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A.0 Arithmetic and numerical computation

Syllabus
2021
Topic
—
Level
A2

Learning objectives

—A.0.1—Units in calculationsRecognise and use appropriate units in calculations, including converting units such as mm³ to cm³ in volumetric calculations and determining units for rates such as breathing rate.—A.0.2—Decimal and standard formRecognise and use decimal and standard form; choose appropriate decimal places; calculate with ordinary and standard form; apply standard form to organelle sizes; convert between forms while retaining significant figures, for example 0.0050 mol dm⁻³ = 5.0 × 10⁻³ mol dm⁻³.—A.0.3—Ratios, fractions and percentagesUse ratios, fractions and percentages, including percentage yields, surface-area-to-volume ratios, measurement scales and phenotypic ratios in monohybrid and dihybrid crosses.—A.0.4—Estimating resultsEstimate results to sense-check whether calculated values are appropriate.—A.0.5—Powers, exponentials and logarithmsUse calculators to find and use power, exponential and logarithmic functions, including estimating bacterial population growth over time.—A.0.P—SI unit prefixesKnow and convert between the SI prefixes kilo, centi, milli, micro and nano, as clarified by Pearson for International A Level Biology candidates.

Derive and convert units in biological calculations

Treat units algebraically: write the equation, convert inputs to a common system, cancel units during substitution and derive the output unit from numerator and denominator. A correct number with an incompatible unit is not a correct biological quantity.

A linear conversion must be raised to match area or volume. Because 1 cm=10 mm1\text{ cm}=10\text{ mm}, 1 cm2=100 mm21\text{ cm}^2=100\text{ mm}^2 and 1 cm3=1000 mm31\text{ cm}^3=1000\text{ mm}^3. Likewise 1 dm3=1000 cm31\text{ dm}^3=1000\text{ cm}^3.

A rate is quantity divided by time, so its unit must preserve both: breathing rate is breaths min−1^{-1}, oxygen uptake may be mm3^3 min−1^{-1}, and mass-specific respiration may be mm3^3 min−1^{-1} g−1^{-1}. For a ratio, check whether units cancel or remain.

Alveolar ventilation with tidal volume 500 cm3^3, dead space 150 cm3^3 and 12 breaths min−1^{-1} is (500−150)×12=4200(500-150)\times12=4200 cm3^3 min−1=4.2^{-1}=4.2 dm3^3 min−1^{-1}. Per hour this is 4.2×60=2.5×1024.2\times60=2.5\times10^2 dm3^3 h−1^{-1} to two significant figures.

Do not multiply a cubic unit by only the linear conversion factor, and never drop the time, area or mass basis when reporting a rate.

Move between decimal and standard form without losing precision

Standard form is a×10na\times10^n with 1≤∣a∣<101\leq |a|<10. Moving the decimal left gives a positive exponent; moving it right gives a negative exponent. The notation changes, but the value and significant figures must not.

Operation Rule Example
Decimal → standard Count decimal-place moves 0.0050=5.0×10−30.0050=5.0\times10^{-3} (two significant figures)
Standard → decimal Shift according to exponent 4.5×10−4=0.000454.5\times10^{-4}=0.00045
Multiply Multiply coefficients; add powers (2×103)(3×10−5)=6×10−2(2\times10^3)(3\times10^{-5})=6\times10^{-2}
Divide Divide coefficients; subtract powers (8×106)/(2×102)=4×104(8\times10^6)/(2\times10^2)=4\times10^4

Keep unrounded calculator values through intermediate steps, then round the final answer to the required significant figures. Decimal places describe position after the decimal point; significant figures begin at the first non-zero digit. Trailing zeros can be significant: 0.0050 records more precision than 0.005.

Standard form makes organelle dimensions, concentrations and population sizes auditable. After using magnification=image size/actual size\text{magnification}=\text{image size}/\text{actual size}, check that the exponent gives a plausible cell or organelle scale.

