Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: sqrt2 and 2 equally mysterious, or not mysterious at all, and proof
that 1 = 0.999...

Replies: 57   Last Post: Jun 10, 2017 3:16 AM

 Messages: [ Previous | Next ]
 vincent64@yahoo.com Posts: 129 Registered: 4/18/07
Re: sqrt2 and 2 equally mysterious, or not mysterious at all, and
proof that 1 = 0.999...

Posted: May 29, 2017 1:52 PM

Here is an rudimentary algorithm that produces all the digits of SQRT(2)/2 in base 2, one by one at each iteration:

http://www.analyticbridge.com/forum/topics/challenge-of-the-week-square-root-of-two

On Friday, April 21, 2017 at 12:02:41 AM UTC-7, Franz Gnaedinger wrote:
> Irrationals not being numbers, and 1 not being 0.999..., are everlasting
> topics in sci.math, popping up all the time. Very simple yet forgotten
> methods of early mathematics can dissolve the problem.
>
> Approximating the square root of 2 by a number column which I reconstructed
> in 1979
>
> 1 1 2
> 2 3 4
> 5 7 10
> 12 17 24
> 29 41 58
> 70 99 140
> and so on
>
> 99/70 is already a fine value for the square root of 2. (Analogous number
> columns approximate the square roots of 3 and 5 and the cube root of 2.)
> Now let us look at the square root of 4 by drawing up the analogous number
> column
>
> 1 1 4
> 2 5 8
> 7 13 28
> 20 41 80
> 61 121 244
> 182 365 728
> and so on
>
> The ratios, for example 41/20 or 121/61, get ever closer to 2. This means
> that 2 as the square root of 4 is equally mysterious as the square root of 2.
> Or that the square root of 2 is no less mysterious than 2.
>
> As for 1 = 0.999... let me begin with the Horus eye series of Ancient Egypt,
> 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 or simply '2 '4 '8 '16 '32 '64
>
> 1 = '1
> 1 = '2 '2
> 1 = '2 '4 '4
> 1 = '2 '4 '8 '8
> 1 = '2 '4 '8 '16 '16
> 1 = '2 '4 '8 '16 '32 '32
> 1 = '2 '4 '8 '16 '32 '64 '64
>
> From this you get
>
> '2 '4 '8 '16 '32 '64 ... = 1
>
> (The Horus eye series had an astronomical function, since a month of 30
> days multiplied by '2 '4 '8 '16 '32 '64 yields 29 '2 '32 days or 29 days
> 12 hours 45 minutes for one lunation of 29 days 12 hours 44 minutes 2.9
> seconds, modern average from 1989 AD - mistake of the old value less than
> one minute per lunation, or half a day in a lifetime. One eye of the Horus
> falcon was the sun, standing for a month of 30 days, his other eye was
> the moon, standing for one lunation or synodic month.)
>
> Another infinite series
>
> 1 = '1
> 1 = '1x2 '2
> 1 = '1x2 '2x3 '3
> 1 = '1x2 '2x3 '3x4 '4
> 1 = '1x2 '2x3 '3x4 '4x5 '5
> 1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6
>
> 1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6x7 '7x8 '8x9 '9x10 '10x11 '11x12 ...
>
> '1x2 '2x3 '5x6 '6x7 '9x10 '10x11 ... = pi/4
>
> Now for 0.999... being 1
>
> 1 = 10/10
> 1 = 9/10 + 10/100
> 1 = 9/10 + 9/100 + 10/1000
> 1 = 9/10 + 9/100 + 9/1000 + 10/10000
> and so on
> 1 = 0.999...
>
> Mathematical education should begin with really simple methods, not on
> the level of Euclid. Would prevent some people from wasting their life
> on kooky notions.