the quantity x squared plus b times x

can be represented geometrically using

an X by x square as pictured here and a

b by X rectangle placed underneath then

x squared plus BX represents this area

but we can take that b by X rectangle

and cut it into two equal pieces so that

one rectangle is a b over 2 by X

rectangle and the other one is a b over

2 by X rectangle but we can place one of

them on the right side of the square

like this now the area represented is

almost an X plus b over 2 by X plus b

over 2 square the only thing missing is

this lower right square that has

Dimensions B over 2 by B over two so the

area we're interested in is X plus b

over 2 squared minus B over 2 squared

this process is known as completing the

square

Completing the Square Visually

This is a short, animated visual proof showing what is meant when we "complete" the square algebraically and justifying why the formula holds for positive real number #math #completingthesquare #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #algebra s. #mathshorts #mathvideo

See the original video here: https://youtu.be/T0HyWIFbsHQ

To learn more about animating with manim, check out:
https://manim.community

______________________________________________
Music in this video:
Prelude No. 9 by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/

Source: http://chriszabriskie.com/preludes/

Artist: http://chriszabriskie.com/


[Music]

suppose that a b c and d are all

positive real numbers and that a over b

is less than c over d the line

connecting the origin to the point b a

has a slope of a over b

the line connecting the origin and the

point dc has a slope c over d

therefore by assumption the line

connecting 0 0 to b a has a slope that's

less than the slope of the line

connecting 0 0 to d c

let's connect the points b over a and d

over c with a line and then highlight

the midpoint between these two points

the x coordinate of this midpoint is

halfway between b and d and the y

coordinate of this point is halfway

between a and c so the coordinates of

this point are given by b plus d over 2

a plus c over 2.

the line connecting the origin to this

point lies between the first two lines

we started with

therefore the slope of the line

connecting 0 0 to b a is less than the

slope of the line connecting 0 0 to the

midpoint and that is less than the slope

of the line connecting 0 0 to dc

as we've already seen the slope of the

line connecting 0 0 to b a is a over b

the slope of the line connecting 0 0 to

d c is c over d and now we can check

that the slope of the line connecting 0

0 to the midpoint is a plus c over b

plus d

therefore the condition that a over b is

less than c over d implies that a over b

is less than a plus c over b plus d

which is less than c over d

the fraction a plus c over b plus d is

the median of a over b and c over d and

this inequality is known as the mediant

inequality

you

Mediant Inequality III (visual proof)

This is a short, animated visual proof demonstrating what one might call the mediant inequality. #math #inequality #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #fractions #mediant  #mathematics #fareyfractions #farey #mathshorts #mathvideo

If you like this video, consider subscribing and checking out my other videos.

For an alternate visual proofs of this fact, see these video: https://youtu.be/RKeBuWL1B4A
https://youtu.be/pODylgdI3p8

This animation is based on a visual proof by Li Changming from the January 1988 issue of Mathematics Teacher (https://www.jstor.org/stable/27965677 - page 63).

To learn more about animating with manim, check out:
https://manim.community
_____________________________
Music in this video:
Melodic Interlude by Alexander Nakarada | https://www.serpentsoundstudios.com
Music promoted by https://www.free-stock-music.com
Attribution 4.0 International (CC BY 4.0)
https://creativecommons.org/licenses/by/4.0/


let's consider the function f of x

equals e to the X and plot it on the

axis here we can draw the tangent line

to the curve which looks like the curve

locally and whose slope is the instant

rate of change of the Curve we can

compute the slope of this line and we

see that here it's .08 as we move the

point along the curve we see that the

slopes of the tangent lines get larger

in fact something even more interesting

happens in this example if we also plot

the y coordinate of the point we see

that as the point moves along the curve

the slope of the tangent line actually

equals the Y value

we can take this one step further where

we plot a new graph on a new set of axes

here the Y value will be the slope of

the tangent line from the first graph as

we move the point along the curve we map

out this graph the function f of x

equals e to the x is a special function

because its derivative or its slope

function f Prime of X is also e to the x

Exponential derivative visual

A visual of the derivative of f(x)=e^x. We show how to think about the derivative of a function visually.

#manim #calculus #derivatives #derivative #tangentline #slope #parabola #mathvideo #mathshorts #math #visualmath #graph #exponential #linearapproximation


why is the sum of a positive real number

and its reciprocal always greater than

or equal to 2 let X be a positive real

number and draw a rectangle with

Dimensions X by 1 /x and create four

rotated copies like this each of these

rectangles has an area of 1x * X which

is 1 so the four rectangles each enclose

an area of one these four rectangles lie

inside of an x + 1X by x + 1X s

enclosing an area of the quantity x +

1x^2 the four rectangles almost never

fill the entire square there's a missing

sub Square inside and that means that x

+ 1X 2 is greater than or equal to 1 + 1

+ 1 + 1 which is equal to 4 taking the

square root of both sides and

remembering that X is positive means

that x + 1X is greater than or equal to

2 can you see how to use the Center

Square to produce a condition to

guarantee x + 1x equal 2

Sum of a positive number and its reciprocal from four rectangles in a square

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 https://www.buymeacoffee.com/VisualProofs
Thanks!

For a wider format and slightly longer version of this proof, see https://youtu.be/IghOHBl0Do8?si=CSvPRS3Dr3AGBWVO

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: https://youtu.be/62G9fak1vyk

This animation is based on a proof by Roger B. Nelsen from the December 1994 issue of Mathematics Magazine, page 374 (https://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:
https://manim.community


start with a cube with side length one

over square root 1. on top of that place

a cube with side length one over square

root of two on top of that place a cube

with the side length of 1 over the

square root of three and continue this

process indefinitely the resulting

structure sometimes called the infinite

GIF consists of an infinite stack of

Cubes each with a side length of 1 over

the square root of n each Cube has a

side with area given by the square root

of one over N squared so the total

surface area of this structure is

greater than or equal to four times the

infinite sum where n ranges from 1 to

Infinity of 1 over root N squared on the

other hand the volume of each cube is

given by the quantity one over the

square root of n cubed so the volume of

this structure is the infinite sum where

n ranges from 1 to Infinity of the

quantity one over root n cubed it turns

out that the area series diverges

because it's the harmonic series times

four and the volume infinite series

converges to a value that's

approximately 2.61 this structure is a

discrete version of Gabriel's horn and

can be used for the painter's paradox

Infinite gift and the painter’s #paradox

In this animation, we show an approximation of the mathematical object colloquially referred to as the "infinite gift." This gift is somewhat paradoxical in nature because it is infinitely long and requires an infinite amount of wrapping paper to cover, yet it only encloses a finite area. This object is a discrete analog of the more famous "Gabriel's horn," which is an object studied in most integral calculus classes as it is an object enclosing finite volume but has an infinite surface area.

One major purpose of this video is to investigate the different, but related, infinite sums of 1/n and 1/sqrt(n cubed). These sums both have interpretations in terms of the infinite gift, and one of them diverges while the other converges.

To buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs
Thanks!

Here is a slower, wide format, high definition alternate version of this video:

https://youtu.be/YCCNWNXXSx4

For related information, I would begin with the Wikipedia article about Gabriel's horn and the painter's paradox: https://en.wikipedia.org/wiki/Gabriel%27s_horn

Here is one of the first places I remember seeing this object (there are earlier ones, but I can't track them down): https://twitter.com/fermatslibrary/status/1166324490760065024?s=20&t=CLJ2N2LDs-yj0ASNTGzK-Q

#manim #infinitegift #paintersparadox #gabrielshorn #harmonicseries #infiniteseries #convergence #divergence

To learn more about animating with manim, check out:
https://manim.community


[Music]

thank you

[Music]

foreign

[Music]

[Music]

foreign

[Music]

[Music]

foreign

[Music]

Halloween Visual Proof: Alternating Sum of Odds #math

This is a short, animated visual proof showing one method to prove that the alternating sum of the first n odd positive integers (starting with 2n-1 positive) is n. #manim #math #mathvideo #mathshorts #sequences #triangles #animation #theorem #pww #proofwithoutwords #visualproof #proof #squarenumbers #squares #oddnumbers #odds #sums #pww #proofwithoutwords  #proof #algebra #finitesums #mathematics #mathvideo #mtbos

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

This animation is based on visual proof by Art Benjamin from the December 2005 issue of Mathematics Magazine, page 385 (https://www.jstor.org/stable/30044195)

To learn more about animating with manim, check out:
https://manim.community
_____________________________
Music in this video:
Halloween Theme 1, written by Alexander Nakarada.
Courtesy of https://freepd.com/horror.php


there are numerous ways that a function

might fail to be differentiable at a

point you might have a removable

discontinuity or a so-called hole at the

point similarly you might have a jump

discontinuity in this case it's

impossible to have a uniquely defined

tangent line in fact any discontinuity

leads to a point of

non-differentiability so vertical ASM

tootes also create problems such as the

one you might find in the function 1 /x

like this you might also have a point

where there's an Abrupt change in slope

this is often called a cusp or a corner

and it appears in functions like the

absolute value function another way is

that you might have a vertical tangent

line like the one appearing in the cube

root function this would correspond to

an infinite slope which is undefined so

we've seen that discontinuities places

with abrupt changes in slope and points

where there is a vertical tangent line

all create problems for

differentiability can you think of any

other situations where a function won't

be differentiable

Non-differentiability

In this short, we show five different graphs of functions with a point where the function is not differentiable.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

