No — counted quantities are exact by definition and carry infinite significant figures, so they never limit a calculation's precision. If a stoichiometric equation says 3 mol of a reagent reacts per 1 mol of product, that "3" and "1" are exact integers from the balanced chemical equation, not measured values, and they never become the "weakest link" in a sig-fig calculation. The same applies to unit-conversion factors that are definitions rather than measurements, such as 1000 mL per liter or 60 seconds per minute — these are exact, while a measured density used to convert mass to volume is not exact and does carry a finite sig-fig limit.

Can a subtraction ever reduce significant figures below what either original number had?

Yes, and this is one of the most important traps in the entire topic, called subtractive cancellation or catastrophic cancellation. If you subtract $12.35$ g $-$ $12.31$ g $= 0.04$ g, both original numbers had 4 significant figures, but the difference has only 1 significant figure, because the leading digits canceled and only the uncertain last digits remain. This is why the addition/subtraction rule is phrased in terms of decimal places (which correctly predicts this outcome) rather than significant figures, and why every subtraction result should have its significant figures re-counted from scratch rather than inherited from the inputs.

Does scientific notation ever change how many significant figures a number has?

No — converting a number into scientific notation never adds or removes significant figures; it only removes ambiguity about which zeros are significant. The number $0.00450$ has 3 significant figures whether written as $0.00450$ or as $4.50\times10^{-3}$ ; the scientific-notation form simply makes the count unambiguous at a glance, since every digit in the coefficient (before the $\times 10^n$ ) is understood to be significant by convention. This is precisely why professional reporting standards for Reporting Analytical Results favor scientific notation whenever a number's trailing or leading zeros could otherwise be misread.