Normalising 15×10315\times10^3 to 1.5×1041.5\times10^4 does not change its value. Do not round away a significant trailing zero during format conversion.

Use ratios, fractions and percentages on the correct basis

Before calculating, name the numerator, denominator and comparison direction. Convert quantities to common units and use the original or total value required by the definition—not whichever number is easiest to divide by.

Biological use Calculation Example or interpretation
Percentage change (new−original)/original×100(\text{new}-\text{original})/\text{original}\times100 40 to 50 is a 25% increase
Percentage yield actual yield/theoretical yield×100\text{actual yield}/\text{theoretical yield}\times100 Yield cannot be interpreted without the theoretical basis
Surface area : volume Calculate each from dimensions, then simplify SA:VSA:V Cube side 22: SA=24SA=24, V=8V=8, so 3:13:1
Scale image length/actual length\text{image length}/\text{actual length} after unit conversion A scale bar remains valid when the whole image is resized
Phenotypic ratio Count each phenotype in a fixed stated order A monohybrid expectation may be 3:13:1; a dihybrid expectation may be 9:3:3:19:3:3:1 under the model's assumptions

A fraction can be converted to a proportion or percentage: 18 affected plants out of 120 gives 18/120=0.15=15%18/120=0.15=15\%. Keep observed ratios separate from theoretical ratios and compare them only after defining categories in the same order.

A 2:1 ratio is ordered, while 'twice as many' must state which category is twice the other. Percentage change uses the original value; percentage of a total uses the total as denominator.

Estimate before calculating so an answer can be checked

An estimate gives the expected scale of a result before exact calculation. It is a reasonableness check, not a replacement for showing the measured or calculated value.

Round inputs to one useful figure, keep powers of ten visible and compare the exact answer with the estimate; investigate a large mismatch.

A 2.0 cm² leaf area with a flux near 5 units cm⁻² h⁻¹ should produce a total near 10 units h⁻¹, not 10,000.

Do not use rough rounding when a small difference is the biological conclusion.

Use powers, exponentials and logarithms for bacterial growth

Multiplicative growth adds the same factor, not the same number, in equal time intervals. If an initial population N0N_0 undergoes nn doublings, N=N0×2nN=N_0\times2^n. If it doubles every gg time units for total time tt, then n=t/gn=t/g.

A culture starts with 500 bacteria and doubles every 20 minutes. After 2 hours, n=120/20=6n=120/20=6, so N=500×26=32,000N=500\times2^6=32{,}000. This estimate assumes constant exponential growth without resource limitation or death.

A logarithm reverses a power: if log⁡10N=5.30\log_{10}N=5.30, then N=105.30≈2.0×105N=10^{5.30}\approx2.0\times10^5. On a base-10 logarithmic axis, equal vertical steps represent equal multiplication; an increase of 1 means tenfold, while a doubling changes log⁡10N\log_{10}N by log⁡102≈0.301\log_{10}2\approx0.301.

Enter brackets explicitly, retain intermediate precision and test the output by substituting back. Use the calculator's 10x10^x, exe^x, log⁡\log or ln⁡\ln function that matches the model; changing log base without changing the equation changes the result.

Exponential growth cannot continue indefinitely in a closed culture. Do not extend an exponential estimate through lag, stationary or death phases without evidence.

Choose SI prefixes before comparing biological measurements

SI prefixes change the scale of a unit: milli is 10⁻³, micro 10⁻⁶, nano 10⁻⁹ and kilo 10³. Convert to one common unit before comparing or substituting.

Write the prefix as a power of ten, cancel units algebraically and check whether the direction of the conversion makes sense.

250 μm = 250 × 10⁻⁶ m = 2.50 × 10⁻⁴ m; it is smaller than 250 mm, not larger.

The prefix belongs to the unit, not to the numerical value; mixing μm and mm can create a thousand-fold error.