The Babylonians (2000 — 1000 BC) used a sexagesimal (base-60) place-value system and were able to indicate fractions. ¹/₂ would be ⁽³⁰⁾/₍₆₀₎ or 0.(30); ¹/_{3 = ⁽²⁰⁾/(60) = 0.(20).}

The Babylonian system benefits from the large number of factors of 60, which may explain its continued use for the measurement of time and angle.

Evidence from the Rhind Papyrus (1650 BC) suggests that the Egyptians used only unit fractions: that is, fractions with numerators of 1. This papyrus lists combinations of unit fractions needed to make up other fractions. For example:

²/₅ = ¹/₃ + ¹/₁₅
²/₇ = ¹/₄ + ¹/₂₈
³/₅ = ¹/₃ + ¹/₅ + ¹/₁₅

Students can experiment with developing techniques for expressing fractions as sums of unit fractions.

It took a long time for the Hindu-Arabic notation to extend to include tenths, hundredths and so on. In most countries the decimal point is used to show the separation between the units and the tenths. In France a comma is used.

Farey Fractions

The Farey sequence of fractions of order N is the set of all simplified fractions between 0 and 1, whose bottom numbers are N or less, arranged in order of size. Thus the Farey sequence of order 4 consists of:

¹/₂, ¹/₃, ²/₃, ¹/₄, ³/₄

These fractions can plotted on a graph and then arranged in order of size, smallest first.

¹/₄, ¹/₃, ¹/₂, ²/₃, ³/₄

Generate other Farey sequences of different orders.

Consider any three consecutive members in a particular Farey sequence. Can you see how to combine the two outer fractions to form the middle fraction? Check that the same rule applies to other sets of three consecutive numbers.

Consider any pair of consecutive members of a Farey sequence. Multiply the numerator of the second by the denominator of the first; then multiply the numerator of the first by the denominator of the second. Subtract the second product from the first. Repeat for other pairs. Try other Farey sequences.

Investigate whether there is a general rule for the number of members of a Farey sequence.

Calendars

A day is the time taken for the Earth to rotate about its axis. This is about 23h 56m 4.1s. The remaining 3m 55.9s is accounted for by the distance it has moved in its orbit round the Sun.

A year is the time taken for the Earth to complete its 580,000,000-mile orbit around the Sun. This is about 365.24 (approximately 365 ¹/₄) days.

A lunar month is the time between one new moon and the next. This is about 29.53 days. The Moon, while orbiting the Earth, completes 12.37 such cycles in a year.

Some calendars take account only of the lunar month, of which there are 12 in about 354 days. The Muslim calendar is an example. A Muslim year consists of 12 lunar months; so the months move to different times of the solar year. The Chinese calendar is also lunar, but an additional month is added every few years to adjust. (Thus the date in the Western calendar of the Chinese New Year varies from year to year). The Jewish calendar, too, follows both the solar and the lunar cycles which determines the months: a thirteenth month is added in some years.

The Roman calendar used to based on a year of 365 days, until Julius Caesar, in about 45 BC, introduced a correction because the months of the year were slipping into the wrong seasons. The Julian calendar was based on a convention worked out by Sosignes. The year was taken to be 365 ¹/₄ days. It was divided into 12 ‘months’, mostly containing 30 or 31 days, which are not actually in step with the moon’s cycle. This convention has lasted to our own times; and this old rhyme helps people remember the number of days in each month:

In the Julian calendar, an extra day is added every four years: the leap day. However by the sixteenth century the calendar had gained about 12 and a half extra days. So in, 1582 Pope Gregory XIII introduced his Gregorian calendar and ordered that ten days should be taken from the calendar. He also decided that from then on the leap year should be missed in the last year of every century unless that year could be divided by 400. That means the year 2000 is a leap year.