The incommensurable

Lengths without a common measure: why side and diagonal of the square stand in no ratio of numbers, why the Euclidean algorithm then never ends, and how the pentagram, the emblem of the Pythagoreans, carries the same discovery within it.

“All is number” is how the teaching of the Pythagoreans has been summed up since Aristotle, and today it is mostly read like this: everything can be expressed by ratios of whole numbers. At some point in the fifth century BC one of them discovered that this is not true. The late tradition in Iamblichus does not name him and tells that for betraying this secret he was expelled or perished at sea. Only modern scholarship links the story with Hippasus of Metapontum, whom Iamblichus elsewhere has drown because of the dodecahedron. Nobody knows whether the story is true. Many see the discovery as a blow to the heart of Pythagorean teaching, other historians doubt that such a crisis ever happened (see Pythagoras: the teaching).

Commensurable

Definition

Two lengths $a$ and $b$ are called commensurable if there is a length $e$ that goes into both exactly:

a = m \cdot e, \quad b = n \cdot e, \qquad m, n \in \mathbb{N}.

Then $a : b = m : n$ is a ratio of whole numbers. Otherwise they are called incommensurable.

Commensurable means “measurable together”. Two strings whose lengths are commensurable stand in an interval as in Chapter 8. The Pythagoreans assumed that this holds for all lengths, since you only have to choose the common measure $e$ small enough.

Side and diagonal

How long is the diagonal of a square? In Plato’s dialogue Meno, Socrates shows a slave that the square on the diagonal is twice as large as the original square: a diagonal divides the small square into two triangles, and the square on the diagonal consists of four such triangles, so it is twice as large. For the diagonal $d$ and the side $s$ we therefore have $d^2 = 2 s^2$ .

Theorem 1

Side and diagonal of a square are incommensurable. Equivalently: $\sqrt{2}$ is not a fraction.

Proof

Suppose $d : s = a : b$ with natural numbers, in lowest terms by Chapter 8, Theorem 1. From $d^2 = 2s^2$ it follows that $a^2 = 2b^2$ . So $a^2$ is even. If $a$ were odd, $a^2$ would be odd too, so $a$ is even, $a = 2c$ . Then $4c^2 = 2b^2$ , so $b^2 = 2c^2$ , and by the same argument $b$ is even too. Then the fraction $a/b$ would not be in lowest terms, a contradiction.

∎

Aristotle already knew this proof. In the Prior Analytics (I 23) he cites it as the model of a proof by contradiction: if side and diagonal were commensurable, the same number would be both even and odd.

The taking away that never ends

How could the Greeks have noticed that there is no common measure? You cannot try out all measures. The way leads through the procedure of Chapter 3.

Theorem 2

Two lengths are commensurable exactly when mutual taking away, the Euclidean algorithm, comes out after finitely many steps.

Proof

If the procedure comes out, the last length measures the one before, hence all previous ones and finally both starting lengths. It is a common measure.

Conversely, if $a = me$ and $b = ne$ , every remainder in the procedure is a whole multiple of $e$ , since it arises by subtracting such multiples. The remainders get smaller at every step, and a decreasing sequence of natural numbers $m, n, \dots$ ends after finitely many steps.

∎

This is Euclid’s Elements X 2 and 3. Now one can try taking away on side and diagonal:

a b

Rectangle in the ratio 1 : √2. After the first square, two squares and a remainder are always left, and the remainder's shape repeats. The taking away never ends.

Theorem 3

In taking away on $1$ and $\sqrt{2}$ , after the first step every further step repeats with the same shape: exactly two squares always fit in. The procedure never ends.

Proof

After the first step a rectangle with sides $1$ and $\sqrt{2} - 1$ remains. Since $(\sqrt{2} - 1)(\sqrt{2} + 1) = 2 - 1 = 1$ , its ratio of sides is

\frac{1}{\sqrt{2} - 1} = \sqrt{2} + 1 = 2 + (\sqrt{2} - 1).

So two squares fit in, and the remainder again has the ratio $1 : (\sqrt 2 - 1)$ , the same as before. It goes on like this without end. By Theorem 2 the lengths are incommensurable.

∎

The pentagram

Many historians, since Kurt von Fritz in 1945, suspect that the discovery was made not on the square but on the pentagon, the figure that later sources ascribe to them as their sign of recognition. In the regular pentagon the ratio of diagonal to side is the golden ratio $\varphi$ , and $\varphi = 1 + \tfrac{1}{\varphi}$ (see The pentagram and the golden ratio).

