The magic and mystery of "pi" | Real numbers and limits Math Foundations 93 | N J Wildberger
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- The number "pi" has been a fascinating object for thousands of years. Intimately connected with a circle, it is not an easy object to get hold of completely rigourously. In fact the two main theorems associated to it--the formulas for the area and circumference of a circle of radius pi--are usually simply assumed to be true, on the basis of some rather loose geometrical arguments in high school which are rarely carefully spelt out.
Here we give an introduction to some historically important formulas for pi, going back to Archimedes, Tsu Chung-Chi, Madhava, Viete, Wallis, Newton, Euler, Gauss and Legendre, Ramanujan, the Chudnovsky brothers and S. Plouffe, and culminating in the modern record of ten trillion digits of Yee and Kondo. And I also throw in a formula of my own, obtained from applying Rational Trigonometry to Archimedes' inscribed regular polygons.
It should be emphasized that the formulas here presented are not ones that can easily be rigorously justified, relying as they do on a prior theory of real numbers and often Euclidean geometry. The lecture ends with some speculations about the future role that "pi" might play in our understanding of the continuum--a huge problem which is not properly appreciated today.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics.
Video Content:
00:00 Intro to the magic of "pi"
3:36 "Pi" is usually defined by area or circumference
7:27 Logical difficulty
10:22 Brief history of "pi"
13:58 The first formulas of "pi"
19:21 Formulas for "pi" discovered by Newton
23:45 "Pi" formula by S. Ramanujan (1914)
28:22 Page 269 of 'Divine Proportions'
30:59 Irrational real numbers
34:30 My attitude to "pi"
38:03 "Pi" is not a real number, it's a meta number
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Dr Wildberger's lecture has opened up a new dimension in my thinking.
Thank you, sir! I have been struggling with the idea of pi for many months and every time I fell into a trap of circular (litterally) reasoning. Now I can see why it happens. Best regards!
I personally appreciate your effort mr. Wildberger! Thanks for your valuable time and efforts.
I want to thank you very much for the valuable information you have exposed in your video. I am fascinated by your knowledge. Please take care of your health, and keep up with your good job. Lenin
Lenin Carrion Thanks Lenin.
Fantastic teaching
Thanks for the illuminating video! After finishing, I have one nagging question. If we went back in time, say 100 years, would our current value of pi with over a trillion decimal places still be said to “exist” as a more accurate version of pi? Similarly, if we find a much more expansive approximation for pi in the future, could it be said to “exist” now even though no one has found it yet? This might be a variation of the question, “is math created or discovered”, but I’m very curious on your take! Thanks for the thought provoking content!
Amazing! Thank you for putting this together!
God bless you soo much for your efforts and devotion to educating the world freely.
You are my hero
Lol god
Loved every minute. You are so clear!
Great stuff!!!!!! Love it. Mathematics is my religion.
this is the video i was searching for. Thank you.
I've been looking forward to this video for a long time!
Dear Sir ,
It happened that I have watched MF93 today 6/10/2014 . I want by this to congratulate you for your precious efforts and simple techniques you are using in presenting your distinguishable ideas . I am in total agreement with your opinions about PI and your expected better future explanation . The different quantum mechanics used to quantify the magnitude of PI were never convincing me . In my opinion there is one relation exist between PI ,Square root of 2 and square root of 3 that we should look at !
Here is the equation :
PI = Square root (3) + Square Root (2)..................... ( A )
And this implies the following :
(PI) + (1/PI ) = 2 x Square root (3).
( PI X PI )+ 1/ ( PI X PI ) = 10
If we are going to assume any magnitude for any of the above three irrationals ( i.e PI , Sqrt (2) and Sqrt (3) ,then the magnitude of the other two irrationals need to be interlocked and calculated through equation ( A ) which is not the case now were every irrational was estimated independently which increases more our uncertainty and distort more quantum accuracy.
I hope to find the above in the same direction and I am looking forward to her from you .
Best regards,
njawabrah@yahoo.com or gmail.com
Dear Sirs Academics (as) and students (as) In the present era Mr Sidney Silva reports with complete truthfulness that the number of "PI" is no longer Irrational, today it is a rational number, made the fraction and 100% to be exact calculations of Modern Mathematics, some time ago I published a video telling all my great discovery (PI" great discovery), and put on my Facebook page, "A Shot in The Dark"?!, is there who do a tour will see all my "Thesis" in relationship to this great discovery, it will be wonderful to meet all visiting my page, my heartfelt hug Mr Sidney Silva.
