The word \u201cirrational\u201d is an element of second level, or semi-technical, vocabulary we all pick up in school.<\/p>\n\n\n\n
First level vocabulary – everyday words\nThird level vocabulary – technical terms specific to the discipline\nSecond level vocabulary – every day words that take on specific meaning in the context of a discipline.<\/p>\n\n\n\n
Second level vocabulary is usually the toughest to acquire because words you already know now take on a different meaning. That new meaning is difficult to pick up until you are familiar enough with its context of use. One example would be how the meaning of the word \u201cdribble\u201d changes once you know we\u2019re talking about basketball, but if you\u2019ve never seen a game played, this meaning will be hard to come by.<\/p>\n\n\n\n
At some point, some math teacher asked us to memorize a definition for \u201cirrational\u201d, but likely didn\u2019t really go into why or why it mattered. In everyday life, irrational means absurd or crazy. And most people tend to bring a bit of this connotation to the math world. And why not? Irrational numbers, and most people remember at least part of this into adulthood, are those that are a bit of a pain in the ass. If you want to write them out, it\u2019s not fun.<\/p>\n\n\n\n
But we need to go back a step in the etymology if we want to really understand \u201cirrational\u201d as it pertains to numbers. One thing that irrational means in both contexts is \u201cnot rational\u201d. It\u2019s just that the word \u201crational\u201d has a technical meaning that we obfuscate.<\/p>\n\n\n\n
The root of \u201crational\u201d is \u201cratio\u201d, what we today see as interchangeable with the notion of fractions. The ratio 1:2 is the same to us as the fraction \\(\\frac{1}{2}\\).1<\/sup><\/a> Then, an irrational number is one that is not a fraction. Let\u2019s say your number is \\(x\\). It is irrational if there are no two whole numbers \\(a\\) and \\(b\\) so that\n\\[x=\\frac{a}{b}.\\]\nAn irrational number is thus one that is not equal to any fraction. i.e. It is impossible to write it as a fraction.<\/p>\n\n\n\n This is already more thought than most school-based work with irrationals gets, except for a mnemonic fact to remember: irrational numbers are those whose decimal expansions go on forever without<\/em> repeating.<\/p>\n\n\n\n If we stop for a minute, we know enough to ask a bunch of questions.<\/p>\n\n\n\n Now some of these questions have subtle, contingent answers, or maybe none at all, but surprisingly for those on the outside, math as a discipline has a very simple answer to question 1. Questions like these are the bread and butter of mathematical methods and epistemology (what math knowledge can be). Here\u2019s the basic gist.<\/p>\n\n\n\n This game is called \u201cproof by contradiction\u201d, and it lets us decide that the very idea of a certain number being a fraction is absurd, sidestepping the need to check every possible fraction in existence. Let\u2019s say that we want to prove that \u221a2 is irrational.<\/p>\n\n\n\n The game starts by pretending the opposite is true. We say, \u201cokay smarty pants, let\u2019s just say that there is a fraction of whole numbers \\(a\\) and \\(b\\) so that \\(\\frac {a}{b}=\\sqrt 2\\).\u201d<\/p>\n\n\n\n Rather than going into the details right away, the game plays out by forming a chain of logically airtight deductions from this initial assumption. We hope to end up claiming something to be true that cannot possibly be true, like \\(0=1\\). If \\(\\frac {a}{b}=\\sqrt 2\\) then this other thing \u201cA\u201d is true, and if \u201cA\u201d is true, then certainly \u201cB\u201d holds, \u2026 , and so this other impossible thing has to be true.<\/p>\n\n\n\n Where does this get us? Well, since the impossible thing, the conclusion at the end of this chain, cannot possibly be true, then the only possibility is that our original assumption, in this case that \\(\\frac {a}{b}=\\sqrt 2\\) is true. So no matter which \\(a\\) and \\(b\\) we choose, that equation never holds. That\u2019s proof by contradiction. Sometimes its called reductio ad absurdum<\/em>.<\/p>\n\n\n\n And the details? Let\u2019s see.<\/p>\n\n\n\n Now, to be careful, we have to take a step back. Often, in these proofs, we have to do some work to make sure something stupid doesn\u2019t get in our way. In this case, the idea of \u201cequivalent\u201d fractions could get in our way if we\u2019re not careful. In most situations, you and I don\u2019t consider \\(\\frac{1}{2}\\) and \\(\\frac{2}{4}\\) to be different; you can cancel a two in the top and bottom of the latter to see their equivalence. Since we\u2019re going to do some manipulation of \\(a\\) and \\(b\\), we should make sure the letters aren\u2019t hiding something important, some critical detail for our proof.