Greek and Hellenistic mathematics (c. 550 BC—AD 300).

Greek mathematics refers to mathematics written in Greek between about 600 BCE (Before the Common Era) and 450 CE. Greek mathematicians lived all over the Eastern Mediterranian from Italy to North Africa, but were united by culture and language. To emphasize that the unity of this body of mathematics is cultural rather than nationalistic, Greek mathematics is sometimes called Hellenistic mathematics.

Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures, such as the Egyptians and Babylonians. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb, not all of them correct. Greek mathematicians, by contrast, used deductive reasoning, that is, the Greeks used logic to derive conclusions from definitions and axioms.

Greek mathematics is thought to have begun with Thales (c.(circa) 624—c. 546 BC) and Pythagoras (c. 582—c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by the ideas of Egypt, Mesopotamia and perhaps India. According to legend, Pythagoras travelled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.

Pythagoras is credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history. Pythagoras also invented a means of translating musical notes into mathematical equations and mathematical equations into musical notes called Pythagorean tuning which used the ratio 3:2 to do this. He was among the first people to recognize that Venus as the morning star and Venus as the evening star were the same planet. Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. According to Proclus' commentary on Euclid, Pythagoras stated the Pythagorean theorem and constructed Pythagorean triples algebraically. It is generally conceded that Greek mathematics differed from that of its neighbors in its insistence on axiomatic proofs. Plato when he started his Academy taught mathematics there and had an inscription upon his academy that read “let none unversed in geometry enter here”.

Greek and Hellenistic mathematicians were the first to give a proof for irrational numbers (due to the Pythagoreans). It is ironic that it was a Pythagorean who discovered the existence of irrational numbers because Pythagoras and his followers denied the existence of irrational numbers because it contradicted their philosophy of mathematics. Once when a bunch of Pythagoreans were on a ship one of them discovered an irrational number when he was working on a division problem because the answer was an irrational number. The other Pythagoreans now realized that this was true but they were not willing to let other people outside their group know that they had been wrong so to prevent any chance of this happening they murdered their member who had discovered the existence of irrational numbers by throwing him overboard into the sea. And they were the first to develop Eudoxus's method of exhaustion, and the Sieve of Eratosthenes for uncovering prime numbers. They took the ad hoc methods of constructing a circle or an ellipse and developed a comprehensive theory of conics; they took many various formulas for areas and volumes and deduced methods to separate the correct from the incorrect and generate general formulas. The first recorded abstract proofs are in Greek, and all extant studies of logic proceed from the methods set down by Aristotle. Euclid, in the Elements, wrote a book that would be used as a mathematics textbook throughout Europe, the Near East and North Africa for almost two thousand years. In addition to the familiar theorems of geometry, such as the Pythagorean theorem, The Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.

Some say the greatest of Greek mathematicians, if not of all time, was Archimedes (287—212 BC) of Syracuse. According to Plutarch, at the age of 75, while drawing mathematical formulas in the dust, he was run through with a spear by a Roman soldier. Roman society has left little evidence of an interest in pure mathematics.

Closure property.

If we add two natural numbers, the sum will also be a natural number. For example, 5 is a natural number and 3 is a natural num­ber. 5 + 3 = 8 and only 8.

The sum, 8, is also a natural number. Following are other exam­ples in which two natural numbers are being added and the sum is another natural number. 19 + 4 = 23 and only 23; 6 + 6 = 12 and on­ly 12; 1429 + 357 = 1786 and only 1786. Actually if any two natural numbers are being added, the sum again is a natural number. Sin­ce this is true we say that the set of natural numbers is closed un­der addition. This is a statement of closure, one of special properti­es of addition.

Notice that we can name the sum in each of the above equati­ons. That is, the sum of 5 and 3 exists, or for example, there is a number which is the sum of 19 and 4. In fact the sum of any two numbers exists. This is called the existence property. Notice also, that when 5 and 3 are being added the sum is 8 and only 8 and not some other number. Since there is one and only one sum for 19 + 4, we say that the sum is unique. This is called the uniqueness property. Both existence and uniqueness are implied in the definition of closure.

If a and b are numbers of a given set, then a + b is also a num­ber of that same set. For example, if a and b are any two natural numbers, then a + b exists, it is unique, and it is again a natural number.

If we use the operation of subtraction instead of the operation of addition, we cannot make the statement we made above. If one na­tural number is being subtracted from another natural number the result produced is sometimes a natural number, and sometimes not. 11 - 6 = 5 and 5 is a natural number. 9—9 = 0 and 0 is not a natural number. Consider the equation 4 - 7 = n. It cannot be solved if we must have a natural number as an answer. Therefore, the set of natural numbers is not closed under subtraction.

When two natural numbers are being multiplied there is always a natural number which is the product of the two numbers. Every pair of natural numbers has a unique product which is again a na­tural number. Thus the set of natural numbers is closed under mul­tiplication.

In general, the closure property may be defined as follows: if x and у are any elements, not necessarily the same, of set A and * (asterisk) denotes an operation *, then set A is closed under the operation * if (x*y) is an element of set A.

It must be pointed out that it is impossible, to find the sum or the product of every possible pair of natural numbers. Hence, we accept the closure property without proof, that is as an axiom.