Numbers that measure: units, notation and change
A measurement combines a number with a unit. Its interpretation depends on the unit, the notation used for large, small or logarithmic values, and whether a change is expressed in percentage points, percentages or folds.
Numbers and units
Counting objects yields an integer, while measuring an amount theoretically yields a real number and a unit. A real number corresponds to a point on a number line. Reporting finitely many decimal places produces a rational number, expressible as a quotient of two integers. Rational decimal expansions terminate or repeat, as in ¾ = 0.75 and ⅓ = 0.333…, while irrational numbers such as π, Euler’s number e and √2 continue indefinitely without repeating.
Outcome measurements should be reported accurately with their units where applicable, because changing the unit changes the numerical value. Scientific writing should use SI units such as metres, kilometres and kilograms; writing for US consumers should probably use familiar units such as feet and inches, miles, pounds and ounces.
The International System of Units (SI) grew from the metric system developed in revolutionary France.
Unit definitions and relationships
A centimetre is one hundredth of a metre, and a litre is 1,000 cubic centimetres. The kilogram originally represented the mass of a litre of water. A newton is the force that accelerates a 1 kg object by 1 metre per second per second, combining mass, distance and time.
A coulomb measures electrical charge, an ampere electrical current and a candela luminous intensity. A mole measures amount of substance and contains 6.02214076 × 10²³ particles, Avogadro’s number.
Time measurement originated in the day, divided into 24 hours, each with 60 minutes and each minute with 60 seconds. The base-60 divisions are a legacy of ancient Mesopotamian numeration; 60 is the smallest number divisible by every integer from 1 to 6. A circle is divided into 360 degrees, a number permitting many exact whole-number divisions.
Powers and scientific notation
Exponentiation raises a base to a power: 2³ = 2 × 2 × 2 = 8, with the exponent 2 called “squared” and 3 “cubed”. For a common base, multiplication adds exponents and division subtracts them: 10² × 10³ = 10⁵ and 10⁵ ÷ 10³ = 10². For nonzero b, b⁰ = 1 and b⁻ⁿ = 1/bⁿ. A negative base raised to an even integer power gives a positive result, while an odd integer power gives a negative result.
Scientific notation expresses 523,000 as 5.23 × 10⁵ and 0.0000523 as 5.23 × 10⁻⁵. E notation writes the same values as 5.23E5 and 5.23E−5.
Logarithmic values
A logarithm reverses exponentiation: log₂(8) = 3 because 2³ = 8. Negative logarithms also occur: log₂(½) = −1 and log₁₀(⅒) = −1. Base-10 logarithms are often written log(x), while the natural logarithm ln(x) has Euler’s number e as its base.
pH is the negative base-10 logarithm of hydrogen-ion activity. Activities of 10⁻⁷ and 10⁻⁶ correspond to pH 7 and pH 6 respectively, a tenfold activity difference.
Viral loads may also be expressed as base-10 logarithms: 5 log₁₀ copies/mL means 100,000 copies/mL. In an editing example, omission of “log” turned that value into an erroneous report of 5 copies/mL.
Fractions and percentage changes
A fraction divides its numerator by its denominator; a percentage has denominator 100, so both quantities need to be identified. A rise from 10% to 20% is an increase of 10 percentage points and a relative increase of 100%.
For a nonzero starting value, percentage change is (final value − starting value) / |starting value| × 100. The same absolute gain and loss therefore produce different percentage changes: rising from 50 kg to 150 kg is a 200% increase, [(150 − 50)/50] × 100, while returning from 150 kg to 50 kg is approximately a 67% decrease, [(50 − 150)/150] × 100 ≈ −67%.
Fold changes
A fold is a ratio between two values, and fold increase is final value / starting value: an increase from 50 kg to 150 kg is threefold, 150/50 = 3. Fold decreases are reported using different conventions. One uses −(starting value / final value), expressing the fall from 150 kg to 50 kg as a −3-fold change or threefold decrease. Others use final value / starting value, giving 50/150 ≈ 0.33 for the same decline, so the intended convention needs to be checked.
Negative values and direction
Subtraction can produce negative values even when removing physical objects cannot: no more than five apples can be removed from a basket of five, but a $200 cheque against a $100 balance produces −$100, requiring a $100 deposit to restore zero. Accountants sometimes indicate a negative value with parentheses instead of a minus sign.
Addition and subtraction can be represented as movement right and left along a number line, involving direction as well as magnitude. A vector has both direction and magnitude; walking one block north, west, south and east covers four blocks but returns to the starting point. Mathematical models can use more than four dimensions to represent relationships among many variables.