1 Arithmetic and numerical computation
- Syllabus
- 2024
- Topic
- 1
- Level
- —
Learning objectives
Use decimal numbers in chemistry calculations
A decimal digit has a value fixed by its position: moving one place left multiplies its value by 10, while moving one place right divides it by 10.
| Number | Meaning |
|---|---|
| 3.2 | 3 ones and 2 tenths |
| 0.32 | 3 tenths and 2 hundredths |
| 0.032 | 3 hundredths and 2 thousandths |
Line up decimal points when adding or subtracting. For multiplication or division, calculate with the full values and check the order of magnitude; keep the unit attached to the final chemical quantity.
\text{mean rate}=\frac{1}{156,\mathrm{s}}=0.00641,\mathrm{s^{-1}}
Zeros before the first non-zero digit locate the decimal point and are not optional. Do not drop a leading zero or move the decimal point without applying the corresponding power of ten.
Write and calculate with standard form
Standard form writes a non-zero number as a×10n, where 1≤∣a∣<10 and n is an integer.
| Decimal form | Standard form | Direction |
|---|---|---|
| 6410 | 6.41×103 | decimal moved 3 places left, so exponent is +3 |
| 0.00641 | 6.41×10−3 | decimal moved 3 places right, so exponent is −3 |
(a\times10^m)(b\times10^n)=ab\times10^{m+n},\qquad \frac{a\times10^m}{b\times10^n}=\frac{a}{b}\times10^{m-n}
After calculating, normalise the coefficient. For example, 12.8×10−4=1.28×10−3. State the unit after the complete standard-form value.
A coefficient such as 12.8 is not final standard form. A negative exponent makes a small positive magnitude; it does not make the number negative.
Use ratios, fractions, percentages, powers and roots
Ratios, fractions and percentages express relative amounts; powers and roots compact repeated multiplication and reverse it.
| Tool | Reusable method | Chemistry-style example |
|---|---|---|
| ratio | divide by the same factor to simplify, or multiply every part by the same scale factor | 2:1 means two reacting units for every one of the other |
| fraction | numerator is the selected part; denominator is the total | fraction of oxygen =total amountoxygen amount |
| percentage | multiply a fraction by 100%; divide a percentage by 100% to recover the fraction | % yield=theoreticalactual×100 |
| power | xn means multiply x by itself n times | area scale uses a square, x2 |
| root | ny reverses raising to power n | if x2=y, then positive length x=y |
If a formula requires a 2:3 ratio and the first amount is 0.40 mol, one ratio part is 0.20 mol and the second amount is 3×0.20=0.60 mol.
Use the same units in numerator and denominator before forming a fraction or percentage. Check whether the context permits values above 100%; percentage yield normally should not, so a value above 100% signals data or method error.
Do not add ratio parts when the task asks for one scaled amount, confuse a percentage with its decimal multiplier, or halve a power when the operation requires a root.