1 Arithmetic and numerical computation
- Syllabus
- 2024
- Topic
- 1
- Level
- —
Learning objectives
11A Decimal-form numbersRecognise and use numbers in decimal form.11B Standard-form numbersRecognise and use numbers in standard form.11C Ratios, fractions, percentages, powers and rootsUse ratios, fractions, percentages, powers and roots.11D Estimate simple calculationsEstimate the results of simple calculations without using a calculator.
Use decimal numbers without losing place value
A decimal records place value on both sides of the decimal point. Keep digits aligned by place value during addition or subtraction, and preserve units before carrying out a biological calculation.
| Move | Example |
|---|---|
| identify place value | in 0.025, 2 is hundredths and 5 is thousandths |
| align or convert units | 90 cm = 0.90 m before combining it with a speed in m s⁻¹ |
| calculate with the full values | 0.25 m ÷ 0.005 s = 50 m s⁻¹ |
| check position of the decimal point | dividing by a number smaller than 1 should increase the numerical value |
| report meaning and unit | write 50 m s⁻¹, not an unlabelled 50 |
A zero before the decimal point makes values below one unambiguous: write 0.5, not .5. Trailing zeros may communicate recorded precision—9.0 mg is expressed more precisely than 9 mg—even though the numerical value is equal.
Do not align numbers by their final digit or remove zeros before deciding what they communicate. A correct decimal with mismatched units is still a wrong biological calculation.
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. The coefficient shows the significant digits; the power of ten shows scale.
| Operation | Rule | Example |
|---|---|---|
| decimal to standard form | move the point until the coefficient is between 1 and 10; count places | 480000000=4.8×108 |
| standard to decimal | positive power moves right; negative power moves left | 2.0×10−1=0.20 |
| multiply | multiply coefficients and add powers | (5.0×109)(5.0×103)=25×1012=2.5×1013 |
| divide | divide coefficients and subtract powers | (6.0×108)/(3.0×102)=2.0×106 |
| add or subtract | first express terms with the same power of ten | 3.2×106+0.5×106=3.7×106 |
After calculating, renormalise the coefficient. 25×1012 has the right value but is not standard form because 25 is not between 1 and 10.
Choose the right proportional calculation
Ratios, fractions and percentages express one quantity relative to another. The denominator or comparison base must match the biological question before any arithmetic is performed.
| Need | Relationship | Worked biological example |
|---|---|---|
| simplify a ratio | divide every part by the same factor | 21 yellow : 10 brown = 2.1 : 1 |
| find a fraction of a total | fraction × total | half of 65 million = 0.5 × 65 million = 32.5 million |
| find a percentage of an amount | percentage ÷ 100 × amount | 16.7% of 28.0 g = 0.167 × 28.0 = 4.68 g |
| express a part as a percentage | part ÷ whole × 100 | 56 farmed out of 146 total = 38.4% |
| percentage change | change ÷ original × 100 | from 0.70 to 10.0 million: 9.30 ÷ 0.70 × 100 = 1330% approximately |
| powers and roots | a power repeats multiplication; a root reverses that power | 32=9 and 9=3 |
For percentage change, divide by the original value, not the final value. A percentage may exceed 100% when the increase is greater than the original. Keep ratios in the requested order and attach units to calculated amounts.
Estimate first to check a calculation
An estimate replaces values with nearby easy numbers to find an approximate result without a calculator. Its purpose is to predict the scale and expose implausible calculator entries, not to replace a required accurate answer.
| Move | Example using (4.8×108)(0.0040) |
|---|---|
| round to one useful significant figure | 4.8×108≈5×108 and 0.0040=4×10−3 |
| calculate mentally | (5×4)×108−3=20×105 |
| write the scale clearly | 20×105=2×106 |
| compare with the accurate result | 1.92×106 is close to 2×106, so its order of magnitude is plausible |
Round enough to make the arithmetic simple but keep the important scale. If one value is just above and another just below a convenient number, note whether both rounding choices push the estimate in the same direction.
An estimate should not contain more apparent precision than the original calculation. A result of 2.000×106 is not a sensible estimate here; write about 2×106 and retain the approximation sign or wording.