Mathematics as an ongoing cultural activity

Prepared by Ooi Li Xuan

Definition• Mathematics is deeply linked to culture and that this

link cannot be considered by isolating it from social, political and economic factors which determine the evolution of thought in general.

• Culture refers to the following ways of life, including but not limited to:a. Languageb. arts and sciencesc. Thoughtd. Spiritualitye. social activityf. interaction

Mathematics and arts…• The ancient Egyptians and ancient Greeks

knew about the golden ratio, regarded as an aesthetically pleasing ratio, and incorporated it into the design of monuments including the Great Pyramid, the Parthenon, and the Colosseum.

• The Golden Ratio, roughly equal to 1.618, was first formally introduced in text by Greek mathematician Pythagoras and later by Euclid in the 5th century BC.

• Aside from interesting mathematical properties, geometric shapes derived from the golden ratio, such as the golden rectangle, the golden triangle, and Kepler’s triangle, were believed to be aesthetically pleasing.

• Many works of ancient art exhibit and incorporate the golden ratio in their design.

• Evidence of mathematical influences in art is present in the Great Pyramids, built by Egyptian Pharaoh Khufu and completed in 2560BC.

• Pyramidologists since the nineteenth century have noted the presence of the golden ratio in the design of the ancient monuments.

• They note that the length of the base edges range from 755–756 feet while the height of the structure is 481.4 feet. Working out the math, the perpendicular bisector of the side of the pyramid comes out to 612 feet. If we divide the slant height of the pyramid by half its base length, we get a ratio of 1.619, less than 1% from the golden ratio.

• This would also indicate that half the cross-section of the Khufu’s pyramid is in fact a Kepler’s triangle.

Mathematics and interactions…

• Measuring activity• Length is the most necessary measurement in

everyday life, and units of length in many countries still reflect humanity's first elementary methods.

• Example, the inch is a thumb. The foot speaks for itself. The yard relates closely to a human pace, but also derives from two cubits (the measure of the forearm).

• For the complex measuring problems of civilization - surveying land to register property rights, or selling a commodity by length - a more precise unit is required.

• The solution is a rod or bar, of an exact length, kept in a central public place. From this 'standard' other identical rods can be copied and distributed through the community. In Egypt and Mesopotamia these standards are kept in temples.

• When a length is standardized, it is usually the king's dimension which is first taken as the norm.

• The measuring activity has developed by more standardize units like the SI units.

Mathematics and language…

• numeral system• Some of the systems for representing

numbers in previous and present cultures are well known.

• Roman numerals use a few letters of the alphabet to represent numbers up to the thousands, but are not intended for arbitrarily large numbers and can only represent positive integers.

• Arabic numerals are a family of systems, originating in India and passing to medieval Islamic civilization, then to Europe, and now standard in global culture—and having undergone many curious changes with time and geography—can represent arbitrarily large numbers and have been adapted to negative numbers, fractions, and real numbers.

• Less well known systems include some that are written and can be read today, such as the Hebrew and Greek method of using the letters of the alphabet, in order, for digits 1–9, tens 10–90, and hundreds 100–900.

Hebrew number system

Greek number system

• A completely different system is that of the quipu, which recorded numbers on knotted strings.