,,meno ,,ii ,a ,,continuation ,,of ,,socrates' ,,dialogue ,,with ,,meno ,,in ,,which ,,the ,,boy ,,proves ,,root #b ,,is ,,irrational ,by ,socrates ,a ,millennium ,fulcrum ,edition ,copyright #aiie ,socrates3 ,well1 here we are at the appointed time1 ,meno4 ,meno3 ,yes1 and it looks like a fine day for it1 too4 ,socrates3 ,and ,i see our serving boy is also here4 ,boy3 ,yes1 ,i am1 and ready to do your bidding4 ,socrates3 ,wonderful4 ,now1 ,meno1 ,i want you to be on your guard1 as you were the other day1 to insure that ,i teach nothing to the boy1 but rather pull out of his mind the premises which are already there4 ,meno3 ,i shall do my best1 ,socrates4 ,socrates3 ,i can ask more of no man1 ,meno1 and ,i am certain that you will do well1 and ,i hope ,i will give you no call to halt me in my saying if ,i should say too much1 in which you would feel ,i was actually teaching the boy the answer to this riddle4 ,meno3 ,no1 ,socrates1 ,i don't think ,i will have to call you on anything you might say today1 for the most wondrously learned men of the group of ,pythagoras have spent many hours1 weeks1 and even months and years toiling in their manner to arrive at the mystic solutions to the puzzles formed by the simple squares with which we worked the other day4 ,therefore1 ,i am certain to regain my virtue1 which ,i lost the other day1 when ,i was so steadfastly proven by you to be in error in my statement that the root of a square with an area of two square feet was beyond this boy1 who is a fine boy1 whom we must make to understand that he should do his best here1 and not feel that he has done any wrongness by causing me to lose my virtue to you the other day4 ,socrates3 ,meno1 my friend1 it is my opinion1 and ,i hope it will soon be yours1 that your virtue was increased the other day1 rather than decreased4 ,meno3 ,i fail to see how1 when ,i was humiliated by seeing this young boy1 of modest education1 arrive in minutes at the highest mystic levels of the magic of the ,pythagoreans4 ,most of all when ,i wagered as many dinners as you could eat at my house that this could not be the case4 ,socrates3 ,first1 friend ,meno1 let me assure you that ,i will promise never to eat you out of house and home1 not that ,i could if ,i tried1 for my tastes are simple and your wallet is large4 ,nevertheless1 ,meno1 my friend1 ,i would hasten to add that ,i will promise1 if you like1 not to ever come to your table uninvited4 ,as a second reason you and your virtue should feel better after the events of the other day1 because you were in error before1 but are less in error now4 ,and the path to virtue1 at least one aspect of the path to virtue1 is in finding and correcting error4 ,meno3 ,socrates1 you know you are always welcome at my table1 except when ,i am suffering from my ulcer1 which you aggravate greatly1 or at times when ,i am entertaining the highest nobles of the land1 and you would appear out of place in your clothing4 7,socrates was known for his simple attire1 and for wearing his garments over and over till they wore out4 ,however1 the only surviving example of his writing is a laundry list1 so we know he kept his clothes clean and somewhat presentable1 though simple7 ,socrates3 ,i would hope you would have me over because ,i was a good influence on your development1 than for any other reason4 ,i notice you did not respond to my claim to have increased your virtue1 through the exorcism of your error4 ,meno3 ,well ,socrates1 you know that it is not always the easiest thing to give up one's ways1 even though one has found them to be in error4 ,therefore1 please forgive me if ,i am not sounding as grateful as you would like for your lessons4 ,socrates3 ,the easier one finds it to give up the ways of error1 the easier it is to replace the error with that which we hope is not in error4 ,is this not the way to virtue8 ,meno3 ,yes1 ,socrates1 and you know the path is hard1 and that we often stumble and fall4 ,socrates3 ,yes1 but is it not true that we stumble and fall over the obstacles which we make for ourselves to trip over8 ,meno3 ,certainly that is most true1 ,socrates1 in some cases4 ,socrates3 ,well1 then1 let us proceed1 for ,i