The square root of 2 is irrational

The main idea behind the proof

By using proof by contradiction, try to express the square root of 2 as a fraction without common factors, i.e. as a rational number. This is not possible and ultimately leads to a contradiction, which means that the square root of 2 cannot be expressed as a rational number.

Historical Notes

The ancient Greeks, the Pythagoreans, studied prime numbers, progressions, and those ratios and proportions, but in contrast to our current understanding, a ratio of two whole numbers was not a fraction, i. e. a distinct kind of number with respect to the whole numbers¹. The (own) discovery of the role of whole numbers in musical harmony inspired Pythagoreans to seek whole-number patterns everywhere². Now, if quantities could have been measured by a common unit using whole numbers, they had a common measure and where called com-mensurable³. The discovery of ratios that were not measurable in this way, i. e. that where in-commensurable, is attributed to Hippasus of Metapontum⁴, and was a turning point in Greek mathematics⁵: It has affected mathematics and philosophy from the time of the Greeks to the present day⁶ and it is assumed that it marked the origin of what is considered the Greek contribution to rigorous procedure in mathematics⁷. The starting point of this scientific event⁸ in Greek mathematics was the Pythagorean theorem, that was discovered independently in several ancient cultures⁹. There is evidence¹⁰ that the Babylonians (1800 BC), the Chinese mathematicians (between 200 and 220 BC) and Indian mathematicians (between 500 and 200 BC) were interested in triangles whose sides where whole-number triples that - denoted in modern notation - satisfy the equation \(a^{2} + b^{2} = c^{2}\)¹¹, such as for instance the following ones, meanwhile referred to as Pythagorean triples¹², e. g. \(\langle 3, 4, 5 \rangle, \langle 5, 12, 13 \rangle, \langle 8, 15, 17 \rangle\):

\(\begin{align*} & 3^{2} + 4^{2} = 5^{2} = 9 + 16 = 25, \\ & 5^{2} + 12^{2} = 13^{2} = 25 + 144 = 169, \\ & 8^{2} + 15^{2} = 17^{2} = 64 + 225 = 289. \end{align*}\)

But it is assumed, that only the Pythagoreans were interested in a special case that eventually led to the discovery of the incommensurable ratios: Given that the two sides \(a\) and \(b\) of a right-angled triangle have the same length, that is \(a = b\), the crucial question - again expressed in modern algebraic symbolism - must have been whether there are whole numbers \(a\) and \(c\) that satisfy the following equation \(c^{2} = 2a^{2}\), that is fulfilling commensurability¹³.

Theorem 1

The square root of 2 is irrational.

Proof: We suppose that the square root of \(2\) \((\sqrt{2} = 2^{1/2})\) is rational.

By definition, a number belongs to the set of the rational numbers \(\mathbb{Q}\), if it can be expressed as a ratio of two integers \(\frac{p}{q}\), where the numerator \(p\) can be any integer and the denominator \(q\) must be a non-zero integer.

If the square root of \(2\) is rational, then it can be expressed as the ratio of two integers \(p\) and \(q\), where \(p\) and \(q\) have no common factor other than \(1\):

\((1) \qquad 2^{1/2} = \frac{p}{q}.\)

Now we square both sides of the equation \((1)\).

On the left-hand side (LHS) it gives:

\((2) \qquad \left(2^{1/2}\right)^{2} = 2^{2/2} = 2^{1} = 2 = ,\)

whereas on the right-hand side (RHS) it gives:

\((3) \qquad = \left(\frac{p}{q}\right)^{2} = \frac{p^2}{q^2}.\)

That is:

\((4) \qquad 2 = \frac{p^2}{q^2},\)

which by basic algebra can be rearranged into:

\((5) \qquad p^{2} = 2q^{2}.\)

Now, by definition, an even integer has the form \((2 \cdot \text{an integer})\) and \(2q^{2}\), having this form, is an even integer, which means that \(p^{2}\) is even. This leads us to the question of whether \(p\) is also even.

