The Square Root of Two is a Real Irrational Number


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Last week’s article, “Cantor Defeated Galileo in the Battle of Infinite Numbers“, noted:

…the square root of two is not rational. No fraction can exactly equal 2^(1/2).

2^(1/2) is, as you know, the square root of two. This claim deserves some proof; let’s ask Pythagoras and Euclid for their expert opinions.

Quick Reviews of Different Types of Numbers

This article almost always deals with positive numbers, since the “square root of two” problem first arose when finding the length of lines in geometry. It would also stretch the article to repeatedly include:

“…when both integers are positive or when both integers are negative.”

Natural Numbers and Integers

The Natural numbers are often defined as the positive integers. In the previous article, we also included zero as a natural number. The natural numbers are sometimes called the “counting numbers.”

Integers, of course, include negative numbers. Sometimes called “whole” numbers because they are not fractions, the integers simply extend the set of natural numbers by including the set formed by multiplying each natural number by ‘-1‘.

Rational Numbers

The Rational numbers are the fractions made by dividing one integer by another. We exclude “division by zero”. Usually the division operation is defined as the inverse of multiplication. For numbers ‘a,’ ‘b’ and ‘c,’ we say “a = b/c” if and only if “b = a*c.” We exclude the cases where ‘c’ is zero because for any ‘a,’ ‘a’ times zero is zero.

Rational numbers include integers; for example 2 = 6/3. In any case where “i = j/k” and {i, j. k} are integers, we say that ‘k’ is a “factor” of ‘j,’

Rational numbers might need a limited number of non-zero digits after the decimal. Examples include “2 = 2.0 = 6/3,” and “2.5 = 2.50 = 5/2.”

Rational numbers might need an infinite series of digits after the decimal. One example is “3.333… = 10/3.”

The infinite series might be a repeated sequence of different digits, not just a single digit. One example is “1.717171… = 170/99.”

For any rational number ‘r’, there is an infinite set of pairs of integers {(i[k], j[k])} such that “r = i[k]/j[k].” For example, “1.717171… = 170/99 = 1700/990 = 17000/9900 =…”. There is only one unique pair (i[1], j[1]) where there is no common factor that can divide both i[1] and j[1]. In this case, the ratio, or fraction, is expressed in its “lowest terms.

Irrational Numbers

The Irrational numbers cannot be expressed as a ratio of two integers.

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