An equation that takes Pythagoras to a new level
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Re: An equation that takes Pythagoras to a new level
#2The 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.
Re: An equation that takes Pythagoras to a new level
#3Ptolemy's theorem is another Pythagoras on steroids
Re: An equation that takes Pythagoras to a new level
#4The 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 equivalent to
16T_nT_n = 16 T_n (1+...n)
which is equvialent to:
T_n=1+...n
But that is the very definition of T_n. Q.e.d.
Also if you want to take Pythagoras to a new level, Edsger W. Dijkstra who is much better known for his work in CS has a very interesting formulation https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/E... proving sgn(alpha+beta-gamma)=sgn(a^2+b^2-c^2) which includes and extends the Pythagoras theorem.
Re: An equation that takes Pythagoras to a new level
#5The 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.
That formatting error is repeated in at least one other place too.
Re: An equation that takes Pythagoras to a new level
#6The 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?
Re: An equation that takes Pythagoras to a new level
#7The 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!
Re: An equation that takes Pythagoras to a new level
#8The 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?
Muckluks. Or the 1890 Russian version
Re: An equation that takes Pythagoras to a new level
#9The 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!
Thanks. I cannot see that on my screen. The powers of 1 caught my eye, though.
Re: An equation that takes Pythagoras to a new level
#10The 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!
Let a=12 then
(a-2)^2 + (a-1)^2 + a^2 + (a+1)^2 + (a+2)^2 /365
and a binomial a+b squared yields a^2 + 2ab + b^2 so
a^2 + 4 + a^2+1 + a^2 + a^2+1 + a^2 + 4
add up all those a^2 's to get 5a^2 + 10
replace a by 12 and 720 + 10
yields 730
2