Let me tell you something, PI is equal to 3.142696805... this is the square root of 2 times 2.22222..... this value for PI is more accurate than the one commmonly accepted . Please try this value for calculating the area of a circle or the circumference and you will notice is possible to square a circle. For example for a circle with diameter equal to the square root of 162 the circumference is exactly 40.
Sir, thank for an awesome lecture! Really an eyeopener beyond anything I've seen in a long time. Thank you very much it made my day so much better!
loved you lecture...... opened up a new thought process in my mind......
Thanks Ilan Vardi for your link to your interesting article discussing difficulties in defining pi. However there is one additional difficulty, probably the most important one, that you don't address: namely how to set up the theory of `real numbers' in which this discussion ostensibly takes place.
At 12:50 or so, you give the Madhava, Gregory, Leibnitz formula of Pi/4=1-1/3+1/5-1/7....However, if you evaluate this formula for at least half-a-million terms, it will NOT give you the accurate decimal expansion of Pi!! But why? In fact, there will be errors in the in the 6th, 16th, 17th, 29th....etc., decimal places. The important question is WHY? Thanks.
why Tan(deta)=opposite/adjacent
Sol Warda You have to be wrong, if you take the power series for arctan x you get:
x - x^3/3 + x^5/5 - x^7/7 +.... convergent for x in [-1, 1]
Taking x = 1, we have pi/4 = 1 - 1/3 + 1/5 - 1/7 +...
Perhaps you are rounding the terms before adding them, I don't know.
You need to provide your value for 500,000 terms, so we can see.
Colin Java No, Colin: If you evaluate it for the first 5,000,000 terms, you get this answer:3.1415924535897932384646433832795027841971693993873058.The underlined digits are wrong!. The reason is explained in a Wikipedia article. Please type the following in Google Search:"Leibniz formula for π" Please read the section under "Unusual Behavior." Sorry, the underlined digits don't appear in the above expansion, but you will see them in the Wikipedia article.Thanks.
Sol Warda Okay I see the confusion now
Its the way you worded it, you put it as the 6th,17th,29th digits are errors.
its not so much that they are errors, as that the 7th,8th,...16th.18th,... digits just happen to match correctly, making the 6th, 17th, 29th.. digits look the errors.
The point is that the first few digits are perfectly correct, the formula itself is just to slow to use, so I guess you would roughly need a trillion terms just to get the first 12 digits right.
Again.. with a trillion terms, some extra digits will be right as well.
So you said "it will not give you the accurate decimal expansion of Pi!!", so that's why I said you were wrong, it will give it as accurately as you require it, provided you give sufficiently many terms.
But its an interesting observation about those extra matching digits, so thanks for sharing.
I love this series. They are amazing.
brilliant video
Thank you. Each video is more fantastic than the next.
Perhaps in time I can bring you around:)
Fascinating and mind blowing! For some reason this makes me realize more that God does in fact exist.
He doesn't, it's all mind, we are mental entities that are necessary for matter to exist, everything is energy that is expressed mathematically, the formula that represents our minds and its processes is eulers formula, with this equation and Fourier transformation, we can create an entire universe, there are multiple of us every human is a mind or monad, every animal and single cell organism, mind is necessary for existence and Matter is a necessary tool for evolution through this game of life, in your dreams you can create entire worlds! This is a function that is natural to our minds, the imagination, we can literally through mathematical processes do anything and create every math form, even when we speak we speak in a mathematical language, this view of living math is called ontological mathematics or real mathematics, sacred geometry is an example of this, and I guess it can be called God but then that includes you and I as God as well, we are God's we are responsible for this planet and its future we are the ones who know we aren't servants. Ad astra friend.
Thank you for your explanation is very useful
Thanks you a lot for the video, very interesting and valuable.
That's probably a good direction for the hair to go.
After reviewing your video on geometric Pi, I think you may find this information on mathematically calculating the accurate value of geometric Pi interesting.
The mathematical proof of the accurate value of Pi is:
applying the equation,
"inverse sin delta theta radians = Pi x theta degrees/6y"
where:
1. delta theta is 1/12 of 2 x Pi radians;
2. theta degrees is 1/12 of 360 degrees = 30 degrees
3. and 'y' is 0.5.
When the value of Pi = 3.141592654 is applied to this equation,
inverse sin 0.523598778 = 3.141592654 x 30 degrees/(6 x 0.5)
31.57396133 degrees = 94.24777961 degrees/3
31.57396133 degrees = 31.415926654 degrees
the left hand side of the equation is not equal to the right hand side.