Proposition 1

For every integer \(s\), if \(s^{2}\) is even then \(s\) is even.

Proof: Suppose \(s\) is any odd integer. By definition, an odd integer has the form \((2 \cdot \text{an integer} + 1)\), so \(s = 2k + 1\) for some integer k. By substitution and basic algebra, we get: \(s^{2} = (2k + 1)^{2} = 4k^{2} + 4k + 1 = 2(2k^{2} + 2k) + 1.\) If we add or multiply integers toghether, the resulting sum or product will still be an integer, i. e. the result belong to the set of integers. This is referred to as the closure properties of addition and multiplication which hold within the set of integers. Due to these properties, the expression \(2k^{2} + 2k\) must be an integer. To simplify the notation we denote as \(2k^{2} + 2k = L\). Hence, also \(s^{2} = 2 \cdot L + 1\) is an integer, and by the definition of an odd integer, \(s^{2}\) is odd. Being proposition \(1\) and its contrapositive logically equivalent, this means that the original statement, that for every integer \(s\), if \(s^{2}\) is even then \(s\) is even, is indeed true. \(\square\)

Getting back to our theorem \(1\), now we know, that also \(p\) must be even. And by definition of an even integer, we denote \(p\) as:

\((6) \qquad p = 2r \qquad \text{for some integer} \; r.\)

Now, by substitution, we insert the RHS of the equation (6) into the LHS of equation (5), and we see that:

\((7) \qquad p^{2} = (2r)^{2} = 4r^{2} = 2q^{2}\).

By dividing both, \(4r^{2}\) and \(2q^{2}\) by \(2\), we get:

\((8) \qquad 2r^{2} = q^{2}\).

As we can see, by the definition of an even integer, \(q^{2}\) is even, and by the proof of proposition 1, also \(q\) must be even. But earlier, we deduced from both, the equation (5) and the proposition 1 that \(p\) is even. And this would mean, that both \(p\) and \(q\) are even, i. e. that they have the common factor \(2\). But this contradicts the initial assumption that \(p\) and \(q\) of the equation (1) had no common factor. It follows that the square root of \(2\) cannot be espressed as a ratio of two integers, which means therefore that the theorem 1 is true. \(\square\)

M. Kline, Mathematical thought from ancient to modern times, New York 1990, p. 32. ↩
J. Stillwell, Mathematics and its history, third edition, New York 2010, p. 11. ↩
M. Kline, Mathematical thought from ancient to modern times, New York 1990, p. 32. ↩
M. Kline, Mathematical thought from ancient to modern times, New York 1990, p. 32. ↩
J. Stillwell, The story of proof: logic and the history of mathematics, New Jersey 2022, p. 1. ↩
R. Courant/H. Robbins/I. Stewart, What is Mathematics? An elemtary approach to ideas and methods, second edition, New York 1996, p. 59-60. ↩
R. Courant/H. Robbins/I. Stewart, What is Mathematics? An elemtary approach to ideas and methods, second edition, New York 1996, p. 59.; J. Stillwell, The story of proof: logic and the history of mathematics, New Jersey 2022, p. 1. ↩
R. Courant/H. Robbins/I. Stewart, What is Mathematics? An elemtary approach to ideas and methods, second edition, New York 1996, p. 59. ↩
J. Stillwell, The story of proof: logic and the history of mathematics, New Jersey 2022, p. 1. ↩
J. Stillwell, Mathematics and its history, third edition, New York 2010, p. 4. ↩
J. Stillwell, The story of proof: logic and the history of mathematics, New Jersey 2022, p. 3-4. ↩
J. Stillwell, Mathematics and its history, third edition, New York 2010, p. 4. ↩
J. Stillwell, The story of proof: logic and the history of mathematics, New Jersey 2022, p. 6-7; R. Courant/H. Robbins/I. Stewart, What is Mathematics? An elemtary approach to ideas and methods, second edition, New York 1996, p. 58. ↩