#math #manim #animation #theorem #iteachmath #mathematics ##mathshorts #mathvideo #mtbos #mathematician #calculus #differentiability  #functions #derivative #discontinuity #cusp #verticalasymptote #tangentline

To learn more about animating with manim, check out:
https://manim.community


[Music]

consider a right triangle with legs of

length a

and b and a hypotenuse of length c

first let's scale this triangle by the

value of c

this creates a right triangle and the

new legs

are length a times c and b times c

the hypotenuse is c squared

next we scale the triangle by the value

of b to obtain another right triangle

this right triangle has leg lengths a

times b and b

squared and a hypotenuse of length c

times b

finally we scale the original triangle

one more time by the value a

this produces a right triangle and in

this instance

the leg lengths are a squared b times a

and hypotenuse ca

notice that the second and third scaled

triangles fit together along their sides

of common length a b

equals b a glued together these

triangles create yet another right

triangle

this can be verified as the two acute

angles are complementary by construction

finally note that this new triangle is

congruent to the first scale triangle as

they have legs of the same lengths and

are both right triangles

therefore the hypotenuses must be equal

so that a squared

plus b squared equals c squared this is

the pythagorean theorem

Pythagorean Theorem I (visual proof)

This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the hypotenuses of scaled triangles. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. #mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof

This animation is based on a visual proof by Frank Burk from the November 1996 issue of The College Mathematics Journal (https://doi.org/10.2307/2687335 page 409).

To learn more about animating with manim, check out:
https://manim.community
__________________________________________________________________
Background music, used according to CC BY-SA 3.0, is "The Road" from Esteban Orlando:

Esteban Orlando:
https://soundcloud.com/orlando-esteban-2
https://www.instagram.com/estebanorla...

https://creativecommons.org/licenses/...
Music provided by RFM: https://youtu.be/b4KCM1bIq50


foreign

[Music]

[Music]

foreign

foreign

[Music]

Summing odds using four stacks

This is a short, animated visual proof demonstrating how to find the sum of the first n positive odd integers. #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #squares #oddintegers #numbertheory #sum #mathshorts #mathvideo

Here is the original widescreen high def version: https://youtu.be/ZeEOgbLo0Rg

This animation is based on a visual proof by Roger B. Nelsen from his first compendium of visual proofs:

Proofs without Words I by Roger B. Nelsen (MAA, 1993) (https://bookstore.ams.org/clrm-1)

To learn more about animating with manim, check out:
https://manim.community


this is one of the ways to prove the

pythagorean theorem of c

squared is equal to a squared plus b

squared using four similar triangles

with sides a b

and c we can arrange the four triangles

in such a way

that the hypotenuse side c would be

touching each other

and this would leave us with a big empty

space in between with a side c

we can compute the area of this empty

space as c times c

or c squared now let's rearrange the

same four right triangles but this time

the sides a and b

will be touching each other unlike the

first one which has one big empty space

left

this time we have two small spaces or

two small squares

one and two the area of one square is a

times a

or a squared and the other is b times v

or b squared

since we didn't change the areas of our

right triangles that means that the big

area c

squared is equal to the sum of b squared

plus a squared or the pythagorean

theorem

Visual Proof of the Pythagorean Theorem | Math Help and Tutorial #shorts

Have you learned this in high school? Here's a visual proof of the popular Pythagorean Theorem #Shorts
This channel is for anyone wanting for math help, algebra help, calculus help, engineering math help, and any tricks on math.

Follow us on Tiktok: https://www.tiktok.com/@engineeringmath?lang=en


here's an amazing diagram due to Rex woo

this is a 3x2 rectangle where we've

inscribed a triangle because these two

highlighted triangles are similar the

inscribed triangle is a right triangle

that happens to be isosceles so these

other two angles are pi over four we can

compute this Arc Length of pi over four

by using these two triangles here which

have angle measurements of arc tangent

one over two and arctangent one over

three using the rectangular diagram we

can compute the other pi over 4 angle as

the difference of the angles pictured in

these two triangles which are arc

tangent of three and arc tangent of one

half respectively using the diagram we

can compute that same pi over 4 angle

using the difference of the angles in

these two triangles which are arc

tangent of two and arctangent of 1 3

respectively we can also compute the

upper right pi over 2 angle as the sum

of the angles in these three picture

triangles so we get that pi over two is

arc tangent one plus arctangent a half

plus arctangent a third and finally we

can compute this Pi angle as the sum of

the angles in these three triangles so

we get that Pi is arctangent of one plus

arc tangent of two plus arctangent of

three this diagram has given us five

formulas for pi

A Five Pi Diagram (pi day short)

In this short, we give five different formulas for pi (or related constants) using only a single diagram consisting of a 3x2 rectangle and a triangle inside. Using the fact that the tangent of an angle is the ratio of the opposite and adjacent sides of an angle (and you can get the angle using the arctangent of the appropriate ratio), we decompose angles (of measure pi/4, pi/2, and pi) using certain triangles with visually computable arctangents using the diagram.

This visual proof is due to Rex Wu and comes from the March 2003 issue of The College Mathematics Journal (https://www.jstor.org/stable/3595783 ) see pages 115 and 138.

For more Pi-related videos, check out my playlist:
https://youtube.com/playlist?list=PLZh9gzIvXQUsGRDvzvXc02lDIKHaVeFwq

#manim #irrational #Pi #mathvideo #math #mtbos  #animation #iteachmath #mathematics #piday #shorts #trigonometry #arctangent #tangent #triangle #rectangle #identities

To learn more about animating with manim, check out:
https://manim.community


[Music]

let's build an expanding triangular

array of squares where the first row has

1 1X one square the second row has 2 2x

two squares the third row has 3 3x3

squares the fourth row has four 4x4

squares and so on until the last row row

N has n and by n

squares the side length of this array is

the sum of the first n positive integers

which is n * n + 1 /

2 the line of slope -2 starting from the

top left hits the upper left corner of

the last Square in every row and

produces a triangle this triangle has

base length n * n +

1 the small triangles excluded from the

large triangle by the line can be

shifted and rotated to fill in miss gaps

at the large triangle like this

therefore the area of the original array

is 12 * n * n + 1 / 2 * n * n + 1 which

is the area of the triangle from this we

can see that the sum of the first n

positive cubes is equal to the square of

n * n + 1 / 2

[Music]

Sum Of Cubes I (visual proof)

This is a short, animated visual proof giving a formula for the sum of the first n positive cubes. #mathshorts #mathvideo #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #cubes #mathematics #sum

This animation is based on a visual proof by Georg Schrage from the June 1992 issue of Mathematics Magazine (https://doi.org/10.2307/2691330 - page 185).

To learn more about animating with manim, check out:
https://manim.community
__________________________________________________________________
Background music, used according to CC BY-SA 3.0, is "The Road" from Esteban Orlando:

Esteban Orlando:
https://soundcloud.com/orlando-esteba...
https://www.instagram.com/estebanorla...

https://creativecommons.org/licenses/...
Music provided by RFM: https://youtu.be/b4KCM1bIq50


let's suppose that you have two finite

sets A and B and you want to find the

cardinality of their Union a union B you

might think that the cardinality of the

Union is simply the sum of the

cardinalities that is that the

cardinality of a union B is the

cardinality of a added to the

cardinality of B but note that elements

in the set a intersect B have been

counted twice in this computation once

in a and once in B so we must remove one

copy of the cardinality of a intersect B

so the cardinal it of a union B is equal

to the cardinality of a plus the

cardinality of B minus the cardinality

of the intersection of A and B an

alternative is to think of the Union in

three separate pieces so that the

cardinality of a union B is the

cardinality of a set minus B plus the

cardinality of a intersect B plus the

cardinality of B set minus a

General Sum Principle #venndiagram  #math

In this video, we show the general sum principle for sets (the formula for the cardinality of A union B). The key feature is that if the sets intersect, then we have to account for elements that may have been double counted.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

For a more dramatic version of this video, check out:
https://youtu.be/wXdoqPMlVRU (which includes the product principle too).

