Sum of positive number and its reciprocal from two triangles

Рет қаралды 22,546

Ай бұрын

This is a short, animated visual proof demonstrating that sum of a positive real number and its reciprocal is always greater than or equal to 2.
To buy me a coffee, head over to www.buymeacoffee.com/VisualPr...
Thanks!
For a wider format and slightly longer version of this proof, see • x plus 1/x is greater ...
It turns out that this theorem is equivalent to the Arithmetic Mean-Geometric Mean inequality. The equivalence is implied by the fact that his video resembles a similar visual proof for the AM-GM inequality found here: • Visual Proof of AM-GM ...
This animation is based on a proof by Roger B. Nelsen from the December 1994 issue of Mathematics Magazine, page 374 (doi.org/10.2307/2690999).
#math #manim #visualproof #mathvideo #geometry #mathshorts #algebra #mtbos #animation #theorem #pww #proofwithoutwords #proof #iteachmath #inequality #amgm #maths #mathematics #education
To learn more about animating with manim, check out:
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Пікірлер: 24

@yb3604 18 күн бұрын

this channel is brilliant when x=1, the whole shape becomes a 1x1 square, with the hypotenuses of the triangles coming together as the diagonal of said square

@MathVisualProofs 18 күн бұрын

@wyattstevens8574 Ай бұрын

When does equality hold based on the diagram? An isosceles right triangle, the case where x=1.

@appel-seed_ Ай бұрын

Math is cool

@aschetos Ай бұрын

Equality holds When x = 1, because inverse of 1 is 1.

@DanielPoupko 18 күн бұрын

Bro *really* likes this result

@LordMarshyMan Ай бұрын

X=1

@prince_bk Ай бұрын

1+1/1=2≥2

@armchair8258 Ай бұрын

Cool! Another way : 1/2(x+1/x)>=sqrt(x times 1/x)=1

@KayneBanham Ай бұрын

Why did he say purple when it's blue

@sohammaheshwari8487 Ай бұрын

Yoooo I just learned this yesterday and the teacher said to memorize it and u showed me the proof thanks for your help ❤❤

@MathVisualProofs Ай бұрын

Glad I could help!

@avijitpramanik5425 Ай бұрын

Visualising is the best way to understand math😊 Do you agree 👍

@ebgurif 24 күн бұрын

I get a feeling that you seem to really love the x + 1/x >= 2 proofs

@MathVisualProofs 24 күн бұрын

Just had four to do. All done now. I do like that this fact is equivalent to AM-GM (two var) so same pics work :)

@raulsantamariadominguez7511 Ай бұрын

The inequality holds that the sum of an integer and its reciprocal is equal or bigger than 2. The sum of the least integer, 1, plus its reciprocal, 1/1, is 2; when x=2, (reciprocal, 1/2) is 2.5, for 3 is 3.3 and so on.

@raulsantamariadominguez7511 Ай бұрын

The bigger the "x" is, the sum of "x" and its reciprocal is equal to (approaches) "x".