History of Mathematics

# History of Mathematics

### Review Midterm 1

You should have no problems with questions like these:
• Order the following mathematicians chronologically: Archimedes; Diophantus; Euclid; Eudoxos; Pythagoras; Thales.
• Which results are due to Thales, which to the Pythagoreans?
• Did Euclid discover all the results of his books himself? Where did Euclid live and teach?
• What is Alexandria famous for?
• The Greeks learned about geometry from which culture? Why was geometry `invented'? From where did the Greeks import their alphabet?
• Explain the difference between the number systems of the Babylonians and the Egyptians. Give examples of cultures using positional (nonpositional) number systems. Why did most cultures develop a number system with base 10; what other bases were used?
• What were the advantages/disadvantages of the Babylonian number system? What is the difference between 0 as a symbol and 0 as a number?
• Write e.g. 3/13 as an Egyptian fraction.
• What are the `elements' of Euclid?
• What are numbers in the sense of Euclid?
• What is a perfect number? List a few. Are there odd perfect numbers?
• What is the difference between the Greek and the Egyptian way of doing geometry?
• What is an axiom (in the sense of Euclid)? Why are axioms necessary in a mathematical theory, and who realized this?
• Describe (broadly) the content of Euclid's elements.
• List Euclid's five postulates of plane geometry (you're not supposed to memorize these literally, but you should be able to explain them in your own words). What is the connection between the axioms and the fact that the Greeks only allowed ruler and compass for constructions in geometry?
• Translate the statement that the side and the diagonal of a square are incommensurable into modern language. [Two sides d and a are commensurable if there exists a length u such that both d and u are integral multiples of this common length u. In other words: if d = ku and a = lu for integers k and l. This is equivalent to d:u = k:l, that is, the ratio of these sides is rational. For the diagonal d and the side a of a square we know that d:a is equal to the square root of 2. Thus the modern statement is that the square root of 2 is not rational.]
You should be familiar with the following proofs:
• Theorem of Thales
• Euclid's Proposition 1
• Irrationality of the square root of 2, 3, 5, ...
• If P = 2m-1 is a prime, then 2m-1P is perfect.
You also should understand Euclid's proof of Proposition 2 (Book 1) and its relevance to collapsing compasses.
Questions from students:
• Translate the statement that the side and the diagonal of a square are incommensurable into modern language.
Two sides d and a are commensurable if there exists a length u such that both d and u are integral multiples of this common length u. In other words: if d = ku and a = lu for integers k and l. This is equivalent to d:u = k:l, that is, the ratio of these sides is rational. For the diagonal d and the side a of a square we know that d:a is equal to the square root of 2. Thus the modern statement is that the square root of 2 is not rational.
• "It is cojectured that every every fraction 4/n (n>5 , odd) is the sum of at most three Egyptian fractions". I divided the proof into two cases n=4k+1 and n=4k+3, I completed the part n=4k+3 but couldn't do the part n=4k+1.
That's why it is a conjecture: if you could do the case n=4k+1, you would be famous. It is possible to do other cases (n = 8k+5?), but the problem is still open.
• What is the difference between the Greek and the Egyptian way of doing geometry?
The geometry of the Egyptians was practical: they needed it for measuring land after the yearly floodings of the Nile. They had formulas for the area of triangles, trapezoids, circles, and the volume of pyramids etc., but they did not prove anything.
The Greeks, starting with Thales, discovered the notion of a proof. This allowed them not only to say for certain whether a certain formula is true or not, it also enabled them to push their mathematics far beyond anything that could be done without the notion of proof.
• What is the difference between 0 as a symbol and 0 as a number?
The Babylonians eventually invented symbols representing missing digits; remember that their system could not distinguish between (I I) 60 + 1 and 3600 + 1 (I I) because they, at first, did not have such a symbol for missing digits. If I use * as such a symbol, then these two numbers would be written I I and I * I.
The Hindus also invented 0 as a symbol in the decimal system, but shortly afterwards they started using 0 as a number: they gave rules for computing with 0, such as a+0 = a, a - 0 = a, a * 0 = 0. This is what I mean by saying they discovered 0 as a number. Symbols are used for writing, numbers for computing.
• In the lecture notes on the web there is a question saying show that for any n>1, 4/(3n+2) can be represented as a sum of (at most) three Egyptian fractions. But I found out that when n is 5, according to greedy algoritm, we write 4/17 as 1/5 + 1/29 + 1/1233 + 1/3039345 which includes four fractions. Am I making an obvious mistake in some place or what?
No. You have just shown that the presentation given by the greedy algorithm does not provide one with three fractions. Now you have to look for other methods. Since 3/17 = 1/6 + 1/102, you might, for example, use 4/17 = 1/6 + 1/17 + 1/102.