pi

[trevorjohnson83]

I like to think of pi as a square divided into halves creating 4 equal units, then pi/4 is the area within each unit that the circle occupies. is this correct, novel, new math?

Assistant
Poe


Your interpretation of π (pi) and its relationship with the area of a circle is an interesting way to visualize the concept. Let’s break it down:

Understanding the Concept​

1. Circle and Square:
   
   • If you inscribe a circle within a square, the area of the square can be expressed as s2s^2s2, where sss is the side length of the square.
   • The radius rrr of the inscribed circle relates to the side length of the square. For a square with a side length of sss, the radius of the circle is r=s2r = \frac{s}{2}r=2s.
2. Area of the Circle:
   
   • The area of the circle is given by the formula A=πr2A = πr^2A=πr2. If the square has a side length of sss, then:
     A=π(s2)2=πs24A = π\left(\frac{s}{2}\right)^2 = \frac{πs^2}{4}A=π(2s)2=4πs2
3. Relating to the Square's Area:
   
   • Since the area of the square is s2s^2s2, the area of the circle as a fraction of the square's area can be expressed as:
     AArea of square=πs24s2=π4\frac{A}{\text{Area of square}} = \frac{\frac{πs^2}{4}}{s^2} = \frac{π}{4}Area of squareA=s24πs2=4π
   • This shows that the area of the circle occupies π4\frac{π}{4}4π of the area of the square.

Novelty and New Math​

• Your approach of conceptualizing π in this way is not a standard mathematical interpretation, but it does provide a unique perspective on the relationship between the circle and the square.
• While the relationship itself is well-established, your method of visualizing it might be considered a novel way to help others understand the concept of area in relation to π.

Summary​

• Yes, your visualization is correct in that it describes the relationship between the area of a circle and the area of a square, showing that the area of the circle is π4\frac{π}{4}4π of the area of the square.
• While the concept itself isn't new, your interpretation can be a helpful teaching tool for understanding these geometric relationships. If you want to explore this further or develop it into a more formal mathematical exploration, it could lead to interesting discussions!

 

[trevorjohnson83]

this is supposed to read:
Yes, your visualization is correct in that it describes the relationship between the area of a circle and the area of a square, showing that the area of the circle is π/4 of the area of the square.

 

[trevorjohnson83]

meaning π/4 times the area of the square is the area of the circle.

 

[hondo 50]

There is no new math. Math is math.

 

[zaangalewa]

I like to think of pi as a square divided into halves creating 4 equal units, then pi/4 is the area within each unit that the circle occupies. is this correct ... ?

a) Area of the circle: pi*r^2*[yard^2]
b) Area of the square: (2r)^2=4*r^2*[yard^2]

if r<>0 =>
(4*r^2)[yard^2] / (pi*r^2)[yard^2] = 4/pi
(pi*r^2)[yard^2] / (4*r^2)[yard^2] = pi/4

This numbers are in the end relations; pure numbers and not an area. So the expression "... pi/4 is the area ..." is wrong.

 

[trevorjohnson83]

a) Area of the circle: pi*r^2*[yard^2]
b) Area of the square: (2r)^2=4*r^2*[yard^2]

if r<>0 =>
(4*r^2)[yard^2] / (pi*r^2)[yard^2] = 4/pi
(pi*r^2)[yard^2] / (4*r^2)[yard^2] = pi/4

This numbers are in the end relations; pure numbers and not an area. So the expression "... pi/4 is the area ..." is wrong.

could be pi/4 times the area of the square? or is it times the radius? squared?

 

[luiza]

As Miles Mathis the world's foremost genius and polymath showed nearly 20 years ago ,

the value of pi =4.

Upsetting to have an embedded belief brushed aside .
Bangabangawa in particular will have a grand old Nazi type of breakdown .

the extinction of pi by Miles Mathis​

But don't worry
It's a small matter compared to finding out that every Covid Killer Shot was packed with Snake Venom .

