Almost Integers, almost incredible

 

I first came into teaching just as computers and programmable calculators did. It was  me, Apple and Texas Instruments side by side against mathematical illiteracy. One of the things I loved doing in the classroom , and still enjoy out of it, were  expressions that were "nearly" an integer, now called "almost integers" in recreational math.

I think my first introduction came a half-dozen or so years before I first became a teacher. It was the famous article by Martin Gardner in April of 1975 when he posted his "Six sensational discoveries that somehow or another have escaped public attention" in his column in Scientific American. One of his six things was his claim that e^π√163 = 262,537,412,640,768,744, a perfect integer. He credited this discovery to Ramanujan, and it has come to be known as Ramanujan's constant, but the nearness to an integer was noted in 1859 by the mathematician Charles Hermite. The number actually ends in ...743.999,999,999,999,25..., but in the age of calculators only given ten to 12 digits, it was enough to be confusing and difficult to check. You can use much smaller numbers to get similar near integers, e^π√43 comes within about 1/5000 of an integer, and my TI-84 silver Edition gives 884736744, an exact integer as the result.

The Golden Ratio, \( \phi =\frac{\sqrt{5}+1}{2} \) will produce almost integers for any power above about ten. \( \phi^{10} =122.991... \) while \( \phi^{18} =5777.9998.. \). Successive powers tend to successively overreach and undershoot an integer. It seems that the twenty-fourth power is sufficient to fool my TI which gives \( \phi^{24} =103,682 \). (I just checked this last one on Desmos online graphing calculator and it gives 103681.99999.  Science marches on.  

The numbers 134903170 and 196418 form a pair that introduce lots of almost integers. 134903170/196418 = 5777.999826..., these are the 45th and 27th Fibonacci numbers, but you get nearer to an integer each time you double the index of the two Fibonacci numbers. Wikipedia tells me that Fib(45*8) /Fib(27*8) will produce a string of about 30 nines after the decimal point.

An unusual one that is more of a mathematical coincidence, and not explained, is the fact that \( e^{\pi} - \pi = 19.99909997... \) The constant \( e^{\pi} \) is called Gelfond's Constant, after Aleksandr Gelfond, who proved that the number was transcendental.

For Trig Students, Sin (11) is an almost integer, (almost -1) and you can totally fool the TI-84 with \( Sin(2017*\sqrt[5]{2})\) which I'm using for my special day number next year on 2/5/2017 . The Sin(11) is nice to introduce when working with half-angle identities since \( sin^2(11)=1/2(1-cos^2(22)\).

\( 163 ( \pi - e ) \) is almost an integer, and powers of pi and e can be fun almost integers, and \( \pi^4 + \pi^5 - e^6 \) is almost zero. Students might search for such an expression that will fool their calculator.

And for Geometry class, the image at the top of the blog shows a triangle with all integer sides and segments except segment d.  Ed Pegg Jr is credited with finding this gem, and showing that the segment marked d is 7.0000000857....

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