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/sci/ - Science & Math


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File: 7 KB, 260x210, 260px-Pythagorean.svg.png [View same] [iqdb] [saucenao] [google]
9562036 No.9562036 [Reply] [Original]

shall we discus Pythagorean Theorem gentlemen?

>> No.9562043

>>9562036
Why is it a theorem

>> No.9562054
File: 39 KB, 1415x679, pythag proof.png [View same] [iqdb] [saucenao] [google]
9562054

>>9562036

>> No.9562056

>>9562043
Why yes it is a theorem you intelligent individual.

>> No.9562058

>>9562054
Are there other proofs in mathematics that can be shown using drawings alone?

>> No.9562061

>>9562036
How is a raven like a Pythagorean theorem old bean?

>> No.9562067

>>9562061
Yes

>> No.9562085

>Let [math]\left(\mathbf V,\,+,\,\cdot,\, \left\langle\cdot \mid \cdot\right\langle \right)[/math] be a euclidean space.
>Choose [math]B[/math] an orthogonal basis of [math]\mathbf V[/math]
>Let [math]\mathbf x\,\in\, \mathbf V[/math].
>Choose [math]\left(x_\mathbf b\right)_{\mathbf b \,\in\, B}[/math] such that [math]\mathbf x\ =\ \sum_{\mathbf b\,\in\,B} x_\mathbf b\,\mathbf b[/math].
>Then [math]\left\|\mathbf x\right\|^2\ =\ \left\langle\mathbf x\mid\mathbf x\right\rangle\ =\ \left\langle\sum_{\mathbf b\,\in\,B} x_\mathbf b\,\mathbf b\mid \sum_{\mathbf b\,\in\,B} x_\mathbf b\,\mathbf b\right\rangle\ =\ \sum_{\mathbf b'\,\in\,B} \left\langle\sum_{\mathbf b\,\in\,B} x_\mathbf b\,\mathbf b\mid x_{\mathbf b'}\,\mathbf b'\right\rangle\ =\ \sum_{\mathbf b'\,\in\,B} \sum_{\mathbf b\,\in\,B} \left\langle x_\mathbf b\,\mathbf b\mid x_{\mathbf b'}\,\mathbf b'\right\rangle\ =\ \sum_{\mathbf b'\,\in\,B} \sum_{\mathbf b\,\in\,B} \delta_{\mathbf b,\,\mathbf b'}\,x_\mathbf b\,x_{\mathbf b'}\, \left|\mathbf b\right\|^2\ =\ \sum_{\mathbf b\,\in\,B} \left(x_\mathbf b\, \left\|\mathbf b\right\|\right)^2[/math].
What's more to say?

>> No.9562091

>>9562085
Oh heavens, much more, my fellow undergraduate!

>> No.9562097

>>9562058
It's not shown by drawing alone.
Greeks used unit lengths instead of numbers. The actual proof all uses algebra.

>> No.9562100

>>9562085
Yes

>> No.9562103
File: 44 KB, 1280x659, 1507992001356.png [View same] [iqdb] [saucenao] [google]
9562103

>>9562036

>> No.9562105

>>9562091
Give me your wisdom, my fellow PhD in Parseval's theorem.

>> No.9562125

>>9562103
very good

>> No.9562133

>>9562103
Good show

>> No.9562895
File: 66 KB, 1200x1600, demorgan.png [View same] [iqdb] [saucenao] [google]
9562895

>>9562058

>> No.9562917
File: 3 KB, 325x356, 6Lp07.gif [View same] [iqdb] [saucenao] [google]
9562917

>>9562058
[math] \forall k>0, \int_{0}^{1} x^{k}\, dx\; +\; \int_{0}^{1} x^{\frac{1}{k}}\, dx\; = \;1 [/math]

>> No.9562925

>>9562097
That's not how the Greeks thought about this stuff. The Pythagorean theorem was fundamentally about areas to them.