When you square a decimal number, you always end up with twice as many significant digits, or one fewer. e.g. 214² = 45796 (3 digits times 2 is 6, but in this case one less, 5), 56² = 3136 (2 digits times 2 is 4). If you think about it, it is obvious. Basically, for the last digit you multiply out not to count as a significant digit, it would have to be 0. You can do that with 2*5, but not by squaring any final significant digit.
But √2 is irrational. It goes on forever. It starts 1.41421356237 (12 significant digits), which squared is be 1.9999999999912458800169 (23 significant digits, i.e. 12*2-1).
Obviously the more digits, the closer... even lots, such as 1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504 squared, is 1.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999843358064061778389918448031979379487768885015043845659822004916377513166730291211571068952082602447585053372271803775212141335211046806468017544012173246168673653215429898752600809076945040478226149804752092795249913294211896685919489781038912339790863836308797672158570101054709837854108990150947671626758096902141232172374611365869462539815599093474631680641002315522198827855254655681178183773171267162025505081207619119154315015019593781997539423459450924437955866358803983089345879502011206070716091422845617882605404958297112110314310433923158845197889301909351312924983769999151087074570462040601631719823168853951704021687236598419124498942731362178121174457523357788222619729955636686231081657732082989686914502778260949270241394967997279088545806588326980598466346571508275959469245048169832037947129583450529101421217940987273080154757322849492641724996747550397932241142493214043106291202086235967891808498928787226259087932847850782010982642427061227177199617553123685346155963957806300620208169830364794225870383219224922644130340677633853727891782934367363062016
Yes, lots of 9's after that 1. but it just gets more complex.
What must blow your mind is that ultimately, to infinite digits, the answer does actually cancel out and actually end up as exactly 2.
How?!?!
This is basically the same principle as the unintuitive but true statement that 0.9recurring is precisely the same number as 1. Words like "infinitesimal" are unhelpful as they lead people to erroneously belive that if you "go on long enough" there's "a tiny bit still left". Infinity is a weird thing to get your head around.
ReplyDeleteI agree with Steve that thinking about infinity makes your brain go funny.
ReplyDeleteThe best solution is to have a break at the Hilbert Hotel.
It seriously took me about 2 seconds to realise this was not a spam advert for an actual hotel before I remembered. I am a plonker.
DeleteI think you can rearrange this into another proof that sqrt(2) is irrational. Not sure if it's just an elaboration on the standard one.
ReplyDelete