This theorem is a storied one from the days of Pythagoras. Legend has it that Pythagoras was so threatened by the result -- it contradicted his religious teachings -- that he had the student responsible for the discovery drowned at sea upon the ruse of its celebration.
A plausible evolution of a dangerous discovery: the irrationality of √2. There is a tale about Pythagoras ordering the murder of a Pythagorean to bury his discovery that the diagonal of a square is incommensurable with any rational number. While this story may be apocryphal, it suggests that the Pythagoreans studied the properties of square roots. One wonders how this study began and how it unfolded. Since we can not know with certainty, we will have fun imagining. We do know the Pythagoreans were quite concerned with ratios. They attempted to account for all magnitudes by establishing ratios between them and positive integers. The modern approach to the proof about the square root of 2 being irrational is quite elegant. However, merely demonstrating or observing that proof does not convey the mathematical experience that led to it. It is more instructive to imagine a recreation of the search for a proof of that fact and then to engage students in that recreation. Both teacher and student will learn far more about the mathematics behind it. If we simply start with the assumption that the square root of 2 is rational, then we run into the problem of having to repeatedly divide common factors in the terms of the ratio. It seems that infinite regress is unavoidable until we realize that the number of factors of an integer is finite and will thus be exhausted once we reach a relatively prime pair of integers. We can get around this problem by making the additional assumption that the terms of the ratio are already relatively prime. However, in the middle of this proof we are confronted with another problem: the relationship between the parities of perfect squares and their square roots. Undoubtedly, this took some time to explore before it was formalized into a simple statement about all perfect squares' roots and included in the proof. One may imagine that it was initially an empirical observation offered to lend credence rather than deductive certainty to the proof. The empirical beginnings of mathematical investigations is something that is rarely discussed with math students. There is little done to convey the scientific process of mathematical experimentation. Mathematics, like the rest of rigorous scientific investigation, proceeds from a question and passes through the stages of hypothesis, documentation of experimental results, analysis, observation, formalization, and related questions. Where mathematics breaks from the rest of science is by leaving the measurable world from which the observations came to enter Plato's Realm of Forms and develop axioms upon which the facts may be deduced rather than merely induced. We are never afforded such treatement in math classrooms. This state of affairs is unfortunate, for it only presents the finale of mathematical investigations and hence hides most of the mathematical experience from students. No wonder we lag in mathematics and science! Math teachers should take care to direct students to empirical observation of the probable truth of propositions they are responsible for correctly applying. Additionally, a contrapositive argument is used in the proof unnecessarily for a direct proof is not difficult. The statement in question is the assertion that a perfect square and its square root have the same parity. Certainly, the contrapositive of that fact leads to a more elegant proof, but it is less instructive because it is non-constructive. On the other hand, the contrapositive leads to a number of cases equal to the value of the prime in the radicand. By taking the time to construct a direct proof of the parity relation we avoid case by case analysis. It also reinforces the import of the Fundamental Theorem of Arithmetic into proofs of irrationality.
What happens if we proceed from the assumption that √2 is rational? You will soon exhaust yourself with tedium if you try to find a rational value for √2. Try it. Suppose there are integers a and b such that √2 = a / b. Where does this lead us? Multiplying both sides by b gives us a = b√2. Squaring both sides gives us a2 = 2b2. We have removed the problem of having a fraction and a confounding square root from our equation, but it encodes the same information. Now, it tells us that a2 is even. What can we infer from this observation? What statements would help us to proceed in the proof? We have arrived at the point of doing mathematics because we must begin to outline desired connections between the last statement and the desired conclusion of the proof. Let's start with mining the last statement for information. If restate by saying there is an integer j such that a2 = 2k, and then take the square root of both sides we get a = √(2j). This is no better; it brings us back to the initial condition if we rename √j as, say, k. What else does it tell us about a? Let's observe some even squares and their roots.
| square | 4 | 16 | 36 | 64 | 100 | 144 | 196 | 256 | 324 | 400 |
|---|---|---|---|---|---|---|---|---|---|---|
| root | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
The table above suggests that the square of an integer is even if and only if the integer is even. This is evident if you also look at the squares of odd integers.
To prove there is no rational equivalent of √2, we first need to prove a helping proposition. Let's call it 'Lemma'.
Lemma: If a square is even, then its root is even.
Proof of Lemma: By contrapositive, the statement "If n is odd, then n^2 is odd." is logically equivalent to Lemma. We will use the contrapositive form of Lemma in this proof. Suppose n = 2k + 1 for some integer k. It follows that n^2 = 4k^2 + 4k + 1. We see that n^2 is odd upon the observations (a) 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 and (b) 2k^2 + 2k is an integer. This concludes the proof.
What does this conclusion mean? If √2 is not rational, then what is it? Where does it belong on the number line? Let's start with a guess. We know that 1 < √2 < 2. 1½ is a reasonable place to start. How close is it? (1½)^2 = 2¼. Solving (3/2)x = 2 gives us 4/3. (4/3)^2 is 1 7/9. Thus, we have improved our bounds: 4/3 < √2 < 3/2. Taking the mean of 3/2 and 4/3 gives us an improved estimate for √2: 17/12. How close is it? (17/12)^2 = 2 1/144, which is extremely close, but we are trying to understand what is going on with √2. The error is found by solving (17/12)x = 2 which gives us x = 24/17. (24/17)^2 = 1 287/289. We are closing in! 1 < 4/3 < 24/17 < √2 < 17/12 < 3/2 < 2. Is there an end to this? If not, it would suggest that the magnitude √2 confounds all of our measurement algorithms.
If you kept improving the bounds, then you would eventually tire of the tedium for it seems to never end. So, now we have expanded the universe of numbers by one object: ℚ ∪ {√2}. Can we do the usual arithmetic in this expanded universe of numbers? If we can't encapsulate √2 in any meaningful way, then we will need a new arithmetic. Without a finite rational expression equivalent to √2, the best we can do is approximate or write things like a ⊛ √2 or √2 ⊛ b, where a and b are rational and ⊛ represents +, −, ×, or ÷. This is not practical. However, further investigation leads to some extremely important mathematical discoveries, like The Calculus.
The diagram above shows that the square built from the diagonal has twice the content of the square upon whose diagonal it was built. Thus, it shows us how to construct a square containing exactly two square units, and so the side of that square will be the square root of 2. This observation is the foundation of the Pythagorean Theorem.
One way to understand √2 is as a Dedekind cut. We redefine √2 as the set of all rational numbers q such that q2 is less than 2. We may do this for other troublesome roots like √3 ∛50. In fact, we can reformulate the number line as the union of the rationals and the totality of cuts. By doing this, we get a new number system, but it has a strange arithmetic and its definition seems self-referential, and therefore paradoxical, which would explain the paradoxes hiding in the bowels of real analysis. There is an ordered field with the least upper bound property. We shall construct it here. Let R be the set of all cuts.