see the hour is upon us when ,i do my best thinking1 and that hour shall be passed soon1 and hopefully with it shall pass a bit of your ignorance4 ,meno3 ,well said1 ,socrates4 ,i am with you4 ,socrates3 ,and shall have we a wager on the events of today8 ,meno3 ,certainly1 ,socrates4 ,socrates3 ,and what shall you wager against this boy proving that the length of the root of a square with an area of two square feet1 cannot be made by the ratio of two whole numbers8 ,meno3 ,you may have anything it is in my power to give1 unless it cause harm to myself or to another to give it4 ,socrates3 ,well said1 my friend ,meno1 and ,i shall leave it at that4 ,and what shall ,i offer you as a return wager8 ,meno3 ,well1 the easiest thing which comes to mind is to wager all those dinners you won from me the other day4 ,socrates3 ,very well1 so be it4 ,meno3 ,now ,socrates1 since you are my friend1 ,i must give you this friendly warning3 you know that the ,pythagoreans jealously guard their secrets with secret meetings1 protected by secret handshakes1 secret signs1 passwords1 and all that1 do you not8 ,socrates3 ,i have heard as much1 friend ,meno4 ,meno3 ,then be sure that they will seek revenge upon you for demystifying the ideas and concepts which they worked so long and hard and secretly to create and protect2 for they are a jealous lot in the extreme1 hiding in mountain caves1 which are hardly fit to be called monasteries by even the most hardened monk4 ,socrates3 ,i take your meaning1 friend ,meno1 and thank you for your consideration1 but ,i think that if ,i lose1 that they will not bother me1 and if ,i win1 it will appear so simple to everyone1 that if would be sheerest folly for anyone to make even the smallest gesture to protect its fallen mystic secrecy4 ,besides1 ,i have a citizen's responsibility to ,athens and to all ,athenians to do my best to protect them and enlighten them4 ,meno3 ,very well1 ,socrates4 ,please do not ever say that ,i did not try to warn you1 especially after they have nailed you to a cross in a public place1 where anyone and everyone could hear you say that the fault of this lay in my name4 ,socrates3 ,do not worry1 friend ,meno1 for if ,i were not to show this simple feat of logic to you1 ,i should just walk down the street and find someone else1 though not someone whose company and conversation ,i should enjoy as much as yours4 ,meno3 ,thank you1 friend ,socrates4 ,socrates3 ,now1 boy1 do you remember me1 and the squares with which we worked and played the other day8 ,boy3 ,yes1 sir1 ,socrates4 ,socrates3 ,please1 ,meno1 instruct the boy to merely call me by my name1 as does everyone else4 ,calling me sir merely puts me off my mental stride1 and1 besides1 it will create a greater distance between me and the boy4 ,meno3 ,you heard what ,socrates1 said1 boy4 ,can you do it8 ,boy3 ,yes1 sir4 7,turning to ,socrates7 ,you know ,i like you very much1 and that ,i call you sir not only out of relation of our positions in society1 but also because of my true respect and admiration1 especially after the events of the other day4 ,socrates3 ,yes1 boy4 ,and ,i will try to live up to your expectations4 7,turning to ,meno7 ,would you allow some reward for the boy1 as well as that which is for myself1 if he should prove to your satisfaction that the square root of two is irrational8 ,meno3 ,certainly1 ,socrates4 ,socrates3 7taking the boy aside7 ,what would you like the most in the whole world1 boy8 ,boy3 ,you mean anything8 ,socrates3 ,well1 ,i can't guarantee to get it for you1 but at least ,i can ask it1 and it shouldn't hurt to ask2 and besides1 as you should know1 it is very hard to expect someone to give you what you want1 if you never let them know you want it4 ,boy3 ,well1 ,socrates444 you know what ,i would want4 ,socrates3 ,do ,i8 ,boy3 ,better than ,i knew the square root of two the other day4 ,socrates3 ,you want to be a free man1 then1 and a citizen4 ,boy3 7looking down7 ,yes4 ,socrates3 ,don't look down1 then1 for that is an admirable desire for one to have1 and speaks highly of him who has it4 ,i will speak to ,meno1 while you hold your tongue4 ,boy3 ,yes1 ,socrates4 7bows to kiss his hand1 ,socrates