A demonstration of Pythagras' Theorem followed by Hippasus's Proof, the points I make being from my concluding questions: Seeing any need for √2, the most interesting question is how does mathematics envelop new numbers? After answering that, what extrensic reason is there for enveloping √2 into a number system? Hippasus' indirect proof merely demonstrated, contrary to our intuition, that √2 cannot be expressed as a rational number p/q; it says nothing about what should replace the assumption from line 8. And I am not satisifed that we have solved this problem for reasons other than the ideal consistency of the real numbers.
I stumbled onto irrational numbers after convincing myself of the veracity of the Pythagorean Theorem and reflecting on the use of the word “irrational”. This reflection led to fundamental questions about how and why mathematics enveloped irrational numbers and whether the rational dilemma presented by Hippasus to the Pythagoreans was thoroughly resolved.
There are many proofs of the Pythagorean Theorem. A common form for this theorem is:
1. c2 = a2 + b2
I used the “areas of plane figures” because I saw the relationship between the common formula and squares; another uses similarity theorems of Euclidean geometry. (Verdina, 130) The Pythagorean Theorem is a good exercise in demonstrating mathematical concepts and the theorem's veracity. Demonstrating the Pythagorean theorem could be made trivial: one needs at least one known, and one unknown as a function of this known; for simplicity's sake, let a = 3 and b = 4. ... Oh this is trivial! I would find the unknown terms and the equation would be satisfied! Rather than trivialize common acceptance of this theorem, ask, “how did Pythagoras arrive as this theorem? what was his rational?
Representing the theorem with no net change made his rational clearer, as:
1'. 0 = a2 + b2 – c2
I reflect on the algebraic and geometric definitions of these symbols and terms. Let the terms a and b be the length of square sides forming a isosceles angle at one end, and c be the length of a square side which forms the secant length of a diagonal drawn from the opposite ends of the sides a and b. When the squares A and B with sides of length a and b respectfully are overlain, the resulting square C has sides of length c. I found that Mathworld calls this method proof by shearing, and can be done as a “Kitchen Experiment” by cutting paper squares A and B and overlaying them to yield square C or by various, online and interactive demonstrations. The "shearing proof" demonstrates that the addition of the area of the squares A and B with sides of length a and b yields the area of the square C with sides of length c: namely, equation 1.
But one cannot rationalize this overlaying process without understanding area, and I found area to be difficult to explain. It merely came down to me defining it: let the fundamental area be a unit square, 1 x 1, and all other areas be the repetitive addition of this fundamental area until the unit square fills the space we confine with geometric lines. For example, a 2 x 3 rectangle has 6 unit squares, calculated by row and column addition of 1 x 1 squares. A fascinating digression is that my definition expresses area pragmatically; it is only valid for whole number areas, not fractional sides. More on that later. Suffice to say, it sounds better to say that we “reasonably” demonstrate the Pythagorean Theorem instead of “rationally” demonstrate it because the terms can be misunderstood as referring to types of numbers. For example, today we express the area of circles “irrationally” (π is involved) but the demonstration that circles contain areas is not irrational.
Eitherway, Pythagoras' rational depends on properties of a triangle, namely the length of any square-side triangle (isosceles triangle) through square areas. These triangular properties are based on Euclidean geometry, where this geometry can be said to be merely the system of postulates used to demonstrate concepts we use in drawing, like side, length, area, etc. (Verdina 290) Perhaps this is boring because of its verbosity but allow this defense: by reasoning through Pythagoras' problem we notice the theorem depends on applying algebraic definitions alongside a geometric demonstration, a geometry which itself is defined. The most interesting algebraic trick was "no net change" seen in 1', which is a fascinating digression in the Theory of Equality, and an interesting relationship between the algebraic representation of a square's area, s2, to the drawing of three squares.
It occurred to me later that another problem, more perplexing than that of defining area, would arise. Similar to the unit square, 1 x 1, that I used to defined the area for squares with sides of length that are whole numbers, I considered the “unit triangle”. Where the unit square S has sides of length one, S: 1 x 1, I say the isosceles, “unit triangle” has legs of length one. According to Pythagoras' theorem,
1. c2 = a2 + b2
where a and b represent the legs of this triangle and c its hypotenuse. Now the problem: given that a = b= 1, find the hypotenuse c.
2. c2 = 12 + 12
3. c2 = 1*1 + 1*1
4. c2 = 1 + 1
5. c2 = 2
2. through 5. use normal algebraic rules that many take for granted, even if acknowledging them here. According to the definition of squares,
6. c * c = 2
6. asks: what number, multiplied by itself, is 2. Using normal rules of addition and multiplication, 1 * 1 = 1, a number less than 2, but 2 * 2 = 4, a number greater than 2, so the value of c shoule be somewhere between 1 and 2.
Another approach to resolving the problem of this unit triangle's hypotenuse is by reflecting on the meaning of symbols, namely, what numbers represent. Given 5. above, we can also denote that:
6'. c = √(2)
This radical sign has become common notation without much elaboration about what it means. This symbol means the square root of a number n, where we define square root as something whose square yields the number n, called the radicand.
SQRT. √(n)*√(n) = n.
