 Pythagoras The Two Mathematics Barry H. Dayton Fibonacci

While Mathematics is often taught as a single consistent and established subject in fact there are two competing views of Mathematics, one taught to math students through the first years of college and the other used by almost everyone else who is not a mathematician. We look at how this came about.

Our story starts 3000 years ago in the middle east where the Egyptians and Babylonians used different writing implements and so found different systems of numeration convenient. Note that we use the word numeral to indicate the name for a number.

The Egyptians used an additive system with different values for their symbols. A numeral was a combination of these symbols similar to the way we could combine pennies, nickels, dimes, quarters, dollar bills, 5 dollar bills, etc. to make a quanity of currency to pay for something. A good reference is from Wikipedia

http://en.wikipedia.org/wiki/Egyptian_numerals

The Egyptians used their numbers for commerce and large numbers in the millions were possible, but eventually new symbols would need to be invented to express larger numbers.

A similar number system that is still sometimes seen today is the Roman Numeral system. There were some improvements to use fewer symbols but, unlike the Egyptians who used their numerals as the basis of their mathematics, the Romans only used their numerals as ordinal numbers, especially dates. Originally Roman numerals only went up to 3999 but in the middle ages the system was expanded to include larger numbers. Again, see Wikipedia or Math Forum with lots of links

http://en.wikipedia.org/wiki/Roman_numerals

On the other hand in Mesopotamia (presently known as Iraq) at the same time Babylonian numbers were used. There is a brief discussion in Section 4.1 of your textbook, a better discussion is in Wikipedia or MacTutor History of Mathematics

http://en.wikipedia.org/wiki/Babylonian_numerals
www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html

The important thing is the Babylonian system was a positional system. Only two separate symbols were used but numbers of any size could be written. Babylonian numerals were used extensively well into the middle ages and our notation for time in hours, minutes and seconds probably was motivated by the sexagesmal, base 60, used in the Babylonian numbers.

A similar early positional system is the Mayan system.

https://en.wikipedia.org/wiki/Maya_numerals

The Mayan's actually had many sytems but one is the Long count which was used to number each day in Mayan history. Like the Babylonian system numbers of any size could be written without the need for a new symbol to be invented.

So far we have seen a big difference between the Egyptian and Babylonian system. But the biggest difference is how they handle division, in particular dealing with parts of a whole.

The Egyptians used fractions. Thus a new symbol was required, but Egyptians were only able to concieve of unit fractions, that is fractions of the form 1/n -- the numerator was always 1. Working with fractions was then a real challenge, see especially the second link below to see how Egyptians had to add fractions.

http://en.wikipedia.org/wiki/Egyptian_fraction
http://en.wikipedia.org/wiki/RMP_2/n_table
It is scary to think that this unit fraction approach was used for over 1000 years, in Greece and Rome as well as Egypt.

On the other hand the Babylonians simply added another place value to the right of the ones place. This should require only the invention of a decimal point but the Babylonians even made do without that for 1000 years relying on context. If a more accurate answer is needed then an additional place value of 1/3600 could be used or even 1/216000 would give answers to modern 6 digit accuracy.

One problem that did arise was division problems where the divisor did not evenly divide some power of 60. For example consider dividing 1 by 7, that is the fraction 1/7. If we give 1 the unit 'hours' then to be accurate to 1/60 of a second we would convert 1 to 216000 1/60'ths of a second. 216000 divided by 7 is just a bit more than 30857 but in segigesimal system this number is 8*602+34*60+17 so 30857/216000 is about 8*1/60+34*1/3600+17/21600 with error less than 1/216000. This was good enough for the Babylonians for most purposes, if not another sexigesmal digit or two could be used. To see how this would continue go to

wolframalpha.com
and enter '1/7 in base 60'. It should respond
0.8 34 17 8 34 17 8 34 17 ...
which means that these digits would repeat indefinitely in any sexigesmal expansion of 1/7. So there was no hope of ever getting a finite exact sexigesmal expansion. But this was the cost of using segigesimals and was worth it to the Babylonians.

In summary, here are the main points.

1. Egyptian mathematics required many different symbols and techniques as numbers got larger or smaller (than one). Babylonian numbers required only 2 symbols and used the same techniques regardless of the kind of number.
2. Egyptian fractions were exact but Babylonian fractions were only approximate.
From point above it is clear that the Babylonian numbers were superior in every way. Engineers and scientists realized this and used the Babylonian system well into the middle ages when it was supplanted by the even better Hindu base 10 positional system that lead to our modern decimal system.

But, perhaps unfortunately for mathematicians, the decision of which direction to go was made by Pythagoras. Pythagoras was a shaman rather than a mathematician and wanted to demonstrate to his followers the existence of non physical truth. Thus it was necessary to have mathematical statements be true or false such as "This triangle has area 13/7" but not "This triangle has area about 1.857". So Pythagoras chose to follow the Egyptian system of fractions. Although he, or his followers, may have had regrets later when they found numbers which could not be expressed as fractions, Plato and later Aristotle and Euclid endorsed Pythagoras' decision and to this day most mathematicians work only with fractions and invent new symbols for special types of numbers. We will call this exact mathematics.

But fractions are impractical for practical computation so non-mathematicians work with approximate decimals. Since two approximate decimals may not be undisputably declared "equal" or "not equal" the rules for this mathematics are different. We call this numerical mathematics. Fibonacci was influencial in introducing the decimal numbers and early numerical methods into western mathematics in 1202. Strangely, his most famous numerical calculation, a solution of an irrational root of a cubic equation, was done using Babylonian numbers to 6 sexagesmal places.