turns7 ,socrates3 ,friend ,meno1 how hard do you think it will be for this boy to prove the irrationality of the square root of two8 ,meno3 ,you know that ,i think it is impossible1 ,socrates4 ,socrates3 ,well1 how long did it take the ,pythagoreans8 ,meno3 ,i should think it took them years4 ,socrates3 ,and how many of them were there8 ,meno3 ,quite a few1 though not all worked equally1 and some hardly at all1 for they were most interested in triangles of the right and virtuous variety1 and not in squares and their roots4 ,socrates3 ,can you give me an estimate8 ,meno3 ,no1 ,i can't say that ,i can4 ,i am sorry1 ,socrates4 ,socrates3 ,no problem1 would you accept five thinkers as an estimate4 ,meno3 ,i think that should be fair4 ,socrates3 ,and shall we assume they worked for two years1 that is the smallest number which retains the plural1 and our assumption was that they worked for years4 ,meno3 ,two years is indeed acceptable to me1 ,socrates4 ,socrates3 ,very well then1 ,meno1 it would appear that the ,pythagoreans spent #aj total years of thinking time to solve the riddles of the square root of two4 ,meno3 ,i agree4 ,socrates3 ,and would you like to hire the ,pythagoreans to run your household1 ,meno8 ,meno3 ,surely ,i would1 ,socrates1 if they were only for hire1 but1 as you well know1 they are a secret lot1 and hire to no one4 ,socrates3 ,well1 if ,i could get you one1 perhaps one of the best of them1 in fact the leader of the group that solved the square root of two1 would you not hire him1 and at high wages8 ,meno3 ,certainly1 ,socrates4 ,i'd be a fool not to4 ,socrates3 ,and you would put him in charge of your house4 ,meno3 ,and all my lands1 too1 ,socrates4 ,socrates3 ,possibly4 ,then ,i would like to propose1 that if this boy should solve the proof of the square root of two being irrational1 in the next few hours of our discussion1 that he be given wages equal to those due to your most highly placed servant for ten years of service1 as he shall perform ten years service for you in the next few hours1 should he succeed4 ,meno3 ,that sounds quite fair1 ,socrates1 ,i like your logic4 ,socrates3 7the boy tugs his tunic1 to complain that he wants his freedom1 not a mere bucket of gold7 ,hush1 boy1 did you not promise to hold your tongue8 ,boy3 ,yes1 ,socrates1 but4444 ,socrates3 7turning to ,meno7 ,and1 of course1 with the monetary rewards for such a position1 go all the rest of it4 ,meno3 ,of course1 ,socrates4 ,i never thought to cheat you4 ,socrates3 ,i know that1 friend ,meno1 but ,i merely ask for the boy's sake1 who is not used to hearing about high finance and the powers and rank which accompany such things4 ,meno3 ,of course1 ,socrates4 ,shall ,i tell the boy what he shall receive8 ,socrates3 ,you are very kind to do so1 my friend ,meno4 ,meno3 7turns to the boy7 ,you are aware that a servant may not own the amount of gold ,i would have to give you1 should you win the day8 ,boy3 ,yes1 sir4 ,meno3 ,therefore1 ,i would have to give to you the freedom to own the money1 before ,i could give you the money1 would ,i not8 ,boy3 ,yes1 sir4 ,meno3 ,and in giving you freedom1 ,i would be remiss if ,i did not give you a job and a coming out party of equal position with your wealth1 would ,i not8 ,boy3 ,i can't really say1 sir1 though ,i suppose so4 ,meno3 ,you suppose correctly4 ,i will feed you for a week of partying1 and dress you in the finest garments1 while you are introduced to the finest ladies and gentlemen of ,athens1 from whom you are free to select for your interests as friends1 business partners1 social acquaintances1 connections1 and perhaps even a wife1 should you find someone you like for that4 ,do you now understand that there is nothing ,i would leave out that you would have to ask for1 or that if you did have to ask1 ,i would give it immediately1 and ask your forgiveness for my error8 ,boy3 ,it is hard to understand1 but ,i take your word4 ,socrates3 ,now don't let this all go to your head1 boy4 ,this is something you could have figured out for yourself1 if you had applied your mind to it as you did to squares the other day4 ,can you do as well1 today8 ,boy3 ,i should think and hope so1 friend ,socrates1 