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An equation that takes Pythagoras to a new level

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Re: An equation that takes Pythagoras to a new level

#11

Post by CliffStoll »

jimhefferon wrote: post #9
CliffStoll wrote: post #7

Earlier quoted context omitted.

I think the equation on the board is ( 10^2 + 11^2 + 12^2 +13^2 + 14^2 ) / 365 which solves for 2. The students are expected solve this with "Mental Arithmetic". Try that in your head!

Thanks. I cannot see that on my screen. The powers of 1 caught my eye, though.

Happy Pi-Day!

Re: An equation that takes Pythagoras to a new level

#12

Post by norrius »

jkinudsjknds wrote: post #6
jimhefferon wrote: post #2

The caption "The equation 1⁰² + 1¹² + 1²² = 1³² + 1⁴², whose answer is that both sides equal 365, was immortalized in a different form in this 1895 painting: “Mental Arithmetic. In the Public School of S. Rachinsky.” (NIKOLAY BOGDANOV-BELSKY)" has funky exponents. Medium requires that I make an account to say that, so I am saying it here.

On the subject of that painting, can anyone tell me what the kids are wearing on their feet?

These are bast shoes [0], known in Russian as lápti, which were the typical footwear in villages until the 20th century. They are strapped to the leg so that they don't fall off.

[0]: https://en.wikipedia.org/wiki/Bast_shoe

Re: An equation that takes Pythagoras to a new level

#13

Post by imglorp »

chx wrote: post #4

The linked paper https://fermatslibrary.com/s/proof-without-words-pythagorean... gives you the formula and I am much more a symbols guy than a geometry guy so I will prove it that way, first rearrange it: (4T_n)^2=(4T_n+1)^2-(4T_n-1)^2+....(4T_n+n)^2-(4T_n-n)^2 Now, (4T_n+k)^2-(4T_n-k)^2=16T_nk (this, of course, shows up in the geometric proof as four rectangles where one side is 4T_n and the other is k) so this is e…

Wow, fermatslibrary has gone further downhill. It's almost unusable now with all the engagement stuff.

Re: An equation that takes Pythagoras to a new level

#17

Post by delhanty »

CliffStoll wrote: post #7
jimhefferon wrote: post #2

The caption "The equation 1⁰² + 1¹² + 1²² = 1³² + 1⁴², whose answer is that both sides equal 365, was immortalized in a different form in this 1895 painting: “Mental Arithmetic. In the Public School of S. Rachinsky.” (NIKOLAY BOGDANOV-BELSKY)" has funky exponents. Medium requires that I make an account to say that, so I am saying it here.

I think the equation on the board is ( 10^2 + 11^2 + 12^2 +13^2 + 14^2 ) / 365 which solves for 2. The students are expected solve this with "Mental Arithmetic". Try that in your head!

Assuming one's memorized a table of squares already.

10^2 = 100

11^2 + 13^2 = 121 + 169 = 290

12^2 + 14^2 = 144 + 196 = 340

100 + 290 + 340 = 730

730 / 365 = 2

Re: An equation that takes Pythagoras to a new level

#20

Post by lonelappde »

delhanty wrote: post #17
CliffStoll wrote: post #7

Earlier quoted context omitted.

I think the equation on the board is ( 10^2 + 11^2 + 12^2 +13^2 + 14^2 ) / 365 which solves for 2. The students are expected solve this with "Mental Arithmetic". Try that in your head!

Assuming one's memorized a table of squares already. 10^2 = 100 11^2 + 13^2 = 121 + 169 = 290 12^2 + 14^2 = 144 + 196 = 340 100 + 290 + 340 = 730 730 / 365 = 2

You can also "walk it up" if you haven't memorized them:

      10^2*5
    + (10+11)*4
    + (11+12)*3
    + (12+13)*2
    + (13+14)
    ------------

       500
     +  84
     +  69
     +  50
     +  27
     -----
       730
It's a bit of detail to juggle in your head but not too bad if you hang on to your current subtotal and remember where you are in the sequence to compute the next addend, and you know it's easier to add multi-digit numbers left-to-right than-right to-left.

You only carry from the ones place twice.

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