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The Irrationality of the Square Roots of 2

There are two parts to this proposition. One part is that there is no rational number such that its square is equal to 2. Second there is a mathematical system in which the square root of 2 makes sense. The second part is fulfilled in geometry. A right triangle with unit sides has a hypotenuse that can be identified as √2.

The positive integers arrive naturally in the human experience. From there the notion of fractions of the form 1/2, 1/3 and so forth arose. It was another step to identify fractions of the 2/3, 3/4, 7/8 and so forth. Still another step led to fractions such as 3/2 and 4/3. Then there was the insight that 6/4 was the same number as 3/2. Thus a rational number is the set of fractions of the form kn/m where k is an integer and n and m are integers with no common divisor. The set is represented by n/m. Addition and multiplication is easily defined for these rational numbers.

Somewhere along the line humans became conscious of zero as being a number. Still further along humans recognized negative integers and hence also negative rational numbers.

The Proposition that
a Square Root of 2 is not Rational
is not True in All Arithmetic Systems

Consider the arithmetic of remainders upon division by 7. This is called arithmetic modulo 7. (Usually modulo is abbreviated as mod.) Thus 1, 8, 15, 22 … are all equal modulo 7 became they all have a remainder of 1 upon division by 7. In this system 5+4 equal 2 modulo 7, and 3*5 equal 1 modulo 7.

Arithmetic modulo 7 is a logically consistent system. Note that 4*4=16=2 modulo 7 so 4 is a square root of 2 modulo 7. So √2 is not only a rational number modulo 7, it is integral. However properly stated the square roots of 2, modulo 7, are ±4. However note that 3*3=9=2 modulo 7, so 3 is a square root of 2 modulo 7. This might seem to be a puzzlement but it is not, because 3=−4 modulo 7.

Note that, modulo 7, 2*2=4 and 4*4=2.

Proof by Contradiction that There Does Not Exist
a Rational Number Such that Its Square is Equal to 2

Assume that such a rational number exists. Let it be denoted by n/m where n and m have no common divisor other than ±1. Then

n²/m² = 2
and hence
n² = 2m²

Thus n² is an even number. The integer n must also be an even number because if it were odd, say 2k+1, then its square would be (4k² + 4k +1), which is 2(2k²+2k)+1, an odd number. But if n=2j, then

n² = 4j² = 2m²
which means that
m² = 2j²

This would mean that m² and also m are even numbers, contrary to the assumption that n and m have no common divisor. Thus no such n/m can exist.

The Existence of √2 as a Dedekind Cut

Richard Dedekind formulated a model for the real numbers that can be represented as partitions of the rational numbers. For a description of that system see Dedekind.

Algebraic Numbers

An algebraic number is a root of a polynomial with rational number coefficients. Therefore √2 is an algebraic number because it is a root of the equation x²−2=0.

It does not matter whether the coefficients are limited to rational numbers or to integers. Any polynomial equation with rational number coefficients can be converted into a polynomial equation with integer coefficients that has the same roots.

There are real numbers which are not algebraic numbers. They are called transcendental numbers. In fact there are vastly more transcendental numbers than there are algebraic numbers. The cardinality of the set of algebraic numbers is the same as the cardinality of the set of rational numbers. Thus the cardinality of the set of transcendental numbers is the same as that of real numbers, the continuum. For more on algebraic number see Algebraic Numbers.


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