Proof that 2 is irrational

Transcription

1 Proof that is irrational When we convert fractions into decimals, we notice two things. Either the decimals () terminate or they () repeat in a recognizale pattern. For example, 3 = 0.75 (decimal expansion terminates at 5) and = where 857 is the repetend (digits that repeat, ad infinitum). 7 It is also possile to construct a fraction given its decimal expansion (whether terminating or repeating). Consider this intriguing thought: Are there decimal expansions that cannot e converted into an equivalent fraction? We can rephrase this question: Are there numers that are not rational? Let s consider this decimal expansion: This expansion has a rule. What is it? There is one more 0 after each successive. With this rule, anyone could continue writing the decimal expansion. Although this expansion has a rule, we cannot find any recurring groups in it. In other words, we cannot find any repeating patterns. This numer, therefore, cannot e the expansion of a fraction. Its partial sums, , 000, 000, 000 0,000,000,000 do not converge to any rational numer. Although the partial sums do not converge to any rational numer, we can show that this numer does converge to some sort of gap in the set of rational numers. You might e saying to yourself, Gap? A gap in the set of rational numers? I thought these numers were everywhere dense. How can there e a gap (or gaps) in them? Consider our numer. First, we know that this numer is less than 0. (or less than ). As we consider the other partial sums, we know that they will all lie etween 0. and 0. on the numer line. Using the same rationale, we know that this numer (in the thousandths position) will lie etween 0.0 and 0.0. We are narrowing down our culprit. Consider the next appearance of in the expansion. From this we know that our numer lies etween Copyright 007 Note: This essay is extracted from a Lesson from the forthcoming textook Mathematics: Building on Foundations. and Next, we know that our numer lies etween and These intervals get small very quickly. It looks like this: The lengths of our intervals can e taulated as follows: Interval Length st 0. nd 0.00 st interval nd 3rd 0. th 0.

2 Interval Length 3 rd th ? Proof that is irrational Copyright 007 These lengths converge to zero. Our larger partial sums will crowd into these ever-decreasing intervals that ecome smaller and smaller at attle speed. Our numer is located at the converging point. We have constructed a numer that does not converge to a rational numer, yet this numer converges to a point on the numer line (which is everywhere dense with rational numers!). Stop and consider that. This point, considered at infinity, is a certain determinale distance from zero. Yet, we cannot measure this distance in whole units or in fractions of whole units. We have no way of fitting this numer in, the set of rational numers. Because of this conundrum, mathematicians have constructed a new set of numers to handle this situation. The set is appropriately called the set of irrational numers (denoted y the letter I). Irrational numers cannot e written as the ratio of two integers. Their decimal expansions are infinite, without pattern, ut given y some rule so that we can say that the expansion represents a point at a definite distance from 0 on the numer line. Irrational numers are non-rational or non-ratio numers. The ancient Greeks, namely the students and followers of Pythagoras (ca. 58-ca. 500 BC), first discovered irrational numers (thanks to some discoveries made y the ancient Egyptians). These numers made their encore appearance as these mathematicians worked with their eloved right angle triangle. Recall the Pythagorean Theorem: the sum of the squares of the two legs of a right angle triangle equal the square of the hypotenuse. If the legs of a right angle triangle measure a and respectively and the hypotenuse measures c, then c = a +. The famous 3--5 right-angled triangle fits this formula like a glove: 5 = 3 + or 5 = Enter Mr. Spoiler. He asks Mr. Pythagoras, What is the length of the hypotenuse when the legs are oth equal to? Mr. Pythagoras responds, Oviously, we can construct such a right angle triangle. I take my compass and ruler and mark off and measure. I draw the two legs of measure. I connect the third edge of the triangle with a straight line. Yes, the hypotenuse is a measurale distance. Mr. Spoiler, Well, Mr. Pythagoras, what is the length of the hypotenuse? Mr. Pythagoras takes fingers to the sand and starts calculating: + = c + = c = c Mr. Pythagoras suddenly lets out an anguished cry. He gras his chest and falls to one knee. Mr. Spoiler cries out, 9! 9! An hour later a chariot rolls up. What s the matter? asks Dr. Bones. Mr. Spoiler answers, Mr. Pythagoras seems to e in a state of distress. Can you help him? Dr. Bones replies, Let me take a look. Dr. Bones gives Mr. Pythagoras a thorough physical exam. After comparing what he oserves with his scroll of medical terms and definitions, he turns to Mr. Spoiler and says, It s serious. We must operate immediately. What are you going to do? queries Mr. Spoiler. Extract the square root of, answers Dr. Bones. So, Dr. Bones places the limp ody of the still conscious Mr. Pythagoras on his chariot and rides off. As they disappear into the sunset, Mr. Spoiler hears a fading, yet heart-rending cry from the lips of Mr. Pythagoras, But you can t! You can t! You can t extract the square root of! Pythagoras was committed to a worldview that saw the natural numers as generating the particular things of the universe. Numers rule the universe or numer is all was the cry of the rotherhood of