for ,i see you are indeed my friend1 and ,i should hope ,i am more capable today1 for having learned some the other day4 ,socrates3 ,we shall see1 boy4 ,let us on to the test4 ,now you remember the squares we dealt with the other day4 ,boy3 ,yes1 ,socrates4 ,socrates3 ,and the one particular square on the diagonal we made1 whose area was two1 do you remember that one8 ,boy3 ,yes1 ,socrates4 ,socrates3 ,and you remember that the length of the side of a square1 when multiplied by itself1 yields the area of the square4 ,boy3 ,everyone at school knows that1 ,socrates4 ,socrates3 ,well1 maybe4 ,however1 it is about that side1 which when multiplied time itself yields an area of two1 that ,i would like to speak further today4 ,how is that with you8 ,boy3 ,that is fine1 ,socrates4 ,i remember that line1 and ,i sort of liked it the best1 if you know what ,i mean4 ,socrates3 ,good1 then we should have a great time4 ,do you know how long that line is1 boy8 ,boy3 ,well1 ,i know that you both thought it wise when ,i said it was of a length which when made a square of1 yielded a square with an area of two1 so ,i suppose ,i should answer that way4 ,socrates3 ,and a good answer it is1 too4 ,we are going to make it an even better answer as we proceed4 ,boy3 ,good4 ,socrates3 ,do you remember when you tripped up and fell on your face the other day1 when you thought that the square of area nine was actually a square of area eight8 ,boy3 ,oh yes1 ,socrates6 ,and ,i am sorely ashamed1 because ,i still do not know enough to make sure ,i never make such an error again1 and therefore ,i know my virtue and rightness are lacking4 ,socrates3 ,they are not lacking so much that they cannot be improved1 are they boy8 ,boy3 ,i should hope and pray not4 ,socrates3 ,well today1 you are going to tell us some things about that number1 which when multiplied by itself gives us two4 ,boy3 ,i will tell you everything ,i know1 or think ,i know1 ,socrates1 and hope that ,i am correct or can be corrected4 ,socrates3 ,to ,meno1 surely he is a fine boy1 eh ,meno8 ,meno3 ,yes1 ,i am proud to own him1 but ,i don't see how he can be smart enough to do the work today that would take a ,pythagorean monk ten years of cloistered life to accomplish4 ,socrates3 ,we shall see4 ,boy1 you are doing fine4 ,i think ,i could even make a scholar of you1 though ,i fear you might turn to wine and women with your new found wealth1 if you succeed1 rather than continue to polish the wit which should get you that reward4 ,boy3 ,i don't think ,i would want to spend that much time with women or with wine1 ,socrates4 ,socrates3 ,you will find something1 no doubt4 ,so1 back to the number which when square gives us two4 ,what can we say about such a number8 ,is it odd or even8 ,well it would have to be a whole number to be one of those1 would it not1 and we saw the other day what happens to whole numbers when they are squared8 ,they give us #a1d1i and #af as square areas1 did they not8 ,boy3 ,yes1 ,socrates1 though ,i remember thinking that there should have been a number which would give eight1 ,socrates8 ,socrates3 ,i think we shall find one1 if we keep searching4 ,now1 this number1 do you remember if it had to be larger or smaller than one8 ,boy3 ,larger1 ,socrates4 ,for one squared gives only an area of one1 and we need and area of two1 which is larger4 ,socrates3 ,good4 ,and what of two8 ,boy3 ,two gives a square of four1 which is too large4 ,socrates3 ,fine4 ,so the square root of two is smaller than the side two which is the root of four1 and larger than the side one which yields one8 ,boy3 ,yes1 ,socrates4 ,socrates3 7,turning to ,meno7 ,so now he is as far as most of us get in determining the magnitude of the square root of two8 ,and getting farther is largely a matter of guesswork1 is it not8 ,meno3 ,yes1 ,socrates1 but ,i don't see how he will do it4 ,socrates3 ,neither does he4 ,but ,i do4 ,watch6 7turning to the boy7 ,now ,i am going to tell you something you don't know1 so ,meno will listen very closely to make sure he agrees that ,i can tell you4 ,you know multiplication1 boy8 ,boy3 ,i thought ,i had demonstrated that1 ,socrates8 ,socrates3 ,so you have1 my boy1 has he