Proof that Square Root of Two is Irrational

First, let us review some basic facts:

1.  N or the set of natural numbers, is the set (collection) {1, 2, 3, …}

2.  Every natural number > 1 can be decomposed into the product of (other) numbers, often smaller numbers, and often in several ways.  For example, 60 = 3·20 or 60 = 6·10.  These other numbers, 3 and 20 or 6 and 10, are called divisors or factors of 60.  P, or the prime numbers, are those natural numbers > 1 that have no divisors other than the number itself and 1, such as 2, 3, 5, and 7.  Other natural numbers greater than 1 (those that have other divisors) are called composite numbers.

3.  The Fundamental Theorem of Arithmetic says that every natural number greater than 1 can be expressed uniquely as a product of primes, up to order.

4.  One consequence of this fundamental theorem is that if there is some number, call it m, and we square it to get m2, then m2 will have exactly the same prime divisors that m has, only the number of occurrences of each prime divisor will be double that of m.  Hence, m will have an even number of prime divisors in total.

5.  The Whole Numbers, or W ,are the numbers in the set {0, 1, 2, 3, …}.

6.  The Integers, or Z, are plus or minus the whole numbers, namely, the set of numbers

{… -3, -2, -1, 0, 1, 2, 3,   }

7.  It is clear that there is a “lot of space” between integers on the number line.  Even though the next integer after 0 is 1, there is room for many other numbers between these two on the number line.  We call this fact “the integers are discrete”, meaning we could draw a small circle (say of radius ¼) around each integer and there would be no other integers inside that circle.

8.  The Rational Numbers, or Q, are all fractions or quotients of the form A/B where A and B are both integers and B is not equal to 0.

9.  Note that if B is plus or minus 1, then A/B is an integer, so we see that all integers are considered rational numbers.

10.  Unlike the integers, the rational numbers are very closely packed on the number line.  The word we give for this fact is dense.  If we draw a circle of very small radius around any fraction on the number line, regardless of how small we make the radius of the circle, there will still be another fraction inside that circle.  For example, if we draw a circle of radius ¼ around the fraction ½, then 5/8 is still inside this circle.  In fact there are many fractions inside this circle.  Even if we make the radius very small, say 1/10 or 1/100 then since ½ = 5/10 = 10/20 = 50/100 we can add one to the numerator and get 11/20 or 51/100 and this new fraction will be inside the circle.  So any circle centered around each and every fraction will contain another fraction or rational number.  There is no “next” rational number.  The next integer after 0 is one, but what is the next fraction after ½?  11/20, 51/100, 251/500?  There is no next fraction, because they are dense.

11.  In spite of the fact that the fractions or rational numbers are dense on the number line, still there is room for more numbers.  There are numbers on the number line that are not fractions, for example, the square root of two or pi.  (For convenience we will write the square root of two as (sr2) in the future.)  All numbers on the number line are called real numbers.  Those numbers on the number line that are not fractions are called irrational numbers.  The Greeks called them irrational because they knew they were there but they could not see how they could fit in since the fractions were dense, so they called them irrational.

12.  We proved in class that if any fraction will be changed to a decimal then the decimal will either terminate, or, if it goes on forever, it will be an infinite repeating fraction.  We also showed the converse, namely that if we start with an infinite repeating decimal, we can change it back to a fraction.  These two facts together mean that the set of fractions or rational numbers is exactly the same as the set of decimals that either terminate or repeat, whereas the set of irrational numbers are those decimals that are infinite but they do not repeat.

This is the end of the review.  Now, we wish to prove that (sr2) is one of the irrationals.  To do we take several steps.  The first step is that we use an indirect proof.  We assume there are integers A and B such that (sr2) = A/B.  We then plan to show that this assumption leads to a contraction.  In mathematics, when an assumption leads to a contradiction, we know it is not true.  Therefore, if we get a contradiction, then our starting assumption that (sr2) = A/B will be false–that is, the square root of 2 cannot be expressed as a fraction or rational number; hence, it must be an irrational number, an infinite, non-repeating decimal.  This method of proof—where we assume the opposite of what we want to prove and show that this assumption results in a contradiction—this method, is called indirect proof.

So the first step is we assume that (sr2) = A/B

Second step:  Multiply both sides of the equation by the same number, B.  We can always multiply both sides of an equation by the same number, and we do so here to get rid of the fraction and use just whole numbers instead.  If we multiply both sides by B, we obtain:

B·(sr2) = B·(A/B)

Since the B’s on the Right-Hand Side (henceforth RHS) will cancel out, this gives us:

B(sr2) = A.

Third step:  Whenever we have an equation, we can square both sides.  For example, if x = y, then x2 = y2.  In general (LHS) (LHS) = (RHS) (RHS).  Let us do that here to get rid of the square root and have only whole numbers:

B(sr2)B(sr2)= A A.

 

Fourth step:  We can rearrange the terms on the left to be (sr2)(sr2)BB.  Since, by definition, the square root of two meets the condition (sr2)(sr2) = 2 and since BB = B2 (and since AA = A2), we can rewrite this equation to be:

2B2 = A2

Fifth step:  Let us use the Fundamental Theorem of Arithmetic, (review facts #3  and #4) and factor this number.  We can factor it by looking at either the right-hand side or the left-hand side.  The Fundamental Theorem says we must get the same result, but in fact, we will not.  Notice that the right-hand side must contain an even number of primes.  On the left side, we factor the number as 2 times B2, and the B2  contains an even number of prime factors.  But there is also the number 2 on the LHS, so the entire left-hand side has one more prime, namely two, for a total of however many primes there are inside of B2 (which is an even number) plus one more prime (for the number 2) giving a total of an odd number of primes.  Thus, we have shown the RHS has an even number of primes, and the LHS contains an odd number of primes.  This fact contradicts the Fundamental Theorem that every number can be factored into primes only one way.  Conclusion:  No such numbers A and B exist, that is, (sr2) must be irrational.

Note, a similar proof will show that (sr5) or any other square root that is not a whole number will be irrational.