Units and Sig Figs Tutor: Method

Units and Sig Figs Tutor

What the tool does

The tool teaches the two habits that every quantitative subject expects: converting a quantity from one unit to another by the factor-label method, and writing a result with the digits it deserves. It has three parts. The conversion part builds the chain of factors for any conversion whose units it knows, prints each factor with its source and its exactness, shows the unit left after every factor, and rounds the result by the reasoning of NIST Special Publication 811. The significant-figures part counts the significant figures of a number with the rule that applies to each zero, works an arithmetic expression by the textbook rules, rounds by the ASTM E29 method (round half to even) and by the school rule side by side, and compares the implied uncertainty of the rounded result with a propagation of the operands' implied uncertainties. The practice part generates problem sets from a seed and scores pasted answers.

The factor-label method

A conversion factor is a fraction whose numerator and denominator are the same quantity written in two units, for example 12 in over 1 ft. Because the two are equal the fraction equals one, and multiplying a quantity by it changes the unit without changing the quantity. The units are treated as algebraic symbols: a unit in a numerator cancels the same unit in a denominator, and a chain of factors is complete when only the target unit is left. The tool writes the chain exactly this way, as a product of stacked fractions, and lists the unit remaining after each factor so that the cancellation can be followed. A compound unit such as the pound per square inch is first written out in its defining units (1 psi = 1 lbf/in²), each of those is converted, and the SI result is recomposed (1 N/m² = 1 Pa, 1000 Pa = 1 kPa). A factor raised to a power converts a unit that appears with that power: (2.54 cm / 1 in)² for square inches.

Two chains are offered. The textbook chain uses only the defining relations a student is expected to carry in memory: 1 mi = 5280 ft, 1 ft = 12 in, 1 in = 2.54 cm, 1 h = 60 min, 1 min = 60 s, 1 lb = 0.453 592 37 kg, 1 lbf = 4.448 221 615 N, 1 gal = 231 in³, 1 atm = 101 325 Pa, 1 Btu = 1055.056 J, the SI prefixes, and the definitions of the SI derived units (1 N = 1 kg·m/s², 1 Pa = 1 N/m², 1 J = 1 N·m, 1 W = 1 J/s). The chain is the shortest path through these relations, found by a breadth-first search over the units of each dimension, so 60 mi/h to m/s takes six factors. The fewest-factors chain adds the direct factors of NIST SP 811 Appendix B.8 (1 mi = 1609.344 m, 1 h = 3600 s, 1 psi = 6.894 757 kPa, 1 hp = 745.699 9 W) and gives the chain an engineer writes. Both chains give the same product; they differ only in how many factors are shown.

Every factor is marked exact or rounded. The inch, foot, yard, mile, pound, pound-force, standard atmosphere, bar, calorie, International Table Btu, horsepower, gallon and acre are exact by definition (NIST SP 811 prints exact factors in boldface); the conventional millimeter and inch of mercury, the inch of water and the thermochemical values that SP 811 prints to seven digits are rounded, and a rounded factor limits the digits of the result when it has fewer than the input. Where SP 811 prints a rounded seven-digit value for a quantity that is exact by definition (1 lbf = 4.448 222 N, 1 slug = 14.593 90 kg), the tool uses the exact value and says what SP 811 prints.

Temperature scales

Degrees Fahrenheit, Celsius, Rankine and kelvins are converted by the affine relations of SP 811 Appendix B.8 (t/°C = (t/°F − 32)/1.8, T/K = t/°C + 273.15), not by a factor, because the zero points differ. A temperature interval, as in Btu/(lb·°F), does convert by a factor (1 °F = 5/9 K), and the tool uses the interval factor whenever the degree appears inside a compound unit.

Rounding a converted value

A measured value written as 36 ft could have come from anything between 35.5 ft and 36.5 ft: it carries a relative maximum rounding error of ±0.5/36 = ±1.4 %. SP 811 Appendix B.7.2 says a converted value should be rounded so that its own rounding error is consistent with that of the input; its example converts 36 ft to 10.9728 m and keeps 11.0 m, because 11 m (±4.5 %) would discard information the input carried and 10.97 m (±0.046 %) would claim precision it never had. The tool prints this comparison for every conversion: the input's number of significant figures with the reason for each digit, the relative error it implies, and a small table of the candidate roundings with their errors. The row kept is the one with the same number of significant figures as the input (fewer when a rounded factor has fewer digits), which is the usual rule and which SP 811's reasoning supports in all but borderline cases; the table lets the reader judge the borderline ones, as SP 811 asks. An input whose trailing zeros are ambiguous (60 mi/h) is flagged, and the results for one more significant figure and for an exact input are given beside it. A quantity that is exact or defined (a count, a definition) does not limit the result at all, and the tool says what the factors alone give.

Counting significant figures

The significant figures of a number are the digits that carry information about its value. Every non-zero digit is significant. A zero between two non-zero digits (a captive zero: 1.013, 405) is significant because it was measured. A zero before the first non-zero digit (a leading zero: 0.00340) is not significant: it only places the decimal point, and the same quantity in another unit would have different leading zeros. A zero after the last non-zero digit (a trailing zero) is significant when the number is written with a decimal point (0.00340 has three significant figures, 100.0 has four, 1200. has four) or in scientific notation (1.200 × 10³ has four), and is ambiguous in an integer written without a decimal point: 1200 may mean two, three or four significant figures. NIST SP 811 §7.9 gives exactly this example and recommends a prefix or scientific notation to remove the doubt (1.200 km). The tool counts such zeros as not significant, which is the common classroom convention, and says so each time. Zero itself is counted as one significant figure.