If you want to learn more about these principles, check out:
https://youtu.be/sbIcWGYNA5I

A related textbook forthis is the open-source textbook by Oscar Levin:
http://discrete.openmathbooks.org/dmoi3.html

#logic #settheory #union #propositions  #intersection #setdifference  #setminus #setconnectives #subsets #venndiagram #visualproof #math #manim #discretemathematics #sumprinciple #inclusionexclusion

To learn more about animating with manim, check out:
https://manim.community


[Music]

let a

b

and c

all be sets

and suppose that f is a function from a

to b

and g is a function from b to c

then the function g composed f

is the function from a to c

defined by the rule that g composed f of

a is equal to g of f a

that is we send a into f

and send the resulting output from b

into g

the composition takes a directly to that

output so that g composed f of a is

equal to g of f of a

let's see this definition in action as

we prove that function composition is

associative

this means that if we have three

functions f g and h

then the composition of h with g

composed f is equal to h composed g

composed with f

in particular no matter what element

from the domain a we plug into these

functions the outputs will be equal

let's see this proof visually let a b

c

and d all be sets

and suppose that f is a function from a

to b g is a function from b to c and h

is a function from c to d

then we know that g composed f is a

function from a to c and h compose g is

a function from b to d

start with an arbitrary element x from

the domain a

we can send x through f

send the output through g

and send that output through h resulting

in the element h of g of f of x

now we end at the exact same element

when we send x through g composed f

and then the resulting output through h

so h composed with a g composed f of x

is equal to h of g composed f of x which

is equal to h of g of f of x

similarly we ended that exact same

element when we send x through f and the

resulting output through h composed g

so h composed g composed with f x is

equal to h composed g of f of x which is

equal to h of g of f x

therefore the two compositions map x to

the same element no matter what x is

so the two compositions are equal as

functions

[Music]

you

Function Composition is Associative  (visual proof)

This is a proof, with a visualization, of the classic set-theoretic proof that the operation of function composition is associative. The video includes a "visual definition" of the composition operation and then shows how to prove the theorem. Implicit in this explanation is the style of proof required when proving a universal statement (here that two functions are equal by showing they behave the same for every element in the domain). #manim #composition #associativeproperty #visualmath #math #mtbos #visualproof #settheory #introductiontoproofs #associative #function #functioncomposition #universalstatement #functiondiagram

To learn more about animating with manim, check out:
https://manim.community


start with a right triangle with legs of

length A and B and a hypotenuse of

length C use the hypotenuse as the

radius of a circle centered at the left

point of the triangle with radius C the

horizontal distance from the right angle

to the circle is then C minus a when we

inscribe a triangle in the top

semicircle like this by the's triangle

theorem this triangle is right angled

one consequence of this fact is that

these two pictured right triangles here

must be similar as their top angles are

compliment because the two triangles are

similar their leg ratios must be equal

that means that C + a / B must equal B /

c - A but this equality means that C + a

* c - A = b^ 2 so that c^2 - A 2 = B ^2

from this we see that c^ 2al A2 + b^2

where A and B are the legs of a right

triangle with hypotenuse C we used a

semicircle to prove the Pythagorean

theorem

Pythagorean theorem from a (semi) circle!

This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the semicircle and Thales triangle theorem. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths.

This animation is based on a proof due to Michael Hardy from the November, 1986 issue of The College Math Journal (https://doi.org/10.2307/2686255).

To buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs
Thanks!

For other proofs of this same fact check out the following playlist:

https://youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rB&si=o_im7-_NHp1zhkcD
#mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof

To learn more about animating with manim, check out:
https://manim.community


to find the area formula for a trapezoid

with bases of length B1 and B2 and a

height of length H we can perform the

following interesting process take one

of the side lengths and bict it cutting

it into two equal length pieces like

this now connect the midpoint to one of

the opposite Corners creating a small

triangle we can cut this triangle off

and rotate it around to create a larger

triangle with base of length B1 plus B2

and height of length H but then this

triangle has an area of/ 12 h * B1 + B2

as this triangle has the same area as

the original trapezoid we see that the

area formula for a trapezoid is 1/2 *

the height times the sum of the two

bases

Trapezoid Area Visual Proof

This is a short animation showing a standard fact about how to find the area of a trapezoid by finding a triangle with the same area.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

Here is the original video in wide-format, wordless, with dramatic music:
https://youtu.be/4ZypJJmd5ZY

Here is a related visual proof for the area of a parallelogram:
https://youtu.be/QjWbBexfJvw

#area #triangle #trapezoid #geometry #visualproof #math #manim

This video was inspired by a beautiful GeoGebra applet created by Tim Brzezinski:

https://www.geogebra.org/m/ZeDN7ZTW

To learn more about animating with manim, check out:
https://manim.community


start with a 4x2 rectangle

and subdivide both sides

half of the width of the rectangle has

length two

we can swing a matching length around

until it intersects the line cutting

through the middle of the rectangle like

this

and we use that line

to create an equilateral triangle

sitting inside the left half of the

rectangle

because the triangle is equilateral and

has side length 2

the horizontal altitude of the triangle

must be the square root of 3.

[Music]

connect the two edges labeled 2

and we create an isosceles triangle as

the two side lengths are equal

the interior obtuse angle is formed from

a 90 degree angle and a 60 degree angle

so it measures 150 degrees

consequently the other two angles in the

isoscele triangle must be equal and so

are 15 degrees and therefore the

complementary angle here is 75 degrees

using a similar argument we can create

another isosceles triangle seen here

one angle is supplementary to 150

degrees so it's 30 degrees this means

the other two angles which are equal

must both be 75 degrees

now we focus only on the lower half of

the rectangle with side lengths of one

using this highlighted right triangle we

determined that the tangent of 75

degrees is the opposite side length over

the adjacent side length of one so the

tangent of 75 is two plus root three

we can use this other highlighted

triangle with angle 15 degrees because

it's complementary to 75 to see that the

tangent of 15 is 2 minus root 3 all

divided by 1.

we have used a rectangle and not the

half angle identity for tangent to

determine that the tangent of 75 is 2

plus root 3 and the tangent of 15 is 2

minus root 3.

[Music]

you

Tangent 75 and tangent of 15 (visual proof; trigonometry)

This is a short, animated visual proof of the showing how to find the tangent of 75 and the tangent of 15 without using the half-angle identities for the tangent function. The idea is to embed some interesting triangles into a 2x4 rectangle. #mathshorts #mathvideo #math #trigonometry #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #tangent #righttriangles #rectangle #identity #halfangle #trigidentity #trig

Please consider subscribing to the channel if you enjoy the content.

Another way to find these measurements is to use a half-angle tangent identity, such as the ones found in this video: https://youtu.be/XIC7SIdl7pI

This animation is based on a proof due to García Capitán Francisco Javier from the January 2017 issue of The College Mathematics Journal (https://doi.org/10.4169/college.math.j.48.1.35) page 35.

To learn more about animating with manim, check out:
https://manim.community


suppose you have a right triangle with

legs of length A and B and hypotenuse of

length C

you can draw squares off the legs

enclosing an area of a squared and of B

squared respectively

you can also draw a square off the

hypotenuse enclosing an area of c

squared let's see an interesting

dissection that'll give a formula for c

squared we can take these two lines off

the C by C square and shift them so that

they pass through the middle of the B by

b square notice we can shift the

original triangle so that it lies along

one of those lines and then we can

rotate the triangle about that line to

form a rectangle that covers part of the

B by b square

the two uncovered rectangles in the B by

b square have the same width by Symmetry

and this width can be computed then as

half the difference between the square

length and the leg length of the

triangle which is B minus a over two so

that the remaining side of the B by b

square has a length of B plus a over two

Again by symmetry this means we've cut

the B by b square into four congruent

trapezoids all with the same dimensions

we can shift these trapezoids as follows

over to the C by C Square

but notice there's a remaining Square

left on the inside the side length of

this Square can be computed as the

difference of B plus a over 2 and B

minus a over two but that's just a so we

can then shift the a by a square up and

now we've covered the entire C by C

Square therefore c squared must be equal

to B squared plus a squared

there is nothing special about the

particular right triangle that we chose

if a and b are the leg lengths of any

right triangle with hypotenuse C then a

squared plus b squared will always equal

c squared here we show a variety of

different right triangles morphing

through and each time we show the

dissection given by this proof it turns

out that the dissection of the B by B

squared and the a by a square will

always cover the C by C Square this is

the Pythagorean theorem As proved by

Henry dudenay

Pythagorean Theorem XI (Dudeney's dissection visual proof)

This is an animated visual proof of the Pythagorean theorem (the right triangle theorem) using a Henry Dudeney's dissection argument (congruent-by-addition proof).

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

This animation is based on a proof from Henry Dudeney and is discussed in Howard Eve's book "Great Moments in Mathematics (before 1650)." You can also find this in Roger Nelsen's first PWW compendium on page 6.

Here is a shorts version of this proof:
https://youtube.com/shorts/azzrmVw8O-8

For other proofs of this same fact check out:
https://youtu.be/q28H5NY5UVU
https://youtu.be/yfGtbNgcrQ8
https://youtu.be/JRjPsuJ7S-g
https://youtu.be/TsKTteGJpZU
https://youtu.be/ZlGaQdNRdqA
https://youtu.be/uqGjs_n-f7U
https://youtu.be/wWEZ-rUlBDQ
https://youtu.be/I75HYC8TRNY
https://youtu.be/LWVog8hg74I?si=4GSzTLZP7XIWeaxJ

#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo

To learn more about animating with manim, check out:
https://manim.community


select 5000 positive integers uniformly

at random from the range one to five

million to plot points in the plane

if the pair is relatively prime plot the

point in red and otherwise plot the

point in blue

here's an animation of one example of

this random process

it turns out that the ratio of the

number of red dots to the total number

of dots gives a good approximation for

the fraction 6 over pi squared

therefore we can use this particular

simulation to get an approximation for

pi as 3.142 not great but not terrible

A random approximation for Pi (pi day short)

In this short, we randomly sample 5000 pairs of positive integers and use the pairs to plot points in the plane. If a pair is relatively prime, we shade the dot red; otherwise we shade the dot blue. The theory behind this process says that roughly 6/pi*pi of the dots will be red. So we can use this simulation to get an approximation for 6/pi*pi and in turn use that to get an approximation for Pi. We get 3.142, which is not that bad, but not really good either.

