Numbers, Numerals and Decimals

Numbers and numerals

A number is an idea of quantity. A numeral is the symbol we use to write it. The words "five", the symbol 5 and the Roman symbol V are three different numerals for the same number.

Kinds of numeral

Roman numerals

The rules are short:

1. Symbols of equal or falling value are added: VIII = 5 + 3 = 8, and XV = 10 + 5 = 15.
2. A smaller symbol placed before a larger one is subtracted: IV = 5 − 1 = 4, IX = 10 − 1 = 9, XL = 50 − 10 = 40.
3. A symbol is not repeated more than three times in a row.

Worked examples:

• XIV = 10 + 4 = 14
• XXIX = 20 + 9 = 29
• XLII = 40 + 2 = 42

On older prescriptions the numbers are often written in small letters, and the last "i" may be written as "j", so iij means 3. The sign ss means one half.

Fractions, decimals and percentages

These are three ways of writing the same amount.

• Fraction to decimal: divide the top by the bottom. 3/8 = 3 ÷ 8 = 0.375.
• Decimal to percentage: multiply by 100. 0.375 = 37.5%.
• Decimal to fraction: write it over 10, 100 or 1000, then simplify. 0.125 = 125/1000 = 1/8.

Place value, notation and standard form

Every digit has a place value. In the number 345.678:

Moving the decimal point one place to the right makes the number ten times bigger. Moving it one place to the left makes it ten times smaller. This is why a misplaced decimal point causes a tenfold error.

Standard form writes a number as a value between 1 and 10 multiplied by a power of ten. It is useful for very large or very small amounts.

• 3 500 = 3.5 × 10³
• 0.0025 = 2.5 × 10⁻³
• 0.00005 = 5 × 10⁻⁵

Working with decimals

Adding and subtracting: line up the decimal points.

12.500
 0.375
------
12.875

Multiplying: multiply as whole numbers, then give the answer as many decimal places as the two numbers had together. 0.25 × 0.4: 25 × 4 = 100, and there are 3 decimal places in total, so the answer is 0.100 = 0.1.

Dividing: move the decimal point in both numbers by the same number of places until you are dividing by a whole number. 4.5 ÷ 0.09 becomes 450 ÷ 9 = 50.

Writing decimals safely

• Always put a zero before the decimal point: write 0.5 mg, never .5 mg.
• Never put a zero after a whole number: write 5 mg, never 5.0 mg.

A missed decimal point can turn 0.5 mg into 5 mg, and a trailing zero can turn 5.0 mg into 50 mg. Both are ten-fold overdoses.

Significant figures

Significant figures are the digits that tell us how precisely a quantity is known.

Rounding

1. Find the last digit you want to keep.
2. Look at the digit after it. If it is 5 or more, round up. If it is less than 5, leave it as it is.

Examples:

• 3.4567 to 2 decimal places = 3.46
• 0.0724 to 2 significant figures = 0.072
• 112.78 to 1 decimal place = 112.8

Round only at the end. Keep extra digits while you calculate, then round the final answer to a quantity that your equipment can actually deliver. A balance that reads to 0.01 g cannot weigh 7.4562 g, so weigh 7.46 g. A measure marked in 1 mL steps cannot measure 112.8 mL, so you measure 113 mL or use a smaller measure.

Estimation

Estimating means rounding the numbers to easy ones to find roughly how big the answer should be. Do this before you calculate and again after.

• 19.8 × 4.1 is about 20 × 4 = 80. The exact answer is 81.18, so it passes the check.
• 5 mL three times a day for 7 days is about 5 × 3 × 7 = 105 mL. A result of 10.5 mL or 1 050 mL would fail the check.
• 0.25 g converted to milligrams must give a number bigger than 0.25, because the milligram is a smaller unit. The right answer is 250 mg.

If your answer is ten times too big or too small, the decimal point is in the wrong place. Estimation catches this before it reaches a patient.