2020-03-14

Pi day

As it is Pi day (14th March) I thought I would say something obvious, but slightly mind blowing about another irrational number.

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?!?!