For more information about why this works, check out the wikipedia site:

https://en.wikipedia.org/wiki/Coprime_integers

For more Pi-related videos, check out my playlist:
https://youtube.com/playlist?list=PLZh9gzIvXQUsGRDvzvXc02lDIKHaVeFwq

#manim #approximation  #relativelyprime #zetafunction #baselproblem #irrational #Pi #mathvideo #math #mtbos  #animation #iteachmath #mathematics #piday

To learn more about animating with manim, check out:
https://manim.community


let A and B be two positive real numbers

and consider an a by B rectangle that

has area a times B

now create four copies of this rectangle

in closing an area of four times a times

B

these four rectangles can be organized

to fit inside of an A plus b by a plus b

square however there's a missing

interior square with a side length of B

minus a

from this diagram we see that the

quantity a plus b squared minus the

quantity B minus a squared must be equal

to the quantity four times a times B

if we remove the inner Square we see

that the quantity a plus b squared is

greater than or equal to 4 times a times

B this is true for all positive Pairs

and we see that we get equality between

a plus b squared and 4 a b if and only

if a equals B taking square roots and

dividing by 2 yields the arithmetic mean

geometric mean inequality

AM-GM inequality visually

This is a short, animated visual proof of the arithmetic mean-geometric mean inequality using areas. This theorem states that the average of two positive numbers is greater than or equal to the square root of the product of the same two numbers (with equality only when the two numbers are the same). #math #amgminequality #area #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #mathshorts #mathvideo

If you like this video, please like, subscribe and check out my others.

This animation is based on a visual proof by Doris Schattschneider from the February 1986 issue of Mathematics Magazine (https://doi.org/10.2307/2690011).

Here are some other visual proof of the AMGM inequality (and others):
https://youtu.be/62G9fak1vyk
https://youtu.be/XJ3643WhKjY
https://youtu.be/Pt8IX_Q5U1A
https://youtu.be/02yM5LqMWUs
https://youtu.be/jNx05gXSblE
https://youtu.be/IJGwvvQdfgg
https://youtu.be/SEPkdPWKdO0

And here is a playlist with AMGM type videos: https://youtube.com/playlist?list=PLZh9gzIvXQUuoRR4kTd628cWdFQ1BDyS6

To learn more about animating with manim, check out:
https://manim.community


[Music]

[Music]

[Applause]

[Music]

[Applause]

[Music]

[Music]

a

[Music]

[Music]

[Music]

[Applause]

[Music]

a

[Music]

Sum of Cubes VII (visual proof without words)

This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes by rearranging two stacks of cubic arrays.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

Here are six other visual proofs of sum of cubes formulas:

https://youtu.be/LYfKapE_g1Q
https://youtu.be/jWpyrXYZNiI
https://youtu.be/YQLicI8R4Gs
https://youtu.be/Ye9OPNqV9FA
https://youtu.be/NxOcT_VKQR0
https://youtu.be/d1yM6Rq7Tfw

This animation is based on a visual proof by Tom Edgar from the September 2024 issue of Mathematics Magazine (https://doi.org/10.1080/0025570X.2024.2379213 page ??).

To learn more about animating with manim, check out:
https://manim.community

#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #calculus #sum #induction #faulhaber #sumofcubes #cubes
___________________________________________
Music in this video:
Epic Cyberpunk | GLORY by Alex-Productions | https://onsound.eu/
Music promoted by https://www.free-stock-music.com
Creative Commons / Attribution 3.0 Unported License (CC BY 3.0)
https://creativecommons.org/licenses/by/3.0/deed.en_US


[Music]

start with a real number a between minus

one and one

to compute the integral of the square

root of one minus x squared between a

and one using the fundamental theorem of

calculus you need a trig substitution

and some trig identities however we can

also remember that this integral

represents the area under the square

root of one minus x squared between x

equals a

and x equals one like this to find this

area we can first find the area of the

full sector covering the region

if this sector has angle alpha then the

area is one half alpha because the

radius is one

the area we want is this area minus the

area of this right triangle

with base a and height square root of

one minus a squared

note that this right triangle also shows

that the cosine of alpha

is a so that alpha is the arc cosine

of a giving the formula for the area we

want in terms of a

but this diagram only works if a is

positive

if a is negative the region that we want

looks a little something like this

luckily this region is now the union of

a sector in a right triangle

again if this sector has angle alpha

then the area

is one-half alpha

this time we add the area of the

triangle

which now has base negative a and height

square root of 1 minus a

squared as before alpha

is the arc cosine of a

and we can think of adding the negative

as subtracting

this yields the same formula as before

so in either of these two cases the

integral that we want

is given by the arc cosine of a over two

minus one half a times the square root

of one minus a squared

you

An Integral without Trigonometric Substitution (visual proof)

This is a short, animated visual proof demonstrating how to find a certain integral that is challenging to do by the fundamental theorem of calculus by instead using an area dissection  . #mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #integral #substitution #trigsub

This animation is based on a visual proof by Marc Chamberland from the February 2001 issue of Mathematics Magazine (https://doi.org/10.2307/2691155).

To learn more about animating with manim, check out:
https://manim.community
__________________________________________________________________
Background music, used according to CC BY-SA 3.0, is "The Road" from Esteban Orlando:

Esteban Orlando:
https://soundcloud.com/orlando-esteba...
https://www.instagram.com/estebanorla...

https://creativecommons.org/licenses/...
Music provided by RFM: https://youtu.be/b4KCM1bIq50


[Music]

imagine two positive real numbers X and

Y represented by lengths of connected

line segments and then draw the

semicircle with diameter given by the

connected segments so that the radius of

the semicircle is x + y/ 2 then let Z be

the length of the half cord from the

point where X and Y meet to the

semicircle connecting the ends of the

diameter to the intersection of Z on the

circle yields a triangle and by the's

triangle theorem this triangle must be

right

angled consequently these two subt

triangles are similar so that the ratios

z overx and y over Z are equal we

conclude z^2 equals x * y so that Z is

the square root of x * y

no matter the lengths of X and Y the

chord length Z is always less than or

equal to the radius of the semicircle by

construction therefore the < TK of x * Y

is less than or equal to x + y / 2 with

equality if and only if xal y this is

the arithmetic mean geometric mean

inequality

[Music]

AM-GM Inequality II (visual proof)

This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using Thales theorem . #mathshorts #mathvideo #math #amgminequality #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #calculus #inequality

This animation is based on a visual proof by Charles D. Gallant from the March 1977 issue of Mathematics Magazine (https://doi.org/10.2307/2689737).

To learn more about animating with manim, check out:
https://manim.community
__________________________________________________________________
Background music, used according to CC BY-SA 3.0, is "The Road" from Esteban Orlando:

Esteban Orlando:
https://soundcloud.com/orlando-esteba...
https://www.instagram.com/estebanorla...

https://creativecommons.org/licenses/...
Music provided by RFM: https://youtu.be/b4KCM1bIq50


[Music]

let's add up the first four perfect

squares

one plus four plus nine plus sixteen

by arranging the l-shaped nomens from

each of this square

we get this tower which consists of a

one by seven

a two by 5 a 3 by 3 and a 4 by 1 stack

of squares

we can then use the sum of squares twice

more

to build a rectangle

notice that this rectangle has width two

times four

plus one

and the length of this rectangle is one

plus two plus three

plus four therefore we see that 3 times

the sum of the first four squares

is 2 times 4 plus 1 multiplied by the

sum of the first four positive integers

there's nothing special about four

squares the analogous geometric

construction will work for the sum of

the first

n squares showing that three times the

sum of the first n squares

is two n plus one multiplied by the sum

of the first n positive integers

applying a known sum formula for the sum

of integers

and then dividing by 3 yields a formula

for the sum of the first

n squares

Sum of Squares I (visual proof)

This is a short, animated visual proof of the formula that computes that sum of the first n squares using 3 copies of the sum of squares to build a rectangle . #mathshorts #mathvideo #math #sumofsquares #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath

This animation is based on (independently discovered) visual proofs by Dan Kalman from the March 1991 issue of The College Mathematics Journal (https://www.jstor.org/stable/2686447) and Martin Gardner from the October 1973 Scientific American (https://www.jstor.org/stable/24923225).

