Little is known about the origins of Greek mathematics, however it is believed that ancient Egyptian and Babylonian mathematics greatly influenced early Greek mathematics. While early Greeks did not have variables or ratios in the modern sense, or use an equation form of Algebra, the Greeks did have variables, ratios and geometric algebra.

Pythagoras's life is not reliably documented. Isocrates in the 4th centry BC is the earliest surviving record that states that Pythagoras traveled to and studied in Egypt. Whether this is true or not, there is considerable evidence that the Greeks knew of the mathematics of other nearby civilizations and it was not unusual for Greek scholars to travel to other ancient centers of learning study. It is also known that Egyptians and the Babylonian mathematic systems had lists or ratios of triangle sides.

As for the proof of the square root of 2 as an irrational number, Hippasus was not trying to prove the existence of irrational numbers. He was simply trying study geometric puzzles and would have presented it as a geometric and logical proof. Pythagoras (or his school) is credited with coming up with the even and odd numbers, and they are also credited with using even and odd numbers in mathematical proofs.

There are two thoughts on the proof of irrational number attributed to Hippasus though. Some believe that Hippaseus did not prove irrational numbers with a square root of two, but that he proved irrational numbers while examining the sides of a pentagram and deducing the nature of Phi.

Regardless of Hippaseus and how he may have proven the existence of irrational numbers, the formulation of Greek proof of the square root of two can through contradiction using even and odd numbers be deduced from Aristotle's statement in Analytica Priora, on proof by contradiction where “the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate”. This statement only makes sense if the details of the proof are worked out and known, but I do not know of any directly attributable ancient Greek texts proof of the square root of 2 being irrational through contradiction.

If we approach this using texts that we do have access to, we can look at Theatetus's theorem as presented in Platonic dialog (as translated by Chris Emilyn-Jones and William Preddy in the Loeb Classical Library):

THEAET. We divided all numbers into two classes. The one, the numbers which can be formed by multiplying equal factors, we represented by the shape of the square and called square or equilateral numbers.

SOC. Well done!

THEAET. The numbers between these, such as 3 and 5 and all numbers which cannot be formed by multiplying equal factors, but only by multiplying a greater by a less or a less by a greater, and are therefore always contained in unequal sides, we represented by the shape of the oblong rectangle and called oblong numbers.

SOC. Very good; and what next?

THEAET. All the lines which form the four sides of the equilateral or square numbers we called lengths, and those which form the oblong numbers we called surds, because they are not commensurable with the others in length, but only in the areas of the planes which they have the power to form. And similarly in the case of solids.

Essentially, Theaetetus's theorem is that the square root of any number that is not a perfect square is irrational, and the cube root of any number that is not a perfect cube is irrational.

Using geographic algebra, you can visually see Thaetetus's theorem.

For simplicity sake, using just two dimensions, make a perfect square with an area of 4 units.

Each side of the square is 2.

4 is a perfect square and 2 is a rational number (length).

Then make an "oblong rectangle" that is 2 spaces (visually a 1x2).

2 is not a perfect square (an oblong), and the square root of 2 is irrational or a "surd".

While a proof of the irrationality of the square root of 2 did appear in some versions of Euclid's Elements, as proposition 117 of Book X, it is considered an addition inserted by someone later. We can look at other propositions in the book at are specifically attributed to Euclid though for an example of how ancient Greek proofs were presented. For instance Elements Book 8 proposition 14 (as translated by Richard Fitzpatrick) which which is directly applicable to Theaetetus's Theorem for the squares (Elements Book 8 Proposition 15 for cubes):

1. If a square measures a square then the side will also measure the side. And if the side measures the side then the square will also measure the square.

2. Let A and B be square numbers, and let C and D be their sides. And let A measure B. I say that C also measures D.

3. For let C make E multiplying D. Thus, A, E, B are continuously proportional in the ratio of C to D. And since A, E, B are continuously proportional, and A measures B, A thus also measures E. And as A is to E, so C to D. Thus, C also measures D.

4. So, again, let C measure D. I say that A also measures B. For similarly, with the same construction, we can show that A, E, B are continuously proportional in the ratio of C to D. And since as C is to D, so A to E, and C measures D, A thus also measures E. And A, E, B are continuously proportional. Thus, A also measures B.

5. Thus, if a square measures a square then the side will also measure the side . And if the side measures the side then the square will also measure the square. The very thing it was required to show.

You can visually see this by making a square that is 16 units (A) and each side is 4 (C).

Then make a square that is 4 units (B) and each side is 2 (D).

A measures B in that there are four B squares in the 16 A square.

C measures D in that there are two Ds to make C.

E equals 2 and it's square is 4, the ratio of A and B.

Change the square to 36 units and 9 units, where A is 36, C is 6, B is 9 and D is 3, you get the same E equals 2 and the its square 4 is the ratio between A and B. You can change C and D to be any value where E equals 2, A and B will still be the same proportion of 4.

If you use different numbers, say a square of 36 units with a 6 unit sides and a square of 4 units and 2 unit sides

C is 6, D is 2 and you get E equal 3.

The square of E, 9, is the proportion between 36, A and 4, B. If you change C and D to any value where E equals 3, A and B will always be in proportion of 9.

If we use Euclids proposition and extend it with Thaetetus's theorem to modern terms then:

If C and D are natural numbers, then A (C^2 ) divides B (D^2 ) if and only if C divides D.

A = XB, C = YD

C^2 = XD^2 , C = YD

C^2 = XD^2 , C^2 = Y^2 D^2

X = Y^2

If X is a perfect square then Y is a rational number. If X is an oblong, then Y is a surd.

If you make X = 2, the you get C^2 = 2D^2

As this is the basic formula to prove the square root of 2 is irrational through the modern method.

C^2 = 2D^2

C^2 must be even because it is a multiple of 2, so C must be even because a square of an odd number is an odd number.

If we solve for 2^-2 we get:

2^-2 = C/D

With the assumption that C/D is the "lowest term", D must be an odd number if C is even.

However if we know that C^2 = 2D^2 where C is even, we know that C is a multiple of 2, therefore C = 2G.

So we get the equation (2G)^2 = 2D^2

4G^2 = 2D^2

2G^2 = D^2

D must be even, which we know it cannot be.

One can try to extrapolate what an Ancient Greek proof would be like based on Euclid's propositions...