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Euclid of Alexandria Circa 325 - 265 BC The library of Alexandria was the foremost seat of learning in the world and functioned like a university. The library contained 600 000 manuscripts. 22 1 1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,,,,,,,

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The Fundamental Theorem of Arithmetic: Every positive integer (excluding 1) can be expressed as a product of primes and this factorisation is unique (Euclid IX.14). The first part of this result is needed for the proof of the infinity of primes (Euclid IX.20) which follows shortly. The type of proof used is a little different and is known as “Reductio ad absurdum”. It was first exploited with great success by ancient Greek mathematicians. The idea is to assume that the premise is not true and then apply a deductive argument that leads to an absurd or contradictory statement. The contradictory nature of the statement means that the “not true” premise is false and so the premise is proven true.

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234567891011 prime 2 2 x 3prime2323 3232 2 x 5prime 12131415161718192021 2 2 x 3prime2 x 73 x 52424 prime2 x 3 2 prime2 2 x 53 x 7 22232425262728293031 2 x 11prime2 3 x 35252 2 x 133 2 2 x 7prime2 x 3 x 5prime 32333435363738394041 2525 3 x 112 x 175 x 72 2 x 3 2 prime2 x 193 x 132 3 x 5prime 42434445464748495051 2 x 3 x 7prime2 2 x 113 2 x 52 x 23prime2 4 x 37272 2 x 5 2 3 x 17 525354 2 2 x 13prime For example: Assume that 54 is the smallest non–prime number that we suspect cannot be expressed as a product of primes. Since it is composite, it can be written as a product of two smaller factors. These factors are either prime or have already been written as a product of primes (6 x 9 or 3 x 18). It is quite easy to see that any number is either prime or can be expressed as a product of primes. Suppose that we check this for all numbers up to a certain number. 2 x 3 3 Any whole number is either prime or can be expressed as a product of its prime factors.

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234567891011 prime 2 2 x 3prime2323 3232 2 x 5prime 12131415161718192021 2 2 x 3prime2 x 73 x 52424 prime2 x 3 2 prime2 2 x 53 x 7 22232425262728293031 2 x 11prime2 3 x 35252 2 x 133 2 2 x 7prime2 x 3 x 5prime 32333435363738394041 2525 3 x 112 x 175 x 72 2 x 3 2 prime2 x 193 x 132 3 x 5prime 42434445464748495051 2 x 3 x 7prime2 2 x 113 2 x 52 x 23prime2 4 x 37272 2 x 5 2 3 x 17 525354 2 2 x 13prime 2 x 3 3 Any whole number is either prime or can be expressed as a product of its prime factors. This argument can obviously be extended to larger numbers. 7038 = 2 x 3 2 x 17 x 23 7038 = 46 x 153 This could be generalised for any whole number N, by using a “reductio” type argument as follows:

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234567891011 2 2 x 32323 3232 2 x 5 12131415161718192021 2 2 x 32 x 73 x 52424 2 x 3 2 2 2 x 53 x 7 22232425262728293031 2 x 112 3 x 35252 2 x 133 2 2 x 72 x 3 x 5 32333435363738394041 2525 3 x 112 x 175 x 72 2 x 3 2 2 x 193 x 132 3 x 5 42434445464748495051 2 x 3 x 72 2 x 113 2 x 52 x 232 4 x 37272 2 x 5 2 3 x 17 525354 2 2 x 13 Any Number Can Be Expressed As a Product of Primes 2 x 3 3 Since N is composite (otherwise it would be prime), N = p x q, both less than N. Since p and q are smaller than N they are either prime or a product of primes. Therefore the assumption is wrong and N can be written as a product of prime factors. Assume N is the smallest number that cannot be expressed as a product of primes. There is no smallest N that cannot be expressed as a product of primes. Any number can be expressed as a product of primes. QED 7038

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In G.H. Hardy’s book “A Mathematician’s Apology”, Hardy discusses what it is that makes a great mathematical theorem great. He discusses the proof of the infinity of primes and the proof of the irrationality of 2. G.H. Hardy (1877-1947) “....It will be clear by now that if we are to have any chance of making progress, I must produce examples of “real” mathematical theorems, theorems which every mathematician will admit to be first-rate.”…. “....I can hardly do better than go back to the Greeks. I will state and prove two of the famous theorems of Greek mathematics. They are “simple” theorems, simple both in idea and execution, but there is no doubt that they are theorems of the highest class. Each is as fresh and significant as when it was discovered – two thousand years have not written a wrinkle in either of them. Finally, both the statements and the proofs can be mastered in an hour by any intelligent reader….” “Two thousand years have not written a wrinkle in either of them.”

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2, 3, 5, 7, 11, 13, 17, The Infinity of Primes 19, 23, 29, 31, 37, 41, …… This again is a “reductio ad absurdum” proof, commonly known as a proof by contradiction. Remember, the idea is to assume the contrary proposition, then use deductive reasoning to arrive at an absurd conclusion. You are then forced to admit that the contrary proposition is false, thereby proving the original proposition true. To prove that the number of primes is infinite. *Assume the contrary and consider the finite set of primes: p 1, p 2, p 3, p 4, …. p n-1, p n Let S = p 1 x p 2 x p 3 x p 4 x …. x p n-1 x p n T = (p 1 x p 2 x p 3 x p 4 …. p n-1 x p n ) + 1 Consider T = S + 1 T is either prime or composite. If T is prime we have found a prime not on our finite list, proving * false. If T is composite it can be expressed as a product of primes by the But T is not divisible by any prime on our finite list since it would leave remainder 1. Euclid Proposition IX.20 (Based on). Therefore there must exist a prime > p n that divides T, also proving * false. The number of primes is infinite. QED “Fundamental Theorem of Arithmetic” (Euclid IX.14).

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The Square Root of 2 is Irrational 22 1 1 This is a “reductio-ad-absurdum” proof attributable to Pythagoras. To prove that 2 is irrational Assume the contrary: 2 is rational That is, there exist integers p and q with no common factors such that: (Since 2q 2 is even p 2 even p even) So p = 2k for some k. (Since p is even even q 2 even q even) So q = 2m for some m. This contradicts the original assumption. 2 is irrational. QED (odd 2 = odd)

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Incommensurable Magnitudes (Irrational Numbers) 1 1 22 The whole of Pythagorean mathematics and philosophy was based on the fact that any quantity or magnitude could always be expressed as a whole number or the ratio of whole numbers. Unit Square The discovery that the diagonal of a unit square could not be expressed in this way is reputed to have thrown the school into crisis, since it undermined some of their earlier theorems. Story has it that the member of the school who made the discovery was taken out to sea and drowned in an attempt to keep the bad news from other members of the school. He had discovered the first example of what we know today as irrational numbers.

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Section 1.7. Definitions A theorem is a statement that can be shown to be true using: definitions other theorems axioms (statements which are given as.

Section 1.7. Definitions A theorem is a statement that can be shown to be true using: definitions other theorems axioms (statements which are given as.

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