A.2 Algebra

Learning objectives

—A.2.1—Mathematical symbolsUnderstand and use the symbols =, <, ≪, ≫, >, ∝ and ~.—A.2.2—Changing the subject of an equationChange the subject of an equation and manipulate equations in biological contexts such as magnification.—A.2.3—Substitution into algebraic equationsSubstitute numerical values into algebraic equations using appropriate physical units, including using a supplied formula to calculate an index of diversity.—A.2.4—Solving algebraic equationsSolve algebraic equations in biological contexts, for example cardiac output = stroke volume × heart rate.

A symbol states a relationship between two quantities. Read the whole expression in words, identify the comparison direction and check whether it claims exact equality, inequality, proportionality or approximation.

Symbol	Meaning	Biological reading
$=$	equal to	$N=200$ states an exact recorded/model value
$<$ / $>$	less than / greater than	$p<0.05$ compares a probability with a threshold
$\ll$ / $\gg$	much less than / much greater than	A virus may be $\ll$ the size of a eukaryotic cell
$\propto$	proportional to	$V\propto r^3$ means a constant ratio $V/r^3$ under the model
$\sim$	approximately or of similar order, as defined by context	$N\sim10^6$ gives an approximate scale, not exact equality

From $y\propto x$ , write $y=kx$ only after introducing the constant $k$ . If $y\propto x^2$ , doubling $x$ multiplies $y$ by four, provided other conditions remain fixed.

$\propto$ does not mean 'is correlated with', and $\ll$ is stronger than $<$ . The symbol $\sim$ is context-dependent, so state whether it means approximate equality or similar order of magnitude.

To make a variable the subject, apply inverse operations to both sides while preserving brackets and powers. Rearrange symbolically first, then substitute values and verify by putting the result back into the original equation.

From $M=I/A$ , where $M$ is magnification, $I$ image size and $A$ actual size: multiply by $A$ to get $MA=I$ , so $A=I/M$ ; alternatively $I=MA$ . Image and actual size must use common units.

For $BMI=m/h^2$ , multiply by $h^2$ : $m=BMI\times h^2$ . With $BMI=36$ kg m $^{-2}$ and $h=1.55$ m, $m=36(1.55)^2=86.5$ kg to three significant figures.

Use dimensions to test the rearrangement: $(\text{kg m}^{-2})(\text{m}^2)=\text{kg}$ . A form that leaves the intended subject on both sides or gives incompatible units is not finished.

Do not move a term across an equals sign by changing it mechanically; identify the inverse operation. Squared denominators require multiplying by the whole square, not just the unsquared variable.

Write the formula, define each symbol, convert every input to the required unit, substitute with brackets, keep guard digits and round only the final answer. This makes both the arithmetic and biological meaning auditable.

For $D=\dfrac{N(N-1)}{\sum n(n-1)}$ , $N$ is the total individuals and each $n$ is one species count. With counts 50, 30 and 20, $N=100$ , numerator $=100\times99=9900$ , denominator $=50\times49+30\times29+20\times19=3700$ , so $D=2.68$ .

Evaluate repeated terms in a table to avoid missing a category. Keep numerator and denominator separate, then divide once. For a supplied rate, ratio or odds equation, preserve category order exactly and place each observed count in the defined position.

Do not use species richness as $N$ in the diversity formula: $N$ is the total number of individuals. A substituted value with the wrong definition can produce tidy arithmetic but an invalid result.

Identify the unknown and isolate it with reversible operations. Preserve unit conversions and brackets, then substitute the solution into the original equation to confirm both equality and biological plausibility.

Cardiac output $CO=SV\times HR$ , so $HR=CO/SV$ . Convert stroke volume to dm $^3$ if output is dm $^3$ min $^{-1}$ . For swimmers, $HR=4.43/0.07440=59.54$ beats min $^{-1}$ ; for controls, $HR=4.21/0.05840=72.09$ beats min $^{-1}$ . The difference is $12.55$ beats min $^{-1}$ .

If the equation contains powers, fractions or several terms, undo outer operations first and keep both sides balanced. Reject extraneous or negative solutions when the biological quantity cannot take them, but state the domain reason.

A correct rearrangement with mixed cm $^3$ and dm $^3$ still gives a thousandfold error. Do not discard a mathematical solution solely because it is unexpected; test whether the model or biological domain excludes it.

A logarithmic scale represents equal multiplication by equal spacing. On a base-10 axis, $10^3$ , $10^4$ and $10^5$ are equally spaced even though their absolute differences grow. This allows cell counts spanning several orders of magnitude to be compared clearly.

If exponential growth follows $N=N_0\times10^{gt}$ , then $\log_{10}N=\log_{10}N_0+gt$ . A plot of $\log_{10}N$ against time is therefore linear during constant exponential growth, with intercept $\log_{10}N_0$ and gradient $g$ .

If the log-count gradient is 0.602 h $^{-1}$ , a doubling is a log increase of $\log_{10}2=0.301$ , so doubling time $=0.301/0.602=0.500$ h. A rise from $10^3$ to $10^6$ cells is three log units but a thousandfold increase.

Use a logarithmic axis when positive values span several orders of magnitude or when multiplicative change is the relationship of interest. Label the actual values or log-transformed quantity clearly.

Zero and negative values cannot be placed on an ordinary logarithmic axis. A straight log-count section supports exponential growth only over that measured interval, not through lag, stationary or death phases.

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—A.3 GraphsRead

A.2 Algebra

Learning objectives

Read comparison and proportionality symbols precisely

Change the subject before substituting biological values

Substitute into biological formulas with traceable units

Solve biological equations and check the result in context

Use logarithmic scales for microbial populations

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