<\/p>\n\n\n\n There\u2019s a lot of ways to be careful. One could be assuming that the fraction is in lowest terms, that we\u2019ve done all the cancelling we can. This is a fine assumption to make as long as we believe that fractions work the way we were taught. If the fraction wasn\u2019t in lowest terms, we\u2019d just cancel a bit until we got one that was.<\/p>\n\n\n\n But maybe you start, now that we\u2019re questioning irrational numbers, to question more. So let\u2019s take a simpler assumption. Let\u2019s assume that \\(a\\) and \\(b\\) aren\u2019t both even. If they are, we can cancel a 2 out of both and get smaller numbers that give an equivalent fraction. If these two smaller numbers are not both even, we\u2019ll use those instead of \\(a\\) and \\(b\\). If they are both even, we cancel again. We can\u2019t cancel forever because each of these numbers, being finite, can only have so many twos in them.<\/p>\n\n\n\n Does that assumption seem watertight enough? That we can assume \\(a\\) and \\(b\\) are not both even?<\/p>\n\n\n\n Good. I knew it would.<\/p>\n\n\n\n Okay so if we\u2019re assuming not both even, what are the other possibilities?<\/p>\n\n\n\n The way we finish the proof is to look at each of these three cases independently. In each case, we will show that our original assumption (that \\(\\frac {a}{b}=\\sqrt 2\\)) and the assumption defining the case (the parity of \\(a\\) and \\(b\\)) lead to a contradiction (an absurdly untrue thing).<\/p>\n\n\n\n If \\(\\frac {a}{b}=\\sqrt 2\\), then it is also true that \\((\\frac {a}{b})(\\frac {a}{b})=\\sqrt 2 \\times \\sqrt 2\\). This is the same as saying<\/p>\n\n\n\n \\[\\frac {a^2}{b^2} = 2.\\]<\/p>\n\n\n\n And since no one likes fractions, we can multiply both sides of this equation by \\(b^2\\), and get something really useful.<\/p>\n\n\n\n \\[a^2=2b^2.\\]<\/p>\n\n\n\n So we will actually use this latter equation instead of the one we first assumed. We\u2019ll do this in all three cases, so that\u2019s why I\u2019m mentioning it here, ahead of time.<\/p>\n\n\n\n The key in this case, along with the other two, is to look at the left and right sides of the equations and ask about their parity. But this is the hard case, so we need to do some extra work first.<\/p>\n\n\n\n \\(a^2\\), with \\(a\\) even, is very even. It is at least divisible by 4.2<\/sup><\/a> The machinery of algebra helps us make something of this realization, using symbols to make the fact stand out.<\/p>\n\n\n\n Since \\(a\\) is even, we could write \\(a=2\\times k\\) for some other whole number \\(k\\). Then \\(a^2 = (2k)^2=4k^2\\). This says\u2014in symbols\u2014that \\(a^2\\) is at least divisible by 4.<\/p>\n\n\n\n But what about the left side of \\(a^2=2b^2\\)? With our substitution, \\(2k\\) for \\(a\\), we can write<\/p>\n\n\n\n \\[4k^2=2b^2.\\]<\/p>\n\n\n\n And we can go ahead and scratch the cancelling itch, and divide both sides by 2. We get<\/p>\n\n\n\n \\[2k^2=b^2.\\]<\/p>\n\n\n\n Now we are ready to find a contradiction. The left hand side of this equation, \\(2k^2\\) is even. The 2 is right there. But the right side, \\(b^2\\) is odd since \\(b\\) is.3<\/sup><\/a> There is no whole number that is both even and odd.<\/p>\n\n\n\n Case 1 is impossible.<\/p>\n\n\n\n Here\u2019s the thing. Since it\u2019s our first proof, I did the hard one for you. But you have to get used to doing some work. So I\u2019m not going to work cases 2 and 3 to their contradictions. You have to. I will say that they yield their contradictions without as much massaging. Just look at the left and right sides of the equation \\(a^2=2b^2\\) in each case.<\/p>\n\n\n\n You can\u2019t learn math without doing it. It\u2019s not about memorizing details but about training a sort of internal VR. By going through the motions, these symbols stop being substations and become their own material reality, something you can directly visualize in your brain.<\/p>\n\n\n\n So go for it!<\/p>\n\n\n\n When you\u2019ve shown that neither of these two cases is possible either, we have together finally arrived at the conclusion that there is no fraction so that \\(\\frac {a}{b}=\\sqrt 2\\). \\(\\sqrt 2\\) is irrational<\/em>!<\/p>\n\n\n\n So if this has been successful, we have now answered question 1 above. But notice, we haven\u2019t had anything to do with the others. We didn\u2019t even mention decimal representations, and we just took the existence of \\(\\sqrt 2\\) for granted. Who is to say we haven\u2019t just twiddled our thumbs in an ivory tower, counting angels on the head of a pin.