To learn more about animating with manim, check out:
https://manim.community
__________________________________________________________________
Background music, used according to CC BY-SA 3.0, is "The Road" from Esteban Orlando:

Esteban Orlando:
https://soundcloud.com/orlando-esteba...
https://www.instagram.com/estebanorla...

https://creativecommons.org/licenses/...
Music provided by RFM: https://youtu.be/b4KCM1bIq50


[音楽]

[音楽]

(a+b) ^3 visual proof. #maths  #formula


[Music]

let's let H sub K represent the harmonic

sum 1 plus a half plus a third up to one

over k

now consider a triangular array with n

minus one rows in the first row place

the harmonics on H1 in the second row

place the harmonic sum H2 and so on so

that the entry in row I and column J is

the fraction one over J

now consider a triangular array of boxes

with n rows take the number n and split

it into n copies of one then equally

distribute 1 into each column using

fractions so that column J contains all

entries of fractions one over J as

pictured

now for the trick we can combine these

two triangular arrays to make an N by n

Square

however if we now look at this Square

row by row we see that each row sums up

to the same value Row 1 is hn Row 2 is

hn and so on each row sums to the value

H sub n

but in this case there are n total rows

and therefore the sum of all the entries

is given by n times H sub n

but now we have an interesting identity

we see that if we sum up the harmonic

sums where K ranges from 1 to n minus 1

we have n times H sub n minus n

this is a formula for the sum of

harmonic sums but wait there's more that

we can do with this identity

we just proved visually that the sum of

harmonic sums where K ranges from 1 to n

minus 1 is equal to n times H sub n

minus n

we can get an equivalent identity by

adding H sub n to both sides

a theorem due to Ernesto Cesaro says

that if we have an infinite series that

converges with some s and we consider

the sequence of partial sums given by S

Sub n then the sequence of average

partial sums shown here also converges

to the same sum s

let's combine these two facts to prove

something interesting about the harmonic

series let's assume that the harmonic

series which is the sum where K ranges

from 1 to Infinity of the fractions 1

over K converges so that it has a sum s

this means that the limit as n

approaches Infinity of H sub n is equal

to s

then according to the theorem we have

that s is equal to the Limit as n

approaches Infinity of the average

partial sums but according to our

formula this limit can be replaced by

the limit as n approaches Infinity of H

sub n minus 1 plus h sub n over n and

this limit is equal to S minus 1 because

H sub n approaches s 1 approaches 1 and

N grows without bound

therefore we've proved that s equals s

minus 1.

but this is a contradiction and that

means that our assumption was incorrect

so that in fact the harmonic series

diverges

Harmonic Sums and Divergence of Harmonic Series (visual proof)

This is a short, animated visual proof showing a formula for the sum of harmonic sums using triangular diagrams. As a bonus feature (watch until the end), we utilize this formula along with a theorem due to Ernesto Cesáro to prove that the Harmonic Series diverges. #math #manim #harmonicseries #harmonicsum #divergent #diverges #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof

If you liked this video, please consider subscribing to the channel and checking out my other videos.

For a static version of the visual proof about Harmonic sums, see Roger Nelsen's "Proof Without Words II: More Exercises in Visual Thinking" compendium (page 116) : https://bookstore.ams.org/clrm-14/. To see the result about the divergence of the Harmonic series, check out this paper by Steven J. Kifowit https://stevekifowit.com/pubs/harm2.pdf (proof 39).

To learn more about animating with manim, check out:
https://manim.community
___________________________________
Music in this video:
Cold Blue by Astron


one way to picture the number n cubed is

to lay down n copies of an N by n Square

side by side flat if we use this

representation here we get a diagram

that represents the sum of the first n

cubes in this case we've shown n equals

four no matter what cubic stack we start

with we can now take four copies and

strategically shift and rotate them as

shown if we do that right we fill in a

perfect square with side length N

squared plus n therefore four copies of

the sum of the first n cubes must be

equal to the square of N squared plus n

this means that the sum of the first n

cubes must be equal to the quantity N

squared plus N squared divided by four

Summing cubes with lines of squares

This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

To see a longer, wide format, wordless (and perhaps more epic) version, see the original: https://youtu.be/Ye9OPNqV9FA

This animation is based on a visual proofs by Warren Lushbaugh (from an article by Solomon Golomb) in the May 1965 issue of the Mathematical Gazette (https://doi.org/10.2307/3612319) and, independently, Antonella Cupillari from the October 1989 issue of Mathematics Magazine page 259 (https://doi.org/10.2307/2689765).

#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #calculus #sum #induction

To learn more about animating with manim, check out:
https://manim.community


hello everyone in this video we are

going to give the visual proof of a + b

square let we draw a square with side

length a definitely the area of this

square is equal to a square and having a

rectangle with the side length A and

B its area is equal to AB we draw

another square with side length as B

then area of this square is equal to b

square and definitely this is is a

rectangle with side length A and B and

having area is equal to a so this

complete figure is a square with side

length as a + b and its area is equal to

a + b s which is equal to sum of these

four areas which are these one and after

simplification we have a + b sare is

equal to a s + 2 a + b² that is the

answer

Visual Proof of (a+b)² 👌👍 #maths

#amsmath #geometry #visualproof #shortsfeed #shortsfeed2024 #education #a+bsquare #mathematicspuzzle


[موسيقى]

y

[موسيقى]

#trending #mathematics Visual Proof: Factors of Quadratic Equation Polynomial PART-4 x^2-2x-3

Other Videos
Fibonacci Sequence, Golden Ratio, How to draw Golden Spiral:
https://youtu.be/yHBa3EvDPco
Radius and Tangent Vectors:
https://youtube.com/shorts/KHSafRET3Sk
De Morgan's laws: Proof by Venn Diagrams:
https://youtube.com/shorts/pR-REo4aBYk?feature=share
Visual Proof of Roots of Quadratic Equation
https://youtube.com/shorts/FdE7rHDZRd8
Visual Proof: Factors of Quadratic Equation PART-1 x^2+3x+2
https://youtube.com/shorts/lIqv5zKAKKE
Visual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2
https://youtube.com/shorts/5pRYbKFWjUM
Visual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3
https://youtube.com/shorts/AzAsz5itOfQ


here's a geometric way to compute the

infinite sum of the powers of 1 6 1 6

plus 1 6 squared plus 1 6 cubed and so

on take a circle of Area 1 and draw a

smaller circle inside with Area 1 6.

then the annulus around that Circle can

be divided into five equal pieces all of

size 1 6. we can shade one of them to

represent the Area 1 6. now we can

repeat the process on the inner 1 6

circle taking 1 6 of 1 6 and decomposing

the annular region into five regions of

area 136 or 1 over 6 squared continuing

this process indefinitely produces a

region whose area represents the

infinite sum of the powers of one-sixth

but we also now see that the Shaded area

can be replicated four more times to

fill in this circle therefore the sum of

the powers of 1 6 represents one-fifth

of the full circle

Infinite sum of powers of sixths!

This is a short, animated visual proof showing the sum of the infinite geometric series of powers of 1/6 (starting with 1/6) is 1/5 using a general method of dissecting a circle into a small circle and annular sectors.

For a different version using polygons (and without words) check out : https://youtu.be/o0AyHmRV8RA

This animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (https://www.jstor.org/stable/27646594).

To learn more about animating with manim, check out:
https://manim.community

#manim #math #mathshorts #mathvideo  #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #calculus #series #geometricseries #infiniteseries  #circle


foreign

[Music]

Visual math

What is the sum of all numbers in a times table?

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start with two positive real numbers a

and b and draw a rectangle with width

root B plus root B and length root a

plus root a inside this rectangular

diagram we can shade two sub rectangles

like this the unshaded area in this

rectangle is then given by two < of B by

of a rectangles for a total unshaded

area of 2 * the of a * B but now we can

cut those two shaded rectangles into two

triangles each and slide those triangles

around the rectangle like this the

resulting unshaded area is now a

parallelogram we can call this angle

Thea and the side lengths are both the

of a plus b by the Pythagorean theorem

using the formula for the area of a

parallelogram we see that the unshaded

area is theun of a plus b^ 2 * the S of

theta but now in this equality we see

that the S of theta is always less than

or equal to 1 so that 2 * the S of a * B

is less than or equal to a + b or

equivalently that the of a * B is less

than or equal to a + b / 2 we get

equality if and only if s of theta

equals 1 so that the angle is 90° and a

equals B this is the arithmetic mean

geometric mean inequality

AM-GM Inequality via negative space

This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using algebraic areas and the Side-Angle-Side formula for a parallelogram.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

For other visual proofs of this fact see:

https://youtu.be/Pt8IX_Q5U1A
https://youtu.be/62G9fak1vyk
https://youtu.be/HFGSsPbYc-s

For a nearly identical visual proof of another inequality (Cauchy-Schwarz) see
https://youtu.be/mKg_gVagHy8

This animation is based on a visual proof that appears in Icons of Mathematics by Claudi Alsina and Roger B. Nelsen (MAA, 2011):

#mathshorts #mathvideo #math #amgminequality #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #calculus #inequality

To learn more about animating with manim, check out:
https://manim.community


let's see two proofs of the Pythagorean

theorem inspired by

uid imagine that we have a right

triangle with legs of length A and B and

a hypotenuse of length C we can draw a

square off the leg of length a and this

Square encloses an area of a * a or a

squ similarly we can draw a square off

the leg of length B enclosing an area of

b^ s we can also draw a square off the

hypoten

and this has an area of c^ 2 in the

picture diagram the Shaded area is a 2 +

b^ 2 ID realized that we can Shear

squares and we maintain the same area we

can also rotate shapes to maintain area

so if we Shear both squares along the

opposite leg of the triangle rotate 90°

and then Shear again down this vertical

line where they meet we end up filling

the C by C Square Off the hypotenuse

this means that A2 + B2 = c^2 where A

and B are the legs of a right triangle

with hypotenuse C this is essentially

ID's proof of the Pythagorean

theorem the shearing and rotation proof

also suggests a dissection proof of the

Pythagorean theorem we can cut the B by

b square like this so that the

Triangular region matches the original

right triangle we can rotate this shape

down and then use the triangle to cut

this region into two shapes dropping

down this top pictured triangle here we

can also cut the a by a square like this

dropping this blue region down into the

original triangle and using the line

shown here to cut another little piece

to fill the last little triangular

region here we rotate 90° and use the

original triangle to create a new small

triangle that then drops down into place

in the C by C Square from this

dissection proof we see that a^2 + b^2

equal c^2 by cutting and moving pieces

around

let's see the original shearing and

rotating proof one more time where we

shear and rotate both rectangles

together you tell me do you like the

shearing and rotating proof better or do

you prefer the dissection that it

implies either way it's nice to have

multiple proofs of one fact and it's

amazing that ukids proof produces the

dissection proof as

well

Pythagorean Theorem XII (visual proof)

This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) following essentially Euclid's proof. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths.  As a bonus, we show how to use Euclid's shear and rotate proof to create a dissection proof of the Pythagorean theorem. Which of these proofs do you prefer?