not ,meno8 ,meno3 ,yes1 ,socrates1 ,i recall he did the other day4 ,socrates3 ,and you know the way to undo multiplication8 ,boy3 ,it is called division1 but ,i do not know it as well as multiplication1 since we have not studied it as long4 ,socrates3 ,well1 ,i will not ask you to do much division1 but rather ,i will ask you only whether certain answers may be called odd or even1 and the like4 ,does that suit you8 ,boy3 ,it suits me well1 ,socrates4 ,socrates3 ,then you know what odd and even are1 boy8 ,boy3 ,yes1 shall ,i tell you8 ,socrates3 ,please do4 ,i would love to hear what they teach4 ,boy3 7the boy recites7 ,a number can only be odd or even if it is a whole number1 that is has no parts but only wholes of what it measures4 ,even numbers are special in that they have only whole twos in them1 with no ones left over1 while odd numbers always have a one left over when all the twos are taken out4 ,socrates3 ,an interesting1 and somewhat effective definition4 ,do you agree1 ,meno4 ,meno3 ,yes1 ,socrates4 ,please continue4 ,socrates3 ,now boy1 what do you get when you divide these odd and even numbers by other odd and even numbers4 ,boy3 ,sometimes you get whole numbers1 especially when you divide an even number by an even number1 but odd numbers sometimes give whole numbers1 both odd and even1 and sometimes they give numbers which are not whole numbers1 but have parts4 ,socrates3 ,very good1 and have your teachers ever called these numbers ratios8 ,boy3 ,sometimes1 ,socrates1 but usually only with simple numbers which make one-half1 one-third1 two-thirds and the like4 ,socrates3 ,yes1 that is usually what people mean by ratios4 ,the learned people call numbers made from the ratios1 rational4 ,does the name rational number suit you to call a number which can be expressed as the ratio of two whole numbers1 whether they be odd or even whole numbers8 ,boy3 ,you want me to call the numbers made from ratios of whole numbers something called rational8 ,a ratio makes a rational number8 ,socrates3 ,yes boy1 can you do that8 ,boy3 ,certainly1 ,socrates4 ,socrates3 ,do you agree with the way ,i told him this1 ,meno8 ,does it violate our agreement8 ,meno3 ,you added -nal to the word ratio1 just as we add -nal to the ,french word jour to create the word journal which means something that contains words of the jour or of today4 ,so we now have a word which means a number made from a ratio4 ,this is more than acceptable to me1 ,socrates4 ,a sort of lesson in linguistics1 perhaps1 but certainly not in mathematics4 ,no1 ,i do not see that you have told him how to solve anything about the square root of two1 but thank you for asking4 ,i give you your journalistic license to do so4 ,socrates3 ,good4 ,now boy1 ,i need your attention4 ,please get up and stretch1 if it will help you stay and think for awhile4 ,boy3 7stretches only a little7 ,i am fine1 ,socrates4 ,socrates3 ,now think carefully1 boy1 what kind of ratios can we make from even numbers and odd numbers8 ,boy3 ,we could make even numbers divided by odd numbers1 and odd numbers divided by even numbers4 ,socrates3 ,yes1 we could4 ,could we make any other kind8 ,boy3 ,well444 we could make even numbers divided by even numbers1 or odd numbers divided by odd4 ,socrates3 ,very good4 ,any other kind8 ,boy3 ,i'm not sure1 ,i can't think of any1 but ,i might have to think a while to be sure4 ,socrates3 7to ,meno7 ,are you still satisfied4 ,meno3 ,yes1 ,socrates4 ,he knows even and odd numbers1 and ratios2 as do all the school children his age4 ,socrates3 ,very well1 boy4 ,you have named four kinds of ratios3 ,even over odd1 odd over even1 even over even1 odd over odd1 and all the ratios make numbers we call rational numbers4 ,boy3 ,that's what it looks like1 ,socrates4 ,socrates3 ,meno1 have you anything to contribute here8 ,meno3 ,no1 ,socrates1 ,i am fine4 ,socrates3 ,very well4 ,now1 boy1 we are off in search of more about the square root of two4 ,we have divided the rational numbers into four groups1 odd_/even1 even_/odd1 even_/even1 odd_/odd8 ,boy3 ,yes4 ,socrates3 ,and if we find another group we can include them4 ,now1 we want to find which one of these groups1 if any1 contains the