Now when Pi is three (3.0), note the answer:
inverse sin 0.5 = 3.0 x 30/(6 x 0.5)
30 degrees = 90/3
30 degrees = 30 degrees
the left hand side of the equation IS EQUAL to the right hand side. This demonstrates that Pi is 3.0. It is also interesting to note that above equation is precisely accurate at 1/12 of the circle at exactly 30 degrees. This information is written a book that explains and proves mathematically; and through a series of "real" life experiments that Pi is equal to three (3.0). The name of the book is "The Great Design Integration of the Cosmic, Atomic, Darmic (Dark Matter) Systems - VOLUME I GEOMETRIC Pi". You can download the PDF version of this book for FREE at www.thegreatdesignbook.com. The book consists of seven (7) volumes, and Volume I focuses on GEOMETRIC Pi.
thank very interesting - u give me inspiration, very helpful
That's a bit difficult to answer. The limit approach has serious problems, which I will be outlining shortly. The usual infinitesimal approach, via so called non-standard analysis, is also highly problematic. However there is a much simpler infinitesimal approach which I will be telling you about in good time. But the key point is that it is possible to derive a lot of calculus purely algebraically-- we'll talk more about that too.
Great video guy. I really thank you for sharing...
Thanks for that.
amazing!!!
Thank you.
Thanks!
I fully agree that the notion of ``number'' is not a precise one in mathematics, and this is an important point. Therefore the statement that ``pi is not a number'' is not really a precise statement. It is meant as a warning that the status of ``pi'', whatever it is, has yet to be properly bedded down.
The real difficulty is that the notion of `length' of a curve is NOT well defined; this is only an approximate concept in general, not an exact concept. Think of it as resolution dependent: the finer the resolution, the more detail you see on the curve, the more complicated the calculation of approximate length. There is, in general, no final exact value for a length, only a range of approximate values.
I realize this is a novel way of thinking: it also happens to be correct.
This reminds me of a picture I drew in high school. It was a drawing of a light bulb. On the left side of the bulb there is a grid background, signifying a scientific approach to this physical thing. As the grid gets closer and closer to the actual light bulb, things start to get weird. Strange ghost like "quantum" phantasm qualities emerge. On the other side of this picture the light bulb is highly stylized. Either black or white with clear sharp edges. Behind this light bulb stylized idea, is a demon, rather daemon, who has a mischievous grin on his face. His joke is that free will spirits, such as himself, are not "suppose to exist" and yet here he is doing the pulling of this "real world object" into an intellectual conception.
Insights into Mathematics ; Excellent video like most of your videos are. I just don't understand why pi is transcendental if it can be expressed as nested square roots. What is the difference between for instance phi which is thought of as being an algebraic expression? What I also don't understand is that all these different formulas generate the same answer namely pi. Isn't there some rounding off going on which doesn't have much influence because the numbers behind the comma become ever more smaller?
This can not be novel thinking. We have measured coastlines and areas of land for ages.
The ACTUAL pi value is 4/sqrt(phi).
It is a common notion that when we model the continuum as the rational numbers, then there are ``gaps'', whatever that means. Please start to consider the possibility that there are no gaps: the rational numbers are a complete and consistent model of the `continuum'. Of course there may be others: But the rational numbers are the best starting point.
Very good！
Drawing A Circle
The most commonly asked question when a person first makes use of a compass to draw a circle and then subdivide it, is, why is there bit left over?.
And no matter how many times we measure or fiddle with the compass radius, we just can't seem to get rid of that extra bit.
When we begin to draw a circle we begin by forcing the sharp tip of the compass down into the drawing surface, this action then in effect causes the radius of the compass to shorten, and the size of the circle to be slightly smaller.
And when the circle is then subdivided, this same shortening of radius happens for each subdivision, leaving a little bit left over.
Which is why the line of a circle never tells us the length of a circle, as the edge of a circle is its length, not the outer drawn line.
More simply, a circle is a shape, and as with the moon we see at night, it has a shape as a silhouette, but not an outer-line.
Diameter 120-centimeters
Multiply by 3, the circle is 360 cm
There are 360 degrees in a circle
Each degree is one centimetre long
www.fromthecircletothesphere.net
:D ...area of a circle with diameter 120 is by that "article" 10800 ... that person who was doing it is a bit nobrain ..
by this result number Pi would be exactly 3 , which is utter BS ...