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

For other proofs of this same fact check out:
https://youtu.be/o2mbN452ywA (compilation)
https://youtu.be/XFHm0Ot3PP0
https://youtu.be/q28H5NY5UVU
https://youtu.be/yfGtbNgcrQ8
https://youtu.be/JRjPsuJ7S-g
https://youtu.be/TsKTteGJpZU
https://youtu.be/ZlGaQdNRdqA
https://youtu.be/uqGjs_n-f7U
https://youtu.be/wWEZ-rUlBDQ
https://youtu.be/I75HYC8TRNY
https://youtu.be/LWVog8hg74I?si=4GSzTLZP7XIWeaxJ

or check out my Pythagorean theorem playlist:

https://www.youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rB

This animation is based on a proof due to Euclid. For a static version of this proof, see Roger Nelsen's first "Proof Without Words: Exercises in Visual Thinking" compendium (page 5) : https://bookstore.ams.org/clrm-1. You can also check out Howard Eves' "Great Moments in Mathematics (Before 1650)" page 31-33.

(I may receive a small commission at no cost to you for these affiliate links):

https://amzn.to/48mjeX1
https://amzn.to/3Tr0OjW

The dissection proof was suggested to me by Glen Whitney (https://studioinfinity.org/), and Glen then found a similar dissection in the compendium of Pythagorean theorem proofs from Elisha Loomis  (Proof 21 on p. 122 ).
(I may receive a small commission at no cost to you for this affiliate link):  https://amzn.to/3RL7rfN

#math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo

To learn more about animating with manim, check out:
https://manim.community


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Surface Area Of Sphere| Proof | Visual | Mathletes |Sunny Sir  #maths #algebra #foryou #satisfying

Surface Area Of Sphere| Proof | Visual | Mathletes | Sunny Sir  #maths #algebra #foryou #satisfying  #cbse #funnyshorts

Do Like Share and Subscribe For more interesting videos

Join our telegram channel: https://t.me/Mathletes_29

Follow us on Instagram: https://www.instagram.com/mathletes19?igsh=MWo2ZzdlY3cxdjg1MA==


oh

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Visual proof of complete the square or vertex form method ￼#maths #visualization #completethesquare


thank you

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Math is easy | math formula visual proof

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given two positive real numbers X and Y

Place lines of length X and Y side by

side to form the diameter of a circle

whose radius is X Plus y over two

at the intersection point of the two

lines create a half chord to the circle

of length Z the triangle formed here

must be a right triangle by thales

triangle theorem and so these two

subtriangles must be similar this means

that Z over X has to be equal to Y over

Z and we can simplify this to see that Z

squared equals x times y so that the

length of Z is equal to the square root

of x times y

notice that no matter the lengths of X

and Y this half chord is always less

than or equal to the radius therefore we

see that the square root of x times Y is

less than or equal to the quantity X

Plus y over 2. the only way that these

two quantities are equal is if x equals

y

this is known as the arithmetic mean

geometric mean inequality

AM-GM Inequality II

This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using Thales theorem . #math #amgminequality #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #calculus #inequality #mathshorts #mathvideo

To see my original, longer form, animation of this proof without words, check out: https://youtu.be/Pt8IX_Q5U1A

This animation is based on a visual proof by Charles D. Gallant from the March 1977 issue of Mathematics Magazine (https://doi.org/10.2307/2689737).

To learn more about animating with manim, check out:
https://manim.community


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hwood that

Summing cubes without words #phonk #phonkmath

This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes by rearranging cubes into a flat square.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

Here is a longer version of this proof : https://youtu.be/YQLicI8R4Gs

Here are three other visual proofs of sum of cubes formulas:

https://youtu.be/Ye9OPNqV9FA
https://youtu.be/NxOcT_VKQR0
https://youtu.be/d1yM6Rq7Tfw

This animation is based on independently discovered, separate visual proofs by J. Barry Love from the March 1977 issue of Mathematics Magazine (https://www.jstor.org/stable/2689727), page 74, and Alan L. Fry from the January 1985 issue of Mathematics Magazine (https://www.jstor.org/stable/2690228) , page 11.

This is my version of a classic visual proof. Here are other renditions:
https://youtu.be/glearwgR1Ls by   @ThinkTwiceLtu
https://youtu.be/YG_gS5vZoyQ by   @Mathocube
https://youtu.be/WY5X_3q80WY by  @Mathologer

#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #calculus #sum #induction

To learn more about animating with manim, check out:
https://manim.community


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Visual proof pythagoras theorem #mathematics #maths Beauty of mathematics


here's a right triangle and let's label

the legs A and B in the hypotenuse C we

can draw and shade an a by a square off

the a leg and we can draw and shade a b

by b square off the B leg like this we

can also draw but leave unshaded a c by

C Square Off the hypotenuse so the

unshaded area in this diagram is given

by c squared but now we're going to

shift and rotate objects around this

shape to produce a different unshaded

area first we use the triangle to

decompose the B by b square into two

pieces like this then we rotate the

original triangle down 90 degrees shift

the original a by a square like this and

then we shift and rotate and then break

along the hinge the original B by b

square filling the diagram now we see

that the unshaded area is given by a

squared plus b squared this means that

in a right triangle with legs of length

A and B and hypotenuse c c squared

equals a squared plus b squared this is

the Pythagorean theorem

Simple Pythagorean Dissection Proof

This is a short, animated (wordless) visual proof of the Pythagorean theorem (the right triangle theorem) using a relatively elementary dissection proof.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

This animation is based on a proof from Grégoire Nicollier, Aljoša Peperko & Janez Šter in the December 2018 issue of Mathematics Magazine (https://doi.org/10.1080/0025570X.2018.1473701 ) pages 382-383.

For other proofs of this same fact check out:
https://youtu.be/q28H5NY5UVU
https://youtu.be/yfGtbNgcrQ8
https://youtu.be/JRjPsuJ7S-g
https://youtu.be/TsKTteGJpZU
https://youtu.be/ZlGaQdNRdqA
https://youtu.be/uqGjs_n-f7U
https://youtu.be/wWEZ-rUlBDQ
https://youtu.be/I75HYC8TRNY

#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo

To learn more about animating with manim, check out:
https://manim.community


let's assume that the square root of 2

is rational so that the square root of 2

equals A over B with A and B integers

this means 2B squared equals a squared

or B squared plus b squared equals a

squared so that there exists an

isosceles right triangle with a side

length of b and a hypotenuse of a we can

swing the side length over to the

hypotenuse like this and create a

subtriangle using the construction shown

we can determine that this subtriangle

is also an isosceles right triangle the

side lengths of this new isosceles right

triangle are a minus B and the

hypotenuse is 2b minus a but now we can

use this new right isosceles triangle as

proof that two copies of a minus B

squared is equal to the quantity 2B

minus a squared but as we saw

geometrically and can verify

analytically two B minus a is a positive

integer less than a and a minus B is a

positive integer less than b so we have

found two smaller positive integers

satisfying the same equation As A and B

we can repeat this process indefinitely

producing a geometric infinite Descent

of positive integers which is a

contradiction so the square root of 2 is

irrational

Root Two is Irrational Visually

In this short, we use a similar triangle argument to prove that the square root of two is irrational by infinite descent.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

For an alternate visual proof of this fact, see this video: https://youtu.be/-dbzvN4jnfY

#irrationalnumbers #realnumbers #manim #math #mtbos  #animation #theorem #visualproof #proof #iteachmath #mathematics #irrational #carpetstheorem #proofbycontradiction #root2 #algebra #infinitedescent

This animation is based on an argument due to Tom Apostol from issue 9 of the 2000 American Mathematical Monthly: https://doi.org/10.1080/00029890.2000.12005280

To learn more about animating with manim, check out:
https://manim.community


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Visual Proof of Pythagorean Theorem