number you found the other day1 the one which squared is two4 ,would that be fun to try8 ,boy3 ,yes1 ,socrates1 and also educational4 ,socrates3 ,i think we can narrow these four groups down to three1 and thus make the search easier4 ,would you like that8 ,boy3 ,certainly1 ,socrates4 ,socrates3 ,let's take even over even ratios4 ,what are they8 ,boy3 ,we know that both parts of the ratio have two in them4 ,socrates3 ,excellent4 ,see1 ,meno1 how well he has learned his lessons in school4 ,his teacher must be proud1 for ,i have taught him nothing of this1 have ,i8 ,meno3 ,no1 ,i have not seen you teach it to him1 therefore he must have been exposed to it elsewhere4 ,socrates3 7back to the boy7 ,and what have you learned about ratios of even numbers1 boy8 ,boy3 ,that both parts can be divided by two1 to get the twos out1 over and over1 until one part becomes odd4 ,socrates3 ,very good4 ,do all school children know that1 ,meno8 ,meno3 ,all the ones who stay awake in class4 7he stretches7 ,socrates3 ,so1 boy1 we can change the parts of the ratios1 without changing the real meaning of the ratio itself8 ,boy3 ,yes1 ,socrates4 ,i will demonstrate1 as we do in class4 ,suppose ,i use #af and #h1 as we did the other day4 ,if ,i make a ratio of #af divided by #h1 ,i can divide both the #af and the #h by two and get #h divided by #d4 ,we can see that #h divided by #d is the same as #af divided by #h1 each one is twice the other1 as it should be4 ,we can then divide by two again and get #d over #b1 and again to get #b over #a4 ,we can't do it again1 so we say that this fraction has been reduced as far as it will go1 and everything that is true of the other ways of expressing it is true of this4 ,socrates3 ,your demonstration is effective4 ,can you divide by other numbers than two8 ,boy3 ,yes1 ,socrates4 ,we can divide by any number which goes as wholes into the parts which make up the ratio4 ,we could have started by dividing by #h before1 but ,i divided by three times1 each time by two1 to show you the process1 though now ,i feel ashamed because ,i realize you are both masters of this1 and that ,i spoke to you in too simple a manner4 ,socrates3 ,better to speak too simply1 than in a manner in which part or all of your audience gets lost1 like the ,sophists4 ,boy3 ,i agree1 but please stop me if ,i get too simple4 ,socrates3 ,i am sure we can survive a simple explanation4 7nudges ,meno1 who has been gazing elsewhere7 ,but back to your simple proof3 we know that a ratio of two even numbers can be divided until reduced until one or both its parts are odd8 ,boy3 ,yes1 ,socrates4 ,then it is a proper ratio4 ,socrates3 ,so we can eliminate one of our four groups1 the one where even was divided by even1 and now we have odd_/odd1 odd_/even and even_/odd8 ,boy3 ,yes1 ,socrates4 ,socrates3 ,let's try odd over even next1 shall we8 ,boy3 ,fine4 ,socrates3 ,what happens when you multiply an even number by an even number1 what kind of number do you get1 even or odd8 ,boy3 ,even1 of course4 ,an even multiple of any whole number gives another even number4 ,socrates3 ,wonderful1 you have answered two questions1 but we need only one at the moment4 ,we shall save the other4 ,so1 with odd over even1 if we multiply any of these times themselves1 we well get odd times odd over even times even1 and therefore odd over even1 since odd times odd is odd and even of even is even4 ,boy3 ,yes4 ,a ratio of odd over even1 when multiplied times itself1 yields odd over even4 ,socrates3 ,and can our square root of two be in that group8 ,boy3 ,i don't know1 ,socrates4 ,have ,i failed8 ,socrates3 ,oh1 you know1 you just don't know that you know4 ,try this3 after we multiply our number times itself1 which the learned call squaring the number which is the root1 we need to get a ratio in which the first or top number is twice as large as the second or bottom number4 ,is this much correct8 ,boy3 ,a ratio which when squared as you called it1 yields an area of two1 must then yield one part which is two times the other part4 ,that is the definition of a ratio of two to one4 ,socrates3 ,so you agree that this is correct8 ,boy3 ,certainly4 ,socrates3 ,now if a number is to be