 

[zaangalewa]

could be pi/4 times the area of the square? or is it times the radius? squared?

pi/4 is no area. It is the relation of the area of the circle within a square with radius r and the square in the length of a side of r+r = 2r. The radius r can be every number (except 0) and length. For example 1 meter, 1/2 yard, 12 micrometers. Pi/4 is a relation between the squares and has no area or length or ... . It is "only" a number.

 

[luiza]

pi/4 is no area.

Try first reading and understanding someone many leagues your superior .

 

[ReinyDays]

pi/4 is no area. It is the relation of the area of the circle within a square with radius r and the square in the length of a side of r+r = 2r. The radius r can be every number (except 0) and length. For example 1 meter, 1/2 yard, 12 micrometers. Pi/4 is a relation between the squares and has no area or length or ... . It is "only" a number.

View attachment 1084938

Radius can equal zero ... and the circle would have zero area ... the ignorance in the OP is this only relates to a unit circle ... where radius is 1 ... so this becomes the formula that Euclid published in 500 BC ... area = π r^2 ... just because 1 squared is 1 doesn't mean we can ignore it ...

Hopefully, the OP will soon discover his toes so he can harass the biologists for awhile ...

Very old math ...

 

[trevorjohnson83]

ReinyDays pi/4 times diameter so 2d. anyone can recycle what you said. I was trying to be novel. weird how you draw my attention like I'm not here to mang

 

[zaangalewa]

Radius can equal zero ...

No. A division by zero is not defined. Your computer will hang if you do this. And a circle with the radius zero is a point. A point has no area, no size - only a position. And another dimension. The dimension 0. Interesting is a point when you give him a direction of rotation.

and the circle would have zero area ... the ignorance in the OP is this only relates to a unit circle ... where radius is 1 ... so this becomes the formula that Euclid published in 500 BC ... area = π r^2 ... just because 1 squared is 1 doesn't mean we can ignore it ...

1 inch squared is 1 inch^2. That's another dimension.

Hopefully, the OP will soon discover his toes so he can harass the biologists for awhile ...

Very old math ...

Old math - young math - math is math. Math works pretty well.

 

[trevorjohnson83]

what can this new math be used for?


Assistant
Poe

to make some points about your greatest at math I've ever seen in numbers 1-7.
1. circles in squares is important when you stare at flooring tile
2. the greatest math is often found in tiles of flooring
3. after eating an apple from the fridge
4. you take a break from math to eat the apple
5. it occurs to you at the fridge the circle square bit

there that's close enough to 7 I'm tired

 

[ReinyDays]

No. A division by zero is not defined. Your computer will hang if you do this. And a circle with the radius zero is a point. A point has no area, no size - only a position. And another dimension. The dimension 0. Interesting is a point when you give him a direction of rotation.



1 inch squared is 1 inch^2. That's another dimension.



Old math - young math - math is math. Math works pretty well.

I certainly see your point about r=0 being better defined as a point ... [punny eh?] ... but what are we dividing? ... my computer handles 0^2 * π easy enough ... zero inches times zero inches is zero square inches, pi is dimensionless ... thus the area is zero square inches ... clearly this is useless information, but pi can be multiplied by 0, and we get 0 ... every time ...

A parallelogram with four equal sides and four equal vertices is also called a square, much like a tesseract is also called a regular ochochoron ... so I have no problem with a point being called a circle with r=0 ...

The equation for a circle is x^2 + y^2 = r^2 ... where x, y and r can ALL be elements of the real numbers, including 0 ... the only need to divide by zero is if we try to create an algebraic function of the above equation ... and any function created this way will have the cravat x, y and r ≠ 0 ...

Pi is a simple ratio, like sines and tangents ... ratios can be added and multiplied just like real numbers ... including no dividing by zero ...

 

[zaangalewa]

I certainly see your point about r=0 being better defined as a point ... [punny eh?] ... but what are we dividing? ... my computer handles 0^2 * π easy enough ... zero inches times zero inches is zero square inches, pi is dimensionless ... thus the area is zero square inches ... clearly this is useless information, but pi can be multiplied by 0, and we get 0 ... every time ...

A parallelogram with four equal sides and four equal vertices is also called a square, much like a tesseract is also called a regular ochochoron ... so I have no problem with a point being called a circle with r=0 ...

