lecture1 - HISTORY OF MATHEMATICS MHF 4404 – FALL 2011...

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Unformatted text preview: HISTORY OF MATHEMATICS MHF 4404 – FALL 2011 BARRY J. GRIFFITHS HISTORY OF MATHEMATICS – LECTURE 1 – MONDAY 22ND AUGUST The process of counting has been used by human beings throughout recorded history. We have evidence from 30,000 years ago (see page 2 of the text) that markings were etched into bones to count the number of items, persons or animals contained within a group, as well as to mark time. While we can use the same marking (usually a line or a dot) to denote an increment of one, to record larger numbers it is convenient to have more symbols. So while we are accustomed to our current decimal system (based on powers of ten) to record numbers, there have been many alternatives used throughout history, and we will look at several examples below. Mayan This is a base 20 system, which uses the symbols dot, dash, and shell to denote the numbers 0‐19: Numbers larger than 19 are written vertically based on powers of 20. Ex. Ex. Attic Greek The earliest notation used by the Greeks was the Attic (or Herodianic) system. Somewhat similar to Roman numerals, the systems used denoted one, five, ten, one hundred, one thousand, and ten thousand. 1 = | 5 = 10 = 100 = H 1000 = X 10,000 = M These symbols can then be combined (see page 19) to obtain numbers such as 50, 500 etc. Ex. Ex. Egyptian Because the Egyptians used a different symbol for each power of 10, the same number could be represented in several different ways (see page 13). Ex. Ex. Roman The Roman numeral system, which is still widely used today, is similar to the Attic Greek system, using letters as symbols for powers of ten, as well as five times each power of ten. 1 = 5 = V 10 = X 50 = L 100 = C 500 = D 1000 = M While the system seems fairly straightforward, there are rules for writing certain numbers that add an additional layer of complexity (see page 20). Ex. Ex. Ionian Greek The (Ionian) Greek system of enumeration was a little more sophisticated than the Egyptian though it was non‐positional. Like the Attic and Egyptian systems it was also decimal. Its distinguishing feature is that it was alphabetical and required the use of more than 27 different symbols for numbers plus a couple of other symbols for meaning. Letter Value Letter Value Letter Value α 1 ι 10 ρ 100 β 2 κ 20 σ 200 γ 3 λ 30 τ 300 δ 4 μ 40 υ 400 ε 5 ν 50 φ 500 ϛ 6 ξ 60 χ 600 ζ 7 ο 70 ψ 700 η 8 π 80 ω 800 θ 9 Ϙ 90 ϡ 900 Note: The symbols corresponding to 6, 90, and 900 are (respectively) called digamma, coppa, and sampi. Note: To denote larger numbers, an accent mark ‘ was placed below and to the left of the corresponding symbol to denote that it was multiplied by 1000. Tens of thousands were written by putting the corresponding symbol above M (which denoted the word “myriad”, i.e. ten thousand). Ex. Ex. Babylonian The Babylonians were the only pre‐Grecian people to use a positional number system, based on a sexagesimal (base 60) scale. Hence the last digit represented a number from 0 to 59, the penultimate digit represented a multiple of 60, the next a multiple of 602 etc. Only two symbols were needed to write down any number, with fifty). Hence being used to count units (up to nine) and to count tens (up to The main drawback with this system is that the Babylonians did not have a symbol for zero, which meant that was used to represent both (though the context of the problem, or an empty space, would usually minimize the ambiguity). From 300B.C. a new symbol, was used to avoid such confusion. However the new symbol was only used for intermediary digits, and not at the end of a number. Hence it was still impossible to distinguish numbers that were exact multiples of a power of 60. Consider for example how one would distinguish 120 from 20. Ex. ...
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