twice as great as another1 it must be two times that number8 ,boy3 ,certainly4 ,socrates3 ,and if a number is two times any whole number1 it must then be an even number1 must it not8 ,boy3 ,yes1 ,socrates4 ,socrates3 ,so1 in our ratio we want to square to get two1 the top number cannot be odd1 can it8 ,boy3 ,no1 ,socrates4 ,therefore1 the group of odd over even rational numbers cannot have the square root of two in it6 ,nor can the group ratios of odd numbers over odd numbers4 ,socrates3 ,wonderful4 ,we have just eliminated three of the four groups of rational numbers1 first we eliminated the group of even over even numbers1 then the ones with odd numbers divided by other numbers4 ,however1 these were the easier part1 and we are now most of the way up the mountain1 so we must rest and prepare to try even harder to conquer the rest1 where the altitude is highest1 and the terrain is rockiest4 ,so let us sit and rest a minute1 and look over what we have done1 if you will4 ,boy3 ,certainly1 ,socrates1 though ,i am much invigorated by the solution of two parts of the puzzle with one thought4 ,it was truly wonderful to see such simple effectiveness4 ,are all great thoughts as simple as these1 once you see them clearly8 ,socrates3 ,what do you say1 ,meno8 ,do thoughts get simpler as they get greater8 ,meno3 ,well1 it would appear that they do1 for as the master of a great house1 ,i can just order something be done1 and it is2 but if ,i were a master in a lesser house1 ,i would have to watch over it much more closely to insure it got done4 ,the bigger the decisions ,i have to make1 the more help and advice ,i get in the making of them1 so ,i would have to agree4 ,socrates3 ,glad to see that you are still agreeable1 ,meno1 though ,i think there are some slight differences in the way each of us view the simplicity of great thought4 ,shall we go on8 ,meno3 ,yes1 quite4 ,boy3 ,yes1 ,socrates4 ,i am ready for the last group1 the ratios of even numbers divided by the odd1 though1 ,i cannot yet see how we will figure these out1 yet1 somehow ,i have confidence that the walls of these numbers shall tumble before us1 as did the three groups before them4 ,socrates3 ,let us review the three earlier groups1 to prepare us for the fourth1 and to make sure that we have not already broken the rules and therefore forfeited our wager4 ,the four groups were even over even ratios1 which we decided could be reduced in various manners to the other groups by dividing until one number of the ratio was no longer even2 then we eliminated the two other groups which had odd numbers divided by either odd or even numbers1 because the first or top number had to be twice the second or bottom number1 and therefore could not be odd2 this left the last group we are now to greet1 even divided by odd4 ,boy3 ,wonderfully put1 ,socrates4 ,it is amazing how neatly you put an hour of thinking into a minute4 ,perhaps we can1 indeed1 put ten years of thinking into this one day4 ,please continue in this manner1 if you know how it can be done4 ,socrates3 ,would you have me continue1 ,meno8 ,you know what shall have to happen if we solve this next group and do not find the square root of two in it4 ,meno3 ,socrates1 you are my friend1 and my teacher1 and a good companion4 ,i will not shirk my duty to you or to this fine boy1 who appears to be growing beyond my head1 even as we speak4 ,however1 ,i still do not see that his head has reached the clouds wherein lie the minds of the ,pythagoreans4 ,socrates3 ,very well1 on then1 to even over odd4 ,if we multiply these numbers times themselves1 what do we get1 boy8 ,boy3 ,we will get a ratio of even over odd1 ,socrates4 ,socrates3 ,and could an even number be double an odd number8 ,boy3 ,yes1 ,socrates4 ,socrates3 ,so1 indeed1 this could be where we find a number such that when multiplied times itself yields an area of two8 ,boy3 ,yes1 ,socrates4 ,it could very well be in this group4 ,socrates3 ,so1 the first1 or top number1 is the result of an even number times itself8 ,boy3 ,yes4 ,socrates3 ,and the second1 or bottom number1 is the result of an odd number times itself8 ,boy3 ,yes4 ,socrates3 ,and an even number is two times one whole