Arithmetic with significant figures

Two rules cover the four operations. In addition and subtraction the result is rounded to the decimal place of the least precise term: 12.11 + 18.0 + 1.013 = 31.123 becomes 31.1 because 18.0 is known only to tenths. In multiplication and division the result keeps the number of significant figures of the factor with the fewest: 1.23 × 4.5 = 5.535 becomes 5.5 because 4.5 has two. A power with an exact integer exponent keeps the base's significant figures. The tool evaluates an expression exactly in decimal arithmetic (so that 1.23 × 4.5 is 5.535 and not a binary approximation) and rounds once at the end, as the rules require; when an intermediate result enters the next operation, it carries the precision the rule gives it (a sum enters a product with the significant figures of its rounded value, a product enters a sum with the decimal place of its rounded value). The steps table shows each operation, its exact value, the rule that applies and the rounded value. Mixed-type expressions, (2.0 + 3.14) × 1.5 for instance, follow the inner-to-outer order in which the rules are taught.

Rounding: half to even against half up

Rounding is unambiguous except when the digits dropped are exactly a 5 followed by nothing or by zeros. ASTM E29 (its Rounding Method, §3.4.3 of the 1967 edition, unchanged in substance since) and NIST SP 811 B.7.1 then keep the digit that makes the result even: 6.974 950 5 to seven digits is 6.974 950, 2.345 to two decimals is 2.34, 56.5 to a whole number is 56. The school rule rounds a 5 up every time (2.35, 57). Half to even is preferred in standards because over many roundings half the ties go down and half go up, so the rounded values carry no bias, whereas half up biases every tie in the same direction. The tool applies the rule chosen in the form to its results and shows both whenever they differ, decided on the exact decimal digits. Two further E29 rules are illustrated: rounding is done in one step from the most precise value (89,490 psi to the nearest 1000 psi is 89,000, not 89,500 and then 90,000), and rounding to the nearest 50, 5, 0.5 or 0.05 is done by doubling the value, rounding to the nearest 100, 10, 1 or 0.1 and halving (6025 to the nearest 50 is 6000; 6075 is 6100). E29's example table (tensile strength 59,950 psi reported to 100 psi becomes 60,000 and conforms to a 60,000 minimum; 56.5 % nickel reported to 1 % becomes 56 and fails a 57 % minimum) is reproduced in the test script.

Implied uncertainty and its propagation

A number written to its last significant digit implies an uncertainty of half a unit in that place: 4.5 means 4.45 to 4.55, an absolute uncertainty of ±0.05 and a relative one of ±1.1 %. Absolute uncertainties add in a sum or difference and relative uncertainties add in a product or quotient; that is why the two rules above have the form they do. The tool propagates the operands' half-units two ways: as a worst-case bound (the sums just described) and in quadrature (the root sum of squares of JCGM 100, the Guide to the Expression of Uncertainty in Measurement, §5.1, which applies when the errors are independent), and sets both beside the half-unit the rounded result itself implies. The comparison shows what the significant-figure rules are: a convention for writing results whose implied precision stays within about a factor of ten of the propagated uncertainty, not an uncertainty analysis. When a real uncertainty statement is needed, the Uncertainty Budget tool does that work.

Practice problems

Problems are generated from the seed with the Park–Miller minimal-standard generator, so a seed, count, level and type always give the same set and a class can work the same problems. Conversion problems draw from pairs chosen by level (single factors; two-factor and compound chains; compound units with all zero patterns); counting problems draw from six patterns of zeros; arithmetic problems draw random operands with two to five significant figures. An answer to a conversion or an arithmetic problem is marked right when its value lies within half a unit of the expected answer's last place and it has the expected number of significant figures; the verdict says which of the two failed. A count must match exactly. To see the worked solution of any problem, paste its line into the conversion or significant-figures part.

Units accepted

Length m, in, ft, yd, mi, nmi, mil; mass g, lb (pound-mass; lbm accepted), oz, slug, ton (short), t (tonne); time s, min, h, d, yr (365 d); temperature K, degC, degF, degR (°C, °F, °R and the single letters C, F, R are accepted); angle rad, deg, rev; force N, lbf, kip, dyn, kgf; pressure Pa, psi, ksi, bar, atm, torr, mmHg, inHg, inH2O; energy J, Btu (International Table), cal (thermochemical, 4.184 J), calIT, eV, erg; power W, hp; volume L, gal, qt, pt, floz, bbl; area acre, ha; frequency Hz; rpm, mph, knot. Any SI unit takes an SI prefix (mm, kPa, MJ, kW, mL, kcal). Products are written with · or * or a space, quotients with /, powers with ^ (lb/ft^3, ft·lbf, Btu/(lb·degF), m^3); kWh, psf, pcf, gpm, cfm, cfs, mpg and kph are expanded. Spelled-out names (inches, pounds, gallons) are accepted for the common units.

Limits

The tool converts by multiplicative factors and the four temperature scales only; it has no electrical, magnetic or photometric units beyond the ampere, mole and candela, no U.S. survey foot (its acre and square mile are those of the international foot, 4046.856 m² rather than SP 811's 4046.873 m²), no British imperial gallon, and no currency. The factor table is the tool's own, kept framework-free so that it can be tested alone; the site's dictionary, which carries other spellings, is not consulted. Significant-figure arithmetic covers + − × ÷ and integer powers; functions (logarithms, trigonometry) follow different rules and are not offered. The implied-uncertainty comparison treats every operand as uniformly uncertain by half a unit in its last place, which is the convention of the rules and not a measured uncertainty.