Theorem 4

Diagonal and side of the regular pentagon are incommensurable. Equivalently: $\varphi$ is not a fraction.

Proof

Since $\varphi = 1 + \tfrac{1}{\varphi}$ , exactly one square fits into the rectangle $1 \times \varphi$ , and the remainder has sides $1$ and $\varphi - 1 = \tfrac{1}{\varphi}$ , so again the ratio $\varphi : 1$ . Every step repeats the previous one, and by Theorem 2 there is no common measure.

∎

In the pentagram one sees this without calculation: in the middle of every pentagram lies a smaller pentagon, into which one can again draw a pentagram, and so on. Diagonal and side of the large pentagon can be expressed by the lengths of the small one, those of the small one by those of a smaller one still, and a last, smallest common measure is never reached. Whoever draws the sign of the Pythagoreans attentively finds in it the limit of their teaching.

a b

Rectangle in the ratio 1 : φ, the ratio of side and diagonal in the pentagon. At every step exactly one square fits in, and the remainder is again a rectangle of the same shape.

Checked by computer: the proof in Lean

root_two says: there are no natural numbers $a$ and $b > 0$ with $a^2 = 2b^2$ . Lean does not prove this with “lowest terms” but by infinite descent, through strong induction on $b$ : from a solution $(a, b)$ with $a = 2c$ comes the solution $(b, c)$ with the smaller $c < b$ . But a natural number cannot get smaller infinitely often. This form of the proof corresponds exactly to the pentagram within the pentagram.

/-! Chapter 9: the square root of 2 is not a fraction -/

/-- There are no natural numbers a, b > 0 with a² = 2b².
    So √2 is not a fraction. -/
theorem root_two (b : Nat) :
    ∀ a, 0 < b → a * a ≠ 2 * (b * b) := by
  induction b using Nat.strongRecOn with
  | ind b ih =>
    intro a hb h
    -- a² is even, so a is even: a = 2c
    have ha : a % 2 = 0 := by
      rcases Nat.mod_two_eq_zero_or_one a with h0 | h1
      · exact h0
      · have : a * a % 2 = 1 := by rw [Nat.mul_mod, h1]
        omega
    obtain ⟨c, rfl⟩ : ∃ c, a = 2 * c := ⟨a / 2, by omega⟩
    -- 4c² = 2b², so b² = 2c²:
    -- the same equation with smaller numbers
    have h' : b * b = 2 * (c * c) := by
      have : 2 * c * (2 * c) = 2 * (2 * (c * c)) := by
        rw [Nat.mul_assoc, Nat.mul_left_comm c 2 c, ← Nat.mul_assoc]
      omega
    have hc : 0 < c := by
      rcases Nat.eq_zero_or_pos c with h0 | hpos
      · subst h0
        have : 0 < b * b := Nat.mul_pos hb hb
        omega
      · exact hpos
    -- c < b, otherwise 2c² ≥ 2b² > b²
    have hcb : c < b := by
      apply Classical.byContradiction
      intro hn
      have : b * b ≤ c * c := Nat.mul_le_mul (by omega) (by omega)
      have : 0 < b * b := Nat.mul_pos hb hb
      omega
    exact ih c hcb b hc h'

Try it in the browser at live.lean-lang.org ↗How to read Lean

What came after

The discovery was not a catastrophe but a beginning. Theodorus of Cyrene showed, as Plato reports in the Theaetetus, that $\sqrt{3}, \sqrt{5}$ and the other roots of non-square numbers up to $\sqrt{17}$ are incommensurable too. In the fourth century BC Eudoxus of Cnidus created a theory of ratios that includes incommensurable magnitudes, and Euclid hands it down in the fifth book of the Elements. Only in the nineteenth century did Richard Dedekind turn it into today’s theory of the real numbers. Its first application, the circle number, is the subject of the next chapter.

Sources and further reading

Sources

Plato, Meno 82b to 85b, and Theaetetus 147d to 148b
Aristotle, Prior Analytics I 23, 41a
Euclid, Elements, around 300 BC, book X
Kurt von Fritz, The Discovery of Incommensurability by Hippasus of Metapontum, Annals of Mathematics 46, 1945