number8 ,boy3 ,of course4 ,socrates3 ,so if we use this even number twice in multiplication1 as we have on top1 we have two twos times two whole numbers8 ,boy3 ,yes1 ,socrates4 ,socrates3 7nudges ,meno7 and therefore the top number is four times some whole number times that whole number again8 ,boy3 ,yes1 ,socrates4 ,socrates3 ,and this number on top has to be twice the number on the bottom1 if the even over odd number we began with is to give us two when multiplied by itself1 or squared1 as we call it8 ,boy3 ,yes1 ,socrates4 ,socrates3 ,and if the top number is four times some whole number1 then a number half as large would have to be two times that same whole number8 ,boy3 ,of course1 ,socrates4 ,socrates3 ,so the number on the bottom is two times that whole number1 whatever it is8 ,boy3 ,yes1 ,socrates4 ,socrates3 7standing7 ,and if it is two times a whole number1 then it must be an even number1 must it not8 ,boy3 ,yes4 ,socrates3 ,then is cannot be a member of the group which has an odd number on the bottom1 can it8 ,boy3 ,no1 ,socrates4 ,socrates3 ,so can it be a member of the ratios created by an even number divided by an odd number and then used as a root to create a square8 ,boy3 ,no1 ,socrates4 ,and that must mean it can't be a member of the last group1 doesn't it8 ,socrates3 ,yes1 my boy1 although ,i don't see how we can continue calling you boy1 since you have now won your freedom1 and are far richer than ,i will ever be4 ,boy3 ,are you sure we have proved this properly8 ,let me go over it again1 so ,i can see it in my head4 ,socrates3 ,yes1 my boy1 er1 ah1 sir4 ,boy3 ,we want to see if this square root of two we discovered the other day is a member of the rational numbers8 ,socrates3 ,yes4 ,boy3 ,so we define the rational numbers as numbers made from the division into ratios of whole numbers1 whether those whole numbers are even or odd4 ,socrates3 ,yes4 ,boy3 ,we get four groups1 even over even1 which we don't use1 odd over even1 odd over odd1 and even over odd4 ,socrates3 ,continue4 ,boy3 ,we know the first number in the squared ratio cannot be odd because it must be twice the value of the second number1 and therefore is must be an even number1 two times a whole number4 ,therefore it cannot be a member of either of the next groups1 because they both have whole numbers over odd numbers4 ,socrates3 ,wonderful6 ,boy3 ,so we are left with one group1 the evens over odds4 ,socrates3 ,yes4 ,boy3 ,when we square an even over odd ratio1 the first number becomes even times even1 which is two times two times some other whole number1 which means it is four times the whole number1 and this number must be double the second number1 which is odd1 as it was made of odd times odd4 ,but the top number cannot be double some bottom odd number because the top number is four times some whole number1 and the bottom number is odd--but a number which is four times another whole number1 cannot be odd when cut in half1 so an even number times an even number can never be double what you would get from any odd number times another odd number444 therefore none of these rational numbers1 when multiplied times themselves1 could possibly yield a ratio in which the top number was twice the bottom number4 ,amazing4 ,we have proved that the square root of two is not a rational number4 ,fantastic6 7,he continues to wander up and down the stage1 reciting various portions of the proof to himself1 looking up1 then down1 then all around4 ,he comes to ,meno7 ,boy3 ,do you see8 ,it's so simple1 so clear4 ,this is really wonderful6 ,this is fantastic6 ,socrates3 7lays an arm on ,meno's arm7 ,tell him how happy you are for his new found thoughts1 ,meno1 for you can easily tell he is not thinking at all of his newly won freedom and wealth4 ,meno3 ,i quite agree with you1 son1 the clarity of your reasoning is truly astounding4 ,i will leave you here with ,socrates1 as ,i go to prepare my household4 ,i trust you will both be happy for the rest of the day without my assistance4 ,the party1 the presentation of #aj years salary to the newly freed young man1 is another story1 as is the original story of the drawing in the sand the square with an area of two4