Visual Proof of Pythagorean Theorem


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Geometric Series (Not my video, Mathematical Visual Proofs on Youtube) #facts #maths #geometry


thank you

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Viviani’s theorem visual proof


here's the sum of the first n positive

squares like you've never seen them

before as the levels get higher the

visual proofs get

Wilder turn it

up turn it

up turn it

up turn

it sometimes it's good to

whisper and keep your voice down

low but sometimes you got to risk it and

let the whole world

know don't want to be too

quiet I'd rather get real

loud we going to start a

riot come on show you

how let's get a little bit

louder let's get a little bit

louder got to get a little bit

louder let's get a little bit turn it up

let's just see how high we can go turn

it up no stopping till we lose our

control turn it up until we feel it down

in our

soul let's get your heartbeat

pumping this is the honest truth this is

the honest truth if we get these bodies

jumped we're going to shake it Loose

Shake It

Loose let's get a little bit like

Lou let's get a little bit

louder got to get a little bit

let's just see how high we can go turn

it up no stopping till we lose our

in our soul turn it

up turn it up let's just see how high we

can go turn it up no stop until we lose

our control turn it up until we feel it

down in our soul turn it

up let's get a little bit

louder let's get a little bit

louder got to get a little bit

louder let's get a little

bit let's get a little bit

louder let's get a little bit louder

let's go come on got get a little bit

louder

yeah let's get a little

bit turn it up let's just see how high

we can go turn it up don't stop until we

lose our control turn it up until we

feel it down in our soul turn it

can go turn it up no stop until we lose

our control turn it up until we feel it

down in our soul turn it

up

turn it up let's just see how high we

can go turn it up don't stop until we

lose our control turn it until we feel

it down in our soul turn

it comment down below which level is

your

favorite

Sum of Squares Visual Proofs: From Level 1 to Level 5

This video is a compilation of five different visual proofs showing the sum formula for the sum of the first n positive squares that can be found on my channel. This is a new style of video for me inspired by  @ThatsAmazing 's trick shots at various levels and  @dudeperfect 's soundtracks. I hope you enjoy it. These visual proofs are likely too fast to follow completely in this format, so the originals are linked below.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

For the original animations, see here:
Level 1, Induction Dissection: https://youtu.be/a8j3sBrXchg?si=zH0Wk9McXyJVxsKc
Level 2, Tokens on Triangular Array: https://youtu.be/UqVmocdLFGc?si=IzMGco38Zv-K7w3L
Level 3, Rectangular Dissection: https://youtu.be/-tJhH_k2LaM?si=EgLkd06b7vk9pt9u
Level 4, Pyramidal Stacks: https://youtu.be/WidzHiUFWNA?si=kbjqvkXtKaGoNXBO
Level 5, Stacking Cuboids: https://youtu.be/VYaEGvClg7Q?si=-2UC2ukk_UnAjQNp

You can see short versions of a levels 3 and 4 here:
https://youtu.be/30TaNnzEMDo?si=CovsP9ZGUfJ7mnhz
https://youtu.be/PVH5uxHFIGQ?si=_fqLjLTUZtua_YMa

Original sources:
Level 1: This animation is based on a visual proof by Tom Edgar from the International Journal of Mathematical Education in Science and Technology (https://doi.org/10.1080/0020739X.2022.2131642 ).
Level 2: This animation is based on a visual proof by Sidney H. Kung from the May 1989 issue of The College Mathematics Journal  (https://doi.org/10.2307/2686767) page 205.
Level 3: This animation is based on (independently discovered) visual proofs by Dan Kalman from the March 1991 issue of The College Mathematics Journal (https://www.jstor.org/stable/2686447) and Martin Gardner from the October 1973 Scientific American (https://www.jstor.org/stable/24923225).
Level 4: This animation is based on a visual proof by Man-Keung Siu from the March 1984 issue of Mathematics Magazine (https://www.jstor.org/stable/2689589), page 92.
Level 5: This animation is based on a visual proof by Sasho Kalajdzievski from the June 2000 issue of The Mathematical Intelligencer (https://doi.org/10.1007/BF03025282 pages 47--49 ).

#math #manim #mathvideo #sumofsquares #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #induction

To learn more about animating with manim, check out:
https://manim.community


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how surface of area of sphere is calculated (visual proof) || Fs bukhari Official #maths #mathlogic


मैथ्स का बहुत ही फेमस फॉर्मूला ए + b² =

a² + b² + 2ab यह आखिर कहां से डेरिव हुआ

समझिए इस वीडियो की हेल्प से वीडियो को

लास्ट तक जरूर देखिए

[संगीत]

[संगीत]

Visual proof of Maths formula (a + b)2. #chalopadhaikare #shorts #maths


start with a regular heptagon and draw

another regular heptagon inside with

Area 1 8 of the original this decomposes

the heptagon into eight pieces of size 1

8. one of them a trapezoid at the bottom

represents 1 8. now let's continue this

again on the inner heptagon now we have

another shaded region of 1 8 times 1 8

or 1 8 squared we can continue this

process indefinitely each time we cut up

the heptagon into eight equal area

pieces and take one of them this means

that each piece we have is 1 8 of the

previous 1 8 of the area

if we do this forever we see that this

shaded region now has an area

represented by the sum of the powers of

1 8. but on the other hand we see that

in terms of the full picture this shaded

area is exactly 1 7 of the entire

heptagon therefore the sum of powers of

1 8 is equal to 1 7.

Summing powers of 1/8 visually!

This is a short, animated visual proof showing the sum of the infinite geometric series of powers of 1/8 (starting with 1/8) is 1/7 using self similar trapezoids in a regular heptagon. #manim #math #mathshorts #mathvideo  #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #calculus #series #geometricseries #infiniteseries  #heptagon #trapezoids

Here is an alternate dissection proof for this series: https://youtu.be/wPgwgoj_aKA

Here is a longer form video with lots of related infinite geometric sums: https://youtu.be/JteQEN1XPyc

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

This animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (https://www.jstor.org/stable/27646594).

To learn more about animating with manim, check out:
https://manim.community


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(a+b)^3 Visual Proof #visualproof #education #mathematics #algebra #identities #algebraicidentities


let's represent the sum of the first n

cubes using cubic stacks of Cubes like

this here we pictured 1 cubed plus two

cubed plus three cubed and so on in this

particular instance we stopped at n

equals 4 but we can imagine a similar

stack for any n now we decompose each

cubic stack into an L-shaped flat nomon

the levels of the odd cubes break up

perfectly to fit around the previous

gnomon but the even cubes don't work

quite as well instead we have to take

one level and split it into two separate

pieces as shown here when n is four once

we've broken up each Cube into a flat

nomon we see that all the small cubes

have been arranged in a perfect square

where the side length is given by the

sum of the first and positive integers

this means that the sum of the first n

positive cubes is equal to the square of

the sum of the first n positive integers

Visual Cube Sums! (from Nicomachus)


imagine two cubes one with a side length

a and one with a side length B the

number a cubed plus B Cubed is given by

the volume of these two cubes let's add

an A plus b by a by B rectangular prism

to this diagram which encloses a volume

of a plus b times a times B now we can

decompose this region into two different

cuboids one of them the shorter and

deeper green one shown can be rotated

and shifted as pictured notice that

these two cuboids sit inside an A plus b

outer shell and if we look at the top of

this diagram we see an a by a square and

a b by B squared so the area is a

squared plus b squared therefore the

volume of these two shapes is a plus b

times a squared plus b squared but now

we can reshift the cuboids together and

remove the region that we originally

added by subtracting that from both

sides of the equation therefore a cubed

plus B Cubed is equal to a plus b times

a squared plus b squared minus a times B

Factor a sum of cubes

This is a short, animated visual proof of showing how to factor a sum of two cubes using two actual cubes sitting inside a larger cube. #manim #math #mathvideo #factor #cubes #mathshorts #geometry  #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #algebra #pww  #proof #areas #mathematics #mtbos
If you like this video, please click "like" and consider subscribing and checking out my other videos.

To buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs
Thanks!

Here is the original wide form video without words (that moves a bit slower and with dramatic music): https://youtu.be/BQBnA9d-wAk

Here are two related videos:
https://youtu.be/2_vqruzQzkc
https://youtu.be/IAsaNUUrfj4

To learn more about animating with manim, check out:
https://manim.community


[Music]

let a and b both be real numbers with a

greater than or equal to b and b greater

than or equal to zero consider a line

segment with length a

and then overlay a line segment of

length b like this

use half of the uncovered length of the

first segment a length of a minus b over

two

to create a circle with a radius of a

minus b over two

[Music]

the yellow line segment created here

connecting the end of the line segment

to the center of the circle has length a

plus b over 2.

draw a radius perpendicular to the line

segment and connect the endpoints of

this radius and the end of the extended

line segment

the resulting triangle is a right

triangle so we can use the pythagorean

theorem to determine that the hypotenuse

has a length of square root of a minus b

over 2 squared plus a plus b over two

squared which simplifies to the square

root of a squared plus b squared over

two

next draw the tangent line to the circle

that passes through the endpoint of the

extended line segment and draw a radius

to this point creating another right

triangle highlighted here

we again use the pythagorean theorem to

determine that the missing leg has a

length of square root of a plus b over 2

squared minus the quantity a minus b

over 2 squared which simplifies to the

square root of a times b

finally drop a perpendicular from the

tangent point on the circle to the

extended line segment

one of the newly created triangles in

the subdivision is similar to the

original triangle

we can see that they're similar because

they're both right triangles and they

share one of the other angles

using the similarity of these triangles

we can find that the leg labeled x

satisfies the equality x over the square

root of a times b equals the square root

of a times b over a plus b over two

we can solve for x to see that x is two

times a times b over a plus b

[Music]

working backward we see that two times a

times b divided by a plus b is the leg

of a right triangle with hypotenuse the

square root of a times b so two a b over

a plus b is less than or equal to the

square root of a times b but the square

root of a times b is the leg of a right

triangle with hypotenuse a plus b over

two so the square root of a times b is

less than or equal to a plus b over two

and then a plus b over two is a leg and

a right triangle with hypotenuse the

square root of a squared plus b squared

over two

so a plus b over two is less than or

equal to the square root of a squared

plus b squared over two

two a b over a plus b is the harmonic

mean

the square root of a times b is the

geometric mean and a plus b over 2 is

the arithmetic mean and the square root

of a squared plus b squared over 2 is

the root mean squared the resulting set

of inequalities is known as the mean

inequalities we note that we get

equality throughout if and only if a

equals b in this case the entire picture

collapses to a single line segment

representing all of the values

[Music]

you

Mean Inequalities (visual proof of AM-GM, HM-GM, and AM-QM inequalities)

This short animated proof demonstrates the two variable mean inequalities for the arithmetic mean, geometric mean, harmonic mean and root mean square (or quadratic mean). #math #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #inequality #iteachmath #mathematics #pythagoreantheorem #similartriangles #means #meaninequality #mathshorts #mathvideo #mtbos

This animation of the inequalities is based on a visual proof by Roger Nelsen from the June 1987 issue of Mathematics Magazine (https://www.jstor.org/stable/2689561) p. 158.