<\/p>\n\n\n\n We will look into these questions too, but not today.<\/p>\n\n\n\n So we have one reason to believe that \\(\\sqrt 2\\) is irrational. For many people that would be enough. But not for math. We have lots more questions, even about the issue we\u2019ve settled.<\/p>\n\n\n\n And why do we look the gift horse in the mouth so? One reason could be fault tolerance of the resulting system of knowledge. Knowing that there are many analytic paths to this same fact is reassuring in a sense that there\u2019s not some hidden mistake in our work. But seriously, it is most often that last questions, the one of aesthetics that drives mathematicians forward from the point of bare sufficiency.<\/p>\n\n\n\n In the case of \\(\\sqrt 2\\) check out this short article<\/a>. It\u2019s from a course blog, an effort of math writing not too dissimilar from what we\u2019re after. So you also may want to read it with that meta context in mind. The post shows many proofs and their origins. I for instance am more convinced by the much simpler proof that makes use of unique prime factorization, the theorem that each number breaks up into a unique set of multiples of prime divisors.<\/p>\n\n\n\n I\u2019ll wait here, go ahead and read that. Better than that, make sure you actually work through at least one of the new proofs.<\/p>\n\n\n\n Done? Good.<\/p>\n\n\n\n Did you find a proof you liked better? On what grounds did it strike the mark? If you feel it was easier to understand, try to figure out the features and adjectives like short, simple, etc. that help get to the bottom of it.<\/p>\n\n\n\n In my own case I said I like the prime factorization based proof. Why? And if I like that one, why not show it to you first?<\/p>\n\n\n\n Well I like it because we all tend to take prime factorization for granted, and that\u2019s a good thing. It\u2019s a wonderful thing to know about numbers. It does make a lot of the other things we\u2019d want to know about numbers work and it certainly generalizes from the case of \\(\\sqrt 2\\) to any other non-square number\u2019s irrationality in a trivial way.<\/p>\n\n\n\n So why not use this one? Well, it\u2019s a primitivism of sorts, that and the fact that this is the proof everyone is shown first, likely because it was the first one we have historically. What I mean by primitive is that our original proof doesn\u2019t take a lot for granted when it comes to what numbers are and how they work. You need to know or figure out a bit about even and odd numbers and their arithmetic, and that\u2019s it. This proof is not built high atop a pile of what could turn out to be cards.<\/p>\n\n\n\n On the other hand, it asks you to follow a rather serpentine argument. You might get lost in it, or get to the end not knowing where you\u2019ve been. There\u2019s nothing worse than a proof that seems logically valid but which makes you none the wiser because you don\u2019t see<\/em> why it\u2019s right.<\/p>\n\n\n\n Now you can go back to the article\u2019s proofs and make some more explicit decisions about how each of them succeeds and fails.<\/p>\n\n\n\n The word \u201cirrational\u201d is an element of second level, or semi-technical, vocabulary we all pick up in school. First level vocabulary – everyday words Third level vocabulary – technical terms specific to the discipline Second level vocabulary – every day words that take on specific meaning in the context of a discipline. Second level vocabulary … Continue reading “Irrationality and $\\sqrt 2$”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":119,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-124","page","type-page","status-publish","hentry"],"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/PbpbST-20","_links":{"self":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/124","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/comments?post=124"}],"version-history":[{"count":1,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/124\/revisions"}],"predecessor-version":[{"id":125,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/124\/revisions\/125"}],"up":[{"embeddable":true,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/119"}],"wp:attachment":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/media?parent=124"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}So Many Questions <\/h2>\n\n\n\n
\\(\\sqrt 2\\) Is Irrational – Our First Proof <\/h2>\n\n\n\n
Being Careful – Equivalent Fractions <\/h3>\n\n\n\n
Case 1 – \\(a\\) is even and \\(b\\) is odd<\/h3>\n\n\n\n
Case 2 and 3 <\/h3>\n\n\n\n
What Else Is Irrational? <\/h2>\n\n\n\n
Other roads to glory <\/h2>\n\n\n\n
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