Every answer in this course is a number, a unit, and a statement about how well you know it. This page is the lookup sheet for the last two. Working through the habits is Significant Figures in Practice; taking the measurements well in the first place is Measuring Well.
Counting significant figures
| Situation | Significant? | Example |
|---|---|---|
| Any non-zero digit | always | 4.72 has three |
| Zeros between non-zero digits | always | 3.05 has three; 1002 has four |
| Zeros before the first non-zero digit | never — they only place the decimal | 0.0042 has two |
| Zeros after the last non-zero digit, when a decimal point is present | yes | 2.50 has three; 0.04200 has four |
| Zeros after the last non-zero digit, with no decimal point | ambiguous | 1500 could be two, three, or four |
| Counted or defined numbers | unlimited — they never limit an answer | 12 in a dozen; the 1000 in 1 L = 1000 mL; the coefficients in a balanced equation |
The ambiguous row is solved by scientific notation, which is the reason scientific notation exists. Writing says two significant figures; says four. Both are 1500 and they make different claims about the measurement.
Calculating with them
| Operation | Rule | Example |
|---|---|---|
| Multiplication and division | the answer keeps the fewest significant figures of any input | |
| Addition and subtraction | the answer keeps the fewest decimal places of any input | |
| Several steps | carry one or two extra digits through and round once, at the end | — |
| Exact numbers | ignore them when deciding — they are infinitely precise | — |
The two rules are genuinely different and mixing them up is the most common error. Multiplication counts significant figures; addition counts decimal places. In the addition example above, 12.11 has four significant figures and 1.4 has two, but the answer has three — because 1.4 is only known to one decimal place, and adding a number you know to the tenth cannot produce an answer good to the hundredth.
Where the rule earns its keep
A balance reads 1.0 g for a paperclip. Divide by the molar mass of iron, 55.85 g/mol, and the calculator returns 0.017905. Multiply by Avogadro’s number and it returns .
The correct answer is atoms — two significant figures, because the mass had two. Writing claims you can distinguish that paperclip from one containing atoms, using a balance that could not tell 1.0 g from 1.04 g. The extra digits are not more accurate; they are a false claim about the instrument.
Reading an instrument
- Analogue scales — record every digit you are certain of, and then estimate one more between the finest graduations. A ruler marked in millimetres gives you readings to a tenth of a millimetre, estimated.
- Digital displays — record every digit shown, including trailing zeros. A balance reading 2.50 g is claiming three significant figures and you must write all three.
- Liquid volumes — read the bottom of the meniscus, at eye level. Looking down at it introduces an error that is consistent, invisible, and entirely yours.
- Burettes — the scale increases downwards, and the volume delivered is the final reading minus the initial one. Both readings need the estimated digit.
Units you will use constantly
| Quantity | Unit | Symbol |
|---|---|---|
| Amount of substance | mole | mol |
| Mass | gram, kilogram | g, kg |
| Volume | litre, millilitre | L, mL |
| Molar mass | grams per mole | g/mol |
| Concentration | moles per litre | mol/L |
| Pressure | kilopascal | kPa |
| Temperature | kelvin, degree Celsius | K, °C |
| Prefix | Symbol | Multiplier |
|---|---|---|
| kilo | k | |
| deci | d | |
| centi | c | |
| milli | m | |
| micro | µ | |
| nano | n |
Conversions worth knowing by heart
- 1 L = 1000 mL, and 1 mL = 1 cm³
- 1 kg = 1000 g
- in kelvins in degrees Celsius
- 1 atm = 101.325 kPa = 760 mmHg
- 1 kPa = 1000 Pa
Kelvin is not optional in a gas law
Every temperature in every calculation in The Gas Laws is in kelvins. Using Celsius does not give an approximate answer; it gives a wrong one, because the gas laws are proportionalities and Celsius has its zero in an arbitrary place.
Carrying units through the calculation
Write the units into the working, not just onto the answer. They cancel like algebra, and when they do not cancel you have caught a mistake before it reached the page:
Grams over grams-per-mole leaves moles, which is what you wanted. Multiplying instead would have given , which is not a quantity anything has. This is the fastest self-check available in Stoichiometry, and it costs nothing.
An answer with no units is not an answer. “29.9” could be grams, moles, or molecules, and a marker cannot award anything for a number that has not said what it is.