|1166425. Sat Jan 02, 2016 7:05 pm
|Natural numbers are counting numbers: 1, 2, 3, etc. The earliest example we have of natural numbers is found on a 30,000-year-old Czechoslovakian wolf bone. It appears to be a quinary (base-5) system, since the 55 notches thereon are grouped into 11 groups of five.
Counting was important to the Ancient Egyptians – unlike the Greeks, who paid their way to the underworld, the Egyptians needed to prove they could count their fingers. The Egyptians, though, had improved on that wolf bone by inventing symbols to stand for various values: an arch represented 10, a finger ten thousand, a pharaoh one million, for instance. They were also notable for extending the number system from the naturals to the rationals, though, as they invented fractions.
Although special fractions were represented by particular symbols – individual sections of the Eye of Horus – Egyptian fractions are most notable these days by their construction, since only fractions whose numerator was one, and denominators distinct, were allowed: for instance, '5/6' would be written as '1/2 + 1/3'. The cumbersome nature of this system was probably the reason that as early as the second century Ptolemy was recommending the use of the Babylonian positional system, even though it was base-60 (indeed, astronomers used sexagesimal notation until the sixteenth century because of this).
Fractions of the Eye of Horus
The decimal point was invented in 1585 by Simon Stevin: private tutor to the son of William the Silent; Inspector of Dykes; Quatermaster-General of the Army, and Minister of Finance of the Netherlands. Inspired by the Ancient Babylonians and his distaste for fractions, he published a new system in which each number was followed immediately by its decimal exponent within a circle; e.g. 4.537 would be written
As people got used to this system, they stopped writing the circled numbers, until only the ⓪ remained. This got smaller and smaller, until it became the decimal point we know and love today.
In first millenium Chinese mathematics a system of counting rods was used for arithmetic. When solving an equation, these rods would be arranged in a certain way, with red rods used to symbolise addition and black rods symbolising subtraction. This led to the concept of negative numbers: cheng fu shu, and a negative number would be represented the same way as its positive equivalent, with the addition of another rod placed diagonally over the top. Hindu mathematicians accepted negative numbers, but weren't happy about it – the twelfth-century mathematician Bhaskara remarked that a negative solution is "not to be taken; people do not approve of [them]". Western mathematicians simply tried to ignore them, however, with even Descartes refusing to accept negative numbers as roots of equations, calling them "false roots" - indeed, he never even extended his eponymous coordinate system below zero.
On the other hand, perhaps Descartes is not a deserving benchmark – it is thanks to his scorn at the idea of a square root of a negative number that we have the term "imaginary numbers". Leibniz, however, was more happy with the notion: he likened i to the Holy Spirit, with their ethereal existence.
If complex numbers are the two-dimensional analogue of real numbers, then quaternions are the four-dimensional analogue of complex numbers. There is also an eight-dimensional analogue – Cayley numbers – but sixteen-dimensional numbers have been proved impossible.
Zero: The Biography of a Dangerous Idea, Charles Seife*
Taming the Infinite: The Story of Mathematics, Ian Stewart
50 Mathematical Ideas You Really Need to Know, Tony Crilly
From Eudoxus To Einstein: A History of Mathematical Astronomy, C. M. Linton
*I purposefully didn't mention zero in this post, because it really merits one of its own!