Reply to Dunce Xardy5806
If You Cannot Reason And Think For Yourself - Go And Read Something Else
CIRCLES
A circle of itself is simply the visible edge of an area of round shape, just as the Sun is a silhouetted round shape against the blue of the sky, and the Moon is a silhouetted round shape against the black of the night.
As such, all circles have a silhouetted area to their round shape, which extends outwards from the centre of their round shape to an equidistant distance - limit - edge from the centre of their round shape.
Therefore, and although, round areas of shape do possess a circular length to their edge, they do not possess a circumferential outline.
THE ROUND TABLE
OF
SQUARING THE CIRCLE
Given a 120-centimeters Diameter Length
1. Multiply the 120-centimeters diameter by 3
2. The Circles length is 360-centimeters long
3. The Circles length is 360-degrees long
4. One Degree is 1-centimeter long
5. The Square of the 120-centimeters diameter is 480-centimeters
6. The Circles 360-centimeter length is three-quarters that of its 480-centimeter square perimeter
Note: To All Mathematicians and Geometer's (And self-professed geniuses of the Social/TV media)
That, if you are unable to disprove or negate the elementary arithmetic as presented above, and all of the elementary arithmetic and differential geometry that follows.
Then it follows, that the differential geometry you are currently teaching is false, and you need to address your consciences (souls) and consider whether you value teaching that which you now know to be the truth, over and above that which you now know to be false.
TWELVE STEPS TO THE VOLUME AND SURFACE AREA OF A SPHERE
CUBE TO CYLINDER
Calculating the surface area and volume of a 6-centimetre diameter sphere, obtained from a 6-centimetre cube.
Note we use 6 centimetres rather than 120 cm, in order to make the numbers easier to follow.
1. Measure cube height to obtain a/its Diameter line = 6 cm
2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of the perimeter to the square face = Length 24 cm, Square area 36 sq cm.
3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm.
4. Divide the cubic capacity by 4, to obtain one-quarter of the cubic capacity of the cube = 54 cubic cm.
5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm.
6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm.
7. Divide the cubes surface area by 4, to obtain one-quarter of the cubes surface area = 54 square cm.
8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm.
CYLINDER TO SPHERE
9. Divide the Cylinders cubic capacity by 4, to obtain one-quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm.
10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere.
11. Divide the Cylinders surface are by 4, to obtain one-quarter of the surface area of the Cylinder = 40 & a half square cm.
12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere.
www.fromthecircletothesphere.net
1. Multiply the 120-centimeters diameter by 3
2. The Circles length is 360-centimeters long
Do you seriously know what you did in point 1 ? So my question is why multiply by 3 ??? By radius of circle you can do perfect inscribed hexagon in that circle ... which means that the hexagon is 6 x radius ...or 3 x diameter ...inscribed hexagon is obviously shorter than circumference ... if you have radius 60 , perimeter of that hexagon will be 360 . Circle is longer obviously than 360 ... not as you stating in point 2 ...
Thankyou
Thanks professor njwildberger
Pi = 22 Over 7
thanks.....you just gave me an idea..:)
The term ``meta-number'', like meta-physics or meta-mathematics, I suppose refers to a vague and imprecise analogy with number, or physics, or mathematics. As for ``irrational numbers'', it has not been clarified to me precisely what this term even means. Should some-one at some point do this, then we might inquire further as to whether or not such things exist.
They are both very good books, I am more familiar with Spivak and especially appreciate his careful and elegant presentation. Having said that, there are still serious logical problems with ALL current treatments, as they all (to the best of my knowledge) rely on real numbers as the basic model of the continuum, and not the rational numbers.
I hope that it is pretty clear that I view most of these formulas with a critical eye. One challenge is to try to make sense of these results in a logically coherent fashion. But it is interesting to know what we might aim at.
It puzzled 16th and 17th century mathematicians a lot too, not to speak of the ancient Greeks. While no one doubts you can roll a circle on a line physically, it is not at all clear how to precisely translate that into mathematics which is completely pinned down.
Or to put it another way, the meaning of `the length of an arbitrary curve' is in fact highly problematic!
I will, hopefully, discuss Apostol's treatment at some future point. Ultimately both Spivak's and Apostol's approaches suffer from serious logical defects, despite the very high quality of both books.
Both books realize that there are serious issues with the current foundations of analysis. Both books try to tackle some of these problems, instead of ignoring them or trying to wave them away with some chit chat.
In your recurrence relation for (pi^2)/16 shouldn't the coefficient of the square root term be 2^(2n-2) instead of 2^n?
