A.2 Algebra

Symbols condense a relationship: = means equality, ≈ approximation, ∝ proportionality, Δ change and Σ a sum. Read the units and definitions before manipulating an expression.

Translate the symbolic statement into words, identify what is held constant and check that both sides have compatible dimensions.

Rate = change/time means a larger change in the same time gives a larger rate; Δtemperature is a difference, not a new absolute temperature scale.

A symbol’s meaning depends on the equation and context; Δ does not automatically mean “final value”.

Changing the subject means isolating one variable while applying the same inverse operation to both sides. Keep brackets and units visible until the rearrangement is complete.

Undo addition before multiplication, undo powers with roots and check the final form by substituting a simple value into both the original and rearranged equations.

From rate = distance/time, multiply by time to obtain distance = rate × time; the units become distance per time × time = distance.

Rearranging is not the same as changing a sign or moving a term without applying an operation to both sides.

Substitution replaces each symbol with a measured or given value. First identify the quantity each symbol represents, then convert units so the equation remains dimensionally valid.

Write the equation, insert values with units, calculate and round at the end. If a value is a rate, preserve its time basis rather than treating it as a total.

For distance = rate × time, 2 m s⁻¹ for 30 s gives 60 m; using 30 min without conversion would inflate the answer by a factor of 60.

A calculator can evaluate the wrong equation perfectly; definition and unit checks come first.

An equation remains true when the same operation is applied to both sides. Isolate the unknown, preserve brackets and check the solution in the original expression.

Undo addition or subtraction, then multiplication or division; for powers use the appropriate root. Keep units attached to quantities when the equation represents a biological measurement.

If 3x + 2 = 11, subtract 2 then divide by 3 to obtain x = 3; substituting 3 restores 11.

Squaring or taking roots can introduce or remove solutions, so verify every candidate.

A logarithm answers “what power gives this value?” Logarithmic quantities are useful for exponential change and compressed biological scales.

Use log(ab)=log a+log b and log(aᵇ)=b log a where the quantities are defined; keep the base consistent and state it when it matters.

If a population grows tenfold, log₁₀(population) increases by 1; a further tenfold increase adds another 1 rather than multiplying the log by ten.

A log cannot be taken of a non-positive quantity, and equal log intervals represent multiplicative—not additive—changes.