If you like visual proofs of inequalities, see these links:

AM-GM inequality via rectangles/squares: https://youtu.be/62G9fak1vyk

AM-GM inequality via circles: https://youtu.be/Pt8IX_Q5U1A

AM-GM-HM-QM via Moments: https://youtu.be/XJ3643WhKjY

To learn more about animating with manim, check out:
https://manim.community


these Stacks each represent sums of n

consecutive positive integers the first

one from 1 to n the second from 2 to n +

1 all the way up to the last one which

goes from n up to 2 N minus one when we

glue these Stacks together we get this

interesting shape here we can take a

second copy of this shape and rotate it

twice once the second copy is rotated

appropriately we see that the two copies

fit together perfectly to form a cuboid

with Dimensions n by n by 2 N that means

the number of cubes in the original

stack we started with is 12 * 2 n * n *

n this equals n cubed this means that

the sum of the consecutive sums of n

positive integers that we started with

equals n cubed so that we've uncovered

an interesting double sum formula for

cubic numbers

Double sum produces cubes

In this video, we demonstrate how each integer cube is a special double sum.  In particular each cube is a sum of n sums of n consecutive integers.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

To see this animation a bit slower (and wordless), check out: https://youtu.be/_dltGrVlW3w

Here are a few other related visual proofs:
https://youtu.be/OECW5qqZN9I
https://youtu.be/936eKowyvR4
https://youtu.be/WidzHiUFWNA
https://youtu.be/BBrtHbRcpXs
https://youtu.be/YQLicI8R4Gs

This animation is based on a visual proof by Roger B. Nelsen from the March 2002 issue of the College Mathematics Journal  (https://www.jstor.org/stable/1559005 page 171 ).

To learn more about animating with manim, check out:
https://manim.community

#math #manim #mathvideo #cubes #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #finitesums #discretemath #induction #doublesum


start with an equilateral triangle and

cut it into nine congruent equilateral

triangles and shade four of them thus

shading 49ths of the area of the

equilateral triangle now repeat this

process on the top four nths cutting it

into nine pieces and shading four of

them so that we've shaded four nths of

four kns of the area we can repeat this

process over and over again each time

taking the unshaded top four kns from

before cutting it into nine pieces and

shading four of them if we imagine

carrying this process out infinitely

often then we get a diagram that

represents the infinite sum the infinite

sum is a limit so in this limiting

diagram we've shaded the infinite sum of

the powers of 49ths on the other hand if

we look at each level along the way we

see that in each level we've shaded four

of five congruent triangles therefore

the infinite sum of the powers of 49ths

must equal 4 fths

Infinite sum of powers of 4/9

This is a short, animated visual proof demonstrating the sum of the infinite geometric series with ratio 4/9 using a decomposition of an equilateral triangle.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

This animation is based on a visual proof by Tom Edgar from the June 2016 issue of Mathematics Magazine page 191 (https://doi.org/10.4169/math.mag.89.3.191).

#mathshorts #mathvideo #math #calculus
#mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricseries #infiniteseries #fractals #sum #induction

To learn more about animating with manim, check out:
https://manim.community


[Music]

[Music]

he

[Music]

(a+b)² Expanded Visual #Proof

This video was made with Clipchamp
#clipchamp #proof #math


[Music]

thank you

[Music]

[Music]

foreign

[Music]

[Music]

[Music]

Sum of First n Even Integers | Visual Proof | Geometric Proof | Proofs Without WORDs

Thanks for watching!

#shorts #viral #viralshorts #shortsvideo #mathtricks #math #mathematics #mathideas #mathanimation #animation #animated #visualization #geometric #proof #proofwithoutword #algebric #algebra #algebricequation #algebricexpression #visualproof #geometricproof #amazingmath #funfacts #mathfacts #abstractart #abstractmath #mathvisualization #mathart #geometry #shapes #equation #expression #square #proofing #proofof #proofofconcept #amazingahorts #amazingfacts #factshorts #intution #intutive #funmath #mathideas #mathworld #magicmath #magicmathtricks #mathmagic #mathzone


to prove that the set of integers is

accountable set we have to find a

one-to-one and onto function from the

natural numbers to the integers to do

that we imagine the integers form a

doubly infinite line of dots and we

start at zero and spiral out like this

we see that every integer will be hit

once and only once to make this more

formal we can Define the function f of x

equal to X plus one over two if x is odd

and negative X over 2 if x is even and

this gives the required one to one and

onto map from the natural numbers to the

integers

we can do something similar to prove

that Z Cross C is a countable set we

plot the integer coordinate points in

the real plane as shown beginning at

zero zero Des McHale showed this spiral

which will eventually visit every single

integer coordinate point in the real

plane therefore this spiral proves the

existence of a one to one and onto map

from the natural numbers to pairs of

integers can you find a formula for the

one to one and onto map guaranteed by

this spiral

Two Countable Sets

This is a short animation with two animations showing that the set of integers is a countable set (in bijection with the set of Natural numbers) and that the set of ordered pairs of integers is also a countable set.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

This animation is based on a visual proof by Des MacHale from the February 2004 issue of Mathematics Magazine  (https://www.jstor.org/stable/3219231 page 55).

Here is a related video proving that function composition is associative:
https://youtu.be/mCnjpnhLeDU

#shorts #visualproof #mathshorts #mathvideo #math #bijection #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #function #injection #surjectivefunction #injectivefunction #surjective #injective  #naturalnumbers #integers #countableset #infinity

To learn more about animating with manim, check out:
https://manim.community


let's play the chaos game in a pentagon

first start with a DOT anywhere inside

the Pentagon then we're going to iterate

a procedure where at any given stage we

move

0.618 of the distance between the

current dot and a randomly chosen vertex

on the Pentagon think of rolling an

imaginary five-sided die and then moving

towards the corresponding vertex what

happens when we iterate this process

30,000 times in this picture we've

shaded the dots corresponding to the

vertex to which we move at any given

stage after 30,000 iterations if we

throw away the first few iterations we

see that the limiting shape will always

look like this diagram no matter where

we started this is an amazing fractal

generated from a simple process note

that the number 0.618 is special it's

approximately one over the golden ratio

Pentagonal Chaos Game

In this short, we show what happens when iterating the procedure of choosing a pentagon vertex at random and moving 0.168 (golden ratio reciprocal) the distance from the current dot to the chosen point.

If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!

If you want to know more about the Chaos game, see the following links:
https://en.wikipedia.org/wiki/Chaos_game
https://mathworld.wolfram.com/ChaosGa...
https://math.bu.edu/DYSYS/chaos-game/...
https://youtu.be/kbKtFN71Lfs
https://youtu.be/IGlGvSXkRGI
https://youtube.com/shorts/yDHkNDrHfIw

#chaos #chaosgame #pentagon #mathvideo #math #mtbos #manim #animation #theorem #iteachmath #mathematics #dynamicalsystems  #iteratedfunctionsystem #dynamics #fractal #fractals

To learn more about animating with manim, check out:
https://manim.community


here's a way to make geometric sense of

the infinite sum of the alternating

powers of one-third start with a one by

one square of area one cut it into three

pieces and remove one of those now cut

the diagram into three pieces vertically

and use that to re-add one third of the

missing area using these triangular

pieces now we repeat the process on the

Central Square created by the first two

divisions we cut that square into three

pieces remove one third and re-add

one-third with triangular diagrams we

continue this process indefinitely each

time decomposing the Central Square

which has one-ninth area of the previous

large Square using this process Shades

the alternating sum of the powers of

one-third on the other hand we see that

the Shaded area represents exactly three

quarters of the entire Square therefore

the infinite sum of the powers of

negative one-third is equal to

three-fourths

[Music]

Alternating infinite sum!

This is a short, animated visual proof demonstrating the sum of the infinite geometric series with ratio -1/3. #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricseries #infiniteseries #fractals #sum #induction #mathshorts #mathvideo

If you like this animation, please consider subscribing.

Here is my original version (longer, wide format, with no words): https://youtu.be/MyCn7gIPxNg

This animation is based on a visual proof by Hasan Unal from the January 2009 issue of The College Math Journal page 39 (https://www.jstor.org/stable/27646717).

Here is an alternating series dissection for powers of -1/2: https://youtu.be/NKFrHfBGDNE

After I made this video, I found the proof animated very nicely on Think Twice's channel too: https://youtu.be/re3Xh74M8g0

To learn more about animating with manim, check out:
https://manim.community