3 Proof that is irrational the Pythagoreans. Pythagoras and his disciples erred in thinking that numers generate the particular things of the universe. Numers are a tool that can only report on the particular things of the universe. For the Pythagoreans, natural numers and fractions (the ratio of counting numers) formed the generating foundation of all existence. Given this worldview, it is no surprise that gave them hearturn. No natural numer (or ratio of natural numers) multiplied y itself exactly equals.. 5 is close ecause (.) = an irrational numer lies hidden in a is close too ecause (.5) =.5. So, kind of cloud of infinity. we know that is etween. and.5. Does Michael Stifel (87-567) exactly equal a rational numer whose decimal expansion either terminates or repeats? No. This is how we prove that is not a rational numer. This proof, first devised y the school of Pythagoras, uses a method of proof called, in Latin, reductio ad asurdum (literally means ringing ack from asurdity ). This method of reasoning is also called indirect proof. To descrie this method, we have to engage the notion of the negation of a statement. For a given statement S that is provale, its negation is a statement meaning the falsity of S. For example, if S is the statement x is a negative integer, then ~S (the negation of S), is the statement x is a positive integer or zero. Recall that one of the transcendental laws of logic, called the law of the excluded middle, asserts that for any given statement that is provale, either the statement itself or its negation is true. Another transcendental law, the law of contradiction, asserts that a provale statement and its negation cannot e true simultaneously. Therefore, according to these two laws, for every statement S, exactly one of the statements, S or ~S, is true. In particular, if ~S is not true, then S must e true. These preliminary laws serve as the asis for the indirect proof or proof y reductio ad asurdum. With this method you start y assuming the opposite of what you want to prove (i.e., the negation of the conclusion). In this case, we start y assuming that is a rational numer. Given this assumption, we proceed to reason, on the asis of the hypothesis, to a point of contradiction. Once we get a contradiction, this shows that the hypothesis and the negation of the conclusion are contradictory. We then recognize that our assumption is false (i.e., the negation of the conclusion) and therefore, y the law of contraposition, the conclusion must e true. Hence, we have proved what we have intended to prove. The fictional detective Sherlock Holmes employed reasoning similar to the reductio ad asurdum method in solving crimes. He stated, When you have eliminated the impossile, whatever remains, however improale, must e the truth. In the context of Holmesian wisdom, reductio ad asurdum means that we eliminate an assumption y reasoning from this assumption to a contradiction. Once we have done this, the assumption is eliminated and we must accept the negation of the assumption as true. In the annals of mathematical proof, there are three ways to prove that is irrational. The method popular in modern texts follows an algeraic line of reasoning. The original proof, thought to e used y the followers of Pythagoras, was ased upon geometric arguments. First, we shall assume that is a rational numer. We must now reason to a contradiction. Second, if is a rational numer, then, y definition, can e written as a ratio of two distinct integers. We get: Equation. a = where a, and 0. Copyright 007 3

4 Proof that is irrational We will also assert that this ratio, a, is a fraction written in lowest terms (it has no common factors). For ex- ample, the fraction 6 8 is not in lowest terms ecause the numerator and denominator have common factors other than, namely. 3 is in lowest terms ecause it has no common factors other than ; the greatest common factor (GCF) of 3 and is. Using numer theory terminology, 3 and are relatively prime. We require that a and are relatively prime; i.e., the GCF of a and is. Third, from Equation, we square oth sides of the equation. We get: a Equation. = Fourth, from Equation, we multiply each side of the equation y. We get: Equation 3. = a What conclusions can we draw from Equation 3? Since is an integer, then must e an integer and must e an even integer. Why? Even integers are multiples of. Since = a, then we can conclude that a must e an even integer. Fifth, since we have estalished that a is an even integer, then what can we say aout a? a must e also an even integer. Why? If we take an odd integer and square it, will we ever get an even integer? Never. Hence, a must e an even integer. If a is an even numer, then must e an odd numer since we agreed that a and had no common factors (other than ). To reiterate, must e an odd integer. Sixth, since a is an even numer it can e written as a multiple of. We shall let a = r where r. Seventh, we sustitute r for a in Equation 3. We get: Equation. ( r) = or = r Eighth, from Equation, we divide oth sides y. We get: Equation 5. = r What conclusions can we draw from Equation 5? By our previous reasoning, that must e an even integer. Ninth, since is an even numer, then must e an even integer. The dénouement of the drama is now upon us. Stating that must e an even integer contradicts what we said earlier that must e an odd integer. Therefore, when we assumed that is a rational numer, we reasoned to a contradiction. This tells us that is a not a rational numer. QED! This logic ehind this proof is classic. The proof that is irrational is considered to e a memer of a special class of mathematical proofs; proofs that are eautiful ecause they are elegant. Sometimes a QED can e lethal. According to legend, Hippasus, a follower of Pythagoras, was thrown overoard and drowned at sea for unveiling a numer that is not equal to a ratio of two numers. Copyright 007

5 Proof that is irrational The Greeks called an incommensurale length ecause it could not e represented as a ratio of two integers. In our construction of a right angle triangle with legs of measure, the hypotenuse definitely has a measurale length. We can park this numer horse of a different color on a point on the numer line just like we could pinpoint on a point on the numer line. 0 on the numer line Copyright 007 5