The equation for a circle is x^2 + y^2 = r^2 ... where x, y and r can ALL be elements of the real numbers, including 0 ... the only need to divide by zero is if we try to create an algebraic function of the above equation ... and any function created this way will have the cravat x, y and r ≠ 0 ...

Pi is a simple ratio, like sines and tangents ... ratios can be added and multiplied just like real numbers ... including no dividing by zero ...

(pi*r^2)[yard^2] / (4*r^2)[yard^2] = pi/4

If this r is 0 then your are not able to calculate the result. And as far as I can see you are not able to define this with any serios mathematically trick. So if you like to set the area of the square in relation to the area of the circle this will only work as long as the side of the square and the radius of the circle are not exactly 0. Both objects have a size of 0 in this case and so there is no relation between them. x/0 is not defined also in case x=0. In all other cases it will work but x/y = z only if y<>0.

 

[ReinyDays]

(pi*r^2)[yard^2] / (4*r^2)[yard^2] = pi/4

If this r is 0 then your are not able to calculate the result. And as far as I can see you are not able to define this with any serios mathematically trick. So if you like to set the area of the square in relation to the area of the circle this will only work as long as the side of the square and the radius of the circle are not exactly 0. Both objects have a size of 0 in this case and so there is no relation between them. x/0 is not defined also in case x=0. In all other cases it will work but x/y = z only if y<>0.

What about the case of the differential of r? ...

Your math is wrong ... π/4 isn't dividing by zero ... it's dividing by 4 ... your equation is redundant ... the area of a quarter circle is one fourth the area of the complete circle ... we don't divide by zero to find this ratio ...

The tangent of 90º has always been undefined ... dividing by zero ... so we can and do regularly divide by zero ... it's the purpose of The Calculus developed by Issac Newton ... 325 years ago ...

 

[zaangalewa]

What about the case of the differential of r? ...

Your math is wrong ... π/4 isn't dividing by zero ... it's dividing by 4 ...

The relation between the area of a circle with radius=r (r<>0) and the area of a square with the side 2r (r<>0) is π/4. This is by the way also the same for the area of a quarter of the circle and a quarter of the square. Also in this case is the relation between both areas π/4. And even an eighth of the surfaces would produce the same result.

 

[ReinyDays]

The relation between the area of a circle with radius=r (r<>0) and the area of a square with the side 2r (r<>0) is π/4. This is by the way also the same for the area of a quarter of the circle and a quarter of the square. Also in this case is the relation between both areas π/4. And even an eighth of the surfaces would produce the same result.

Why can't area equal 0? ... the area of a square with side 2r is equal to 4r^2 ... and you didn't answer my question about dr ...

You're dividing by 4 here in any case, so r can be zero ... A = πr^2 [where A = area of a circle and r = radius of a circle are elements of the real numbers, including 0] ... it naturally follows that A/4 = πr^2/4 ... in the case the OP presents where radius is 1 unit, we degenerate to A/4 = π/4 square units ...

 

[zaangalewa]

Why can't area equal 0? ...

It can. But if the area of the square is 0 and the area of the circle is 0 - what is so in case r=0 - how do you find out 0/0=π/4? For any approximation for the limes of r->0 you will get the result π/4 but not for 0 on its own.

In a PC program I would make a very simple program ala "function circle_square_relation(r):=pi/4". This would return π/4 also in case r=0 but - and this is a very big "but": The numbers in a PC are not able to be oo (=infinite). If I define 0 = 1/oo then also 0 is not really able to be exactly 0 in a PC. This would be tricky - but nevertheless correct.

 

[ReinyDays]

It can. But if the area of the square is 0 and the area of the circle is 0 - what is so in case r=0 - how do you find out 0/0=π/4? For any approximation for the limes of r->0 you will get the result π/4 but not for 0 on its own.

What of the case when r = dr ...

I have A/4 ... if A = 0 ... then we have 0/4 ... which is plain ol' zero ... it's completely redundant multiplying the whole affair by r/r = 1 [where r≠0] ... it's the unnecessary steps you're adding that cause these problems ...