For our example, 6', √(2)*√(2)=2. People would later notice problems when n < 0, or when “taking the square root of a negative number”, and these were settled elsewhere (by imaginary numbers) We will exclude such problems by confining our analysis to the expected length of c. Dividing a number line of domain [ 1 .. 2 ] demonstrates that there are an infinite amount of fractional numbers between the whole numbers 1 and 2, so the Greeks expected to find the value of c somewhere in there. Any fractional number is represented as a ratio of p to q:
7. RN: p/q.
where p and q are non-factorable integers, and with the normal rules for division apply. So if p=9 and q=8, then 9/8 falls between 1 and 2 because, represented as a mixed number, 9/8 = 1 1/8 where 1 1/8 is greater than 1 and less than 2. It is important to reiterate that p/q is reduced, or cannot be simplified by a common factor. For example, a/b=19/9 is not reduced because both a=19 and b=9 have a common factor of 9, reducing the ratio to 2 1/9.
Expecting the hypotenuse to be such a fraction, we can represent c as some integer p divided by some integer q. Let us assume c is such a fractional number on the number line between 1 and 2,
8. c = p/q, c: [1 .. 2]
Substitution of 8. into 5. yields,
9. (p/q)2 = 2
10. p2/q2 = 2
11. p2 = 2*q2
Because the coefficient of the q term is now 2, the p term must be even. For example: let a be unknown and b = 3 for the equation a = 2*b. Solving the equation, 6 = 2 * 3, we notice a is an even number, 6. This is not a proof, although one can be written, but merely exemplifies why the p term in 11. must be even. Also, by assuming 5. and the definition of SQRT, we can infer that q will be odd.
To include the deduction that p is even, one can represent p as a factor of 2: given that any integer r multiplied by the factor 2 is even, then 2*r is even. So, let p = 2 * r for 11. above as:
12. (2 * r)2 = 2*q2
13. 4 * r2 = 2*q2
Now, intuition might tell you that something is amiss. 14. looks like 11; this is because they are saying the same thing, although about different terms. As already stated, 11. identifies p as the even term. Using the same, deductive logic, 14. identifies q as an even term. But, if both p and q are even, then the fraction c has a factor of 2. We explicitly demanded in our assumption for 8. that c be non-factorable. This assumption is necessary because of the expected value of c, or in math-lingo, the domain limits the value of c. If there is a common factor, say 2, for p/q, then c is not where we expect to find it. c would be outside the bounded number line [ 1 .. 2 ]. So we have a problem.
The Pythagorean's Dilemma
Does one abandon 5. by insensibly allowing the hypotenuse of the triangle to be longer than geometrically demonstrable? or does one maintain that both the deduction of 11. where p is even and expected q to be odd, andthe deduction of 14 that says the converse are both valid? The Greek response was: either the hypotenuse of the unit triangle is longer than drawn OR q is both odd and even.
The latter possibility – that q is both odd and even -- is, by the Law of (Non-)Contradiction, irrational. The Law of (Non-)Contradiction is a fundamental aspect of rational thinking because it ensures against holding opposite conclusions. It assures against answering, “Yes and No” to a “Yes or No” question. The former possibility – that the hypotenuse of the triangle is longer than expected -- is insensible, meaning impractical for construction of whatever. Even another approach to demonstrating the Pythagorean Theorem -- one I did not see as evident, namely the proof by similar triangles -- more closely depends on the postulates of Euclidean geometry, and these postulates have been proven, if for any other geomatric shape, consistent for such a triangle on a plane.
So the Pythagorean dilemma was: does one abandon Euclidean geometry or the Law of (Non-)Contradiction?
This dichotomy is a sizable quandary in the relationship between what one expects, as drawn, against what one expects, as deduced. Hence the Greeks called the length of this hypotenuse “incommensurable”, meaning it was not congruent with measuring tools. “This result disturbed the logic-loving Greeks, who felt that the existence of such a number was proved by geometry and disproved by algebra. Some historians conjectured that this paradox of the 'incommensurable', as it was called, retarded Greek science, and consequently world science, for centuries.” (187, Denbow and Goedicke) “The early Greeks believed that every measurable quantity had to be a rational number. However, this idea was overturned in the fifth century B.C. by Hippasus of Metapontum * who ... using geometric methods,  showed that the length of the hypotenuse of the isosceles triangle could not be expressed as the ratio of two integers.” (A1, Anton, Bivens and Davis) This proof was the very one demonstrated above, or at least a contemporary version of it. “According to legend, Hippasus made his discovery at sea and was thrown overboard by fanatic Pythagoreans because his result contradicted their doctrine.” (ff A1, Anton, Bivens and Davis), see also (129 Smith) So the stakes were high when the dilemma arose and how did we resolve them?
Should we consider another case: Neither? We neither abandon Euclidean geometry nor deduction and the Law of (Non-)Contradiction by restating that the dilemma is not a “Yes or No”, bipolar question?
Denbow and Goedicke in Foundations of Mathematics explain the eventual solution surmounted the intuitive notion of the number line and our expectations of it, or as I would like to say, the "fix" recognized the demands this problem placed upon the mathematical system. As they illustrate with the number line [1 .. 2] mentioned above: ”We can then select any two of these [number line] points, no matter how close, and insert an infinite number of further points between them ... If we depend on our intuition we are inclined to believe that this process will give us all the points on the line. ... This intuitive conclusion is false.” (184) Seeing any need for √2, the most interesting question is how does mathematics envelop new numbers? After answering that, what extrensic reason is there for enveloping √2 into a number system? Hippasus' indirect proof merely demonstrated, contrary to ourintuition, that √2 cannot be expressed as a rational number p/q; it says nothing about what should replace the assumption from line 8. And I am not satisifed that we have solved this problem for reasons other than the ideal consistency of the real numbers.