Some reminders of basic mathematics

Errors in calculations sometimes appear because of faults in mathematics rather than computational errors. For reference purposes, Tables 39.1 and 39.2 give some basic mathematical principles that may be useful.

Exponents
Exponential notation is an alternative way of expressing numbers in the form aⁿ ('a to the power n'), where a is multiplied by itself n times. The number a is called the base and the number n the exponent (or power or index). The exponent need not be a whole number, and it can be negative if the number being expressed is less than 1. See Table 39.2 for other mathematical relationships involving exponents.

Scientific notation
In scientific notation, also known as 'standard form', the base is 10 and the exponent a whole number. To express numbers that are not whole powers of 10, the form c × 10ⁿ is used, where the coefficient c is normally between 1 and 10. Scientific notation is valuable when you are using very large numbers and wish to avoid suggesting spurious accuracy. Thus if you write 123000, this suggests that you know the number to ±0.5, whereas 1.23 × 10⁵ might give a truer indication of measurement accuracy (i.e. implied to be ±500 in this case). Engineering notation is similar, but treats numbers as powers of 10 in groups of three, i.e. c × 10⁰, 10³, 10⁶, 10⁹, etc. This corresponds to the SI system of prefixes.

A useful property of powers when expressed to the same base is that when multiplying two numbers together, you simply add the powers, while if dividing, you subtract the powers. Thus, suppose you counted eight molecules in a 10⁻⁷ dilution, there would be 8 × 10⁷ in the same volume of undiluted solution; if you now dilute this 500-fold (5 × 10²), then the number present in the same volume would be 8/5 × 10⁽⁷⁻²⁾= 1.6 × 10⁵ = 160000.

Logarithms
When a number is expressed as a logarithm, this refers to the power n that the base number a must be raised to give that number, e.g. 1og₁₀(1000) = 3, since 10³ = 1000. Any base could be used, but the two most common are 10, when the power is referred to as 1og₁₀ or simply log, and the constant e (2.718282), used for mathematical convenience in certain situations, when the power is referred to as log, or In. Where a coefficient would be used in scientific notation, then the log is not a whole number.

To obtain logs, you will need to use the log key on your calculator, or special log tables (now largely redundant). To convert back, use:

• the 10^x key, with x = log value;
• the inverse then the log key; or
• the y^x key, with y = 10 and x = log value.

With log tables, you will find complementary antilogarithm tables to do this.

There are many uses of logarithms in chemistry, and in particular physical chemistry, including pH ( = −log[H⁺]), where [H⁺] is expressed in mol L⁻¹), and chemical kinetics, e.g. rate constants, where a plot of In(reactant) against time produces a straight line if the reaction is first order.