As we shall see, weakness with the foundations of analysis has major repercussions throughout modern pure mathematics.
@Greg Jacques Lucifer's Jizz Gargler Sorry if you want to comment, please get a proper name.
The essential problem is: how long must you go through that possibly infinite string of nines before you conclude that it actually IS an infinite string of nines?
If you can show they are all 9's some logical way then there is an infinite string of them.
Sorry, I don't understand the question. If you are referring to a particular section of the video, please let us know what this is, thanks.
Dear Prof. Wildberger,
I still keep in the back of my mind the principle you espoused so succinctly in two pages of your book-I believe it was in the introduction.
All of those "infinite formulas" for π are just algorithms, ones that happen to converge-however we wish to define such notion-and whose convergence "target" we call π, or some multiple of it.
Therefore π is not a number, but a "target" of convergence for a-possibly infinite-class of algorithms. The same goes for curve lengths, curve areas, and other such integrals, in the general case: they are not numbers, but targets of convergence for algorithms.
As you wrote, we are currently unable to prove in the general case whether any two such algorithms converge onto the same target (equality), let alone doing arithmetic with them. Therefore calling them numbers-a word with strong algebraic implications-feels dubious or at least premature.
Given this, to have the gall of stating that all numbers (aka. rationals) plus all such "targets" (aka. computable irrationals) are an order of infinity beneath "real" numbers! That is a global, collective academic hallucination if I have ever seen one.
How do you know if in fact it does ultimately repeats after we discover the next 10 trillion 50 decimal points? Then it becomes a "rational" number.
This is not a good question, but I will answer it anyway. The modern conception of mathematics as built on axiomatic foundations, with infinite sets featuring as central objects, is in fact contrary to the thinking of most of the great mathematicians who lived before the middle of the 19th century. I will be discussing historical evidence to support my positions later, but you should understand that the way we think about foundations now, in 2012, is a historical aberration.
In nature I doubt if there is a perfect circle, at least one that is visible to us. But that is only speculation. The point of this lecture is that the true nature of ``pi'' is not at all very clear. It is a big assumption to suppose that such a number exists! What does exist are rational numbers that have certain properties close to what an ideal ``pi'' would have, if it existed.
I'm not a mathematician, or even particularly good at maths. But the whole infinity business gives me the shivers. So much of pure maths and quantum mechanics feels like an exercise in dodging infinities by finding clever arguments to push them out of calculations so your results DURING a calculation don't depend on an infinity (I'm explaining this very badly) because it's never going to end. Once you accept that there's no such thing as infinitely small, or infinitely large... it all falls away. What's PI? It's the result you get when your 'circle' is drawn at the smallest resolution in our universe (whatever that is) and you can't keep dividing up space smaller than that. As you say, no such thing as a perfect circle - because at the smallest level it ends up being discrete. Anyway, that's enough badly explained fuzzy thought out drivel from me.
I think circle (and any kind of curve) is not a 2or3dimentional object, but a 2d passing to 4d(time) hopping above the 3rd one. Explaining myself, circle doesnt fit in a 2d diagramme but....you make a 3d one (width,heigh,deph) it starts to seem like a sphere but also giving you the image of something in progress...but again not be fitted well. So i thing pi is the ryth(time) of change between the points in a straight line when you try to curve it...cause as Einstein proved, when the space is curved you get time effects,and it will go on and on because you have to stretch the gap between the points of line more and more but....without making the line bigger than when was a straight one.It shows in my eyes how fast(deep in the 4d) a 2d or a 3d obgect transforms to a 4d statical object. It is like an entropy effect with ice cubes in a glass of a water. Imagin the line as the cubes and the temprature of the water as the effect of curving the line...once they come together...the straight line wil be curved for good...there is no way back...pi is the ryth of the transformation, and you can never calculate back with accuracy. I see it like that, maybe a silly thought...
The ratio of circle to diameter varies according to the actual length of the diameter.
For a diameter close to zero mm, the ratio is 3.164
For a diameter close to infinity mm, the ratio is 3.1416
See TH-cam ... Pi Revolution...by Aetzbar
img2.timg.co.il/forums/2/2357afa1-c99d-4e2c-9979-41506e265a07.pdf
All differential equations involving the formulae Pi π are infinitely irrational because you cannot equally divide a “Disproportionate - Irrational - Asymmetrical” amount into a “Proportional - Rational - Symmetrical” amount of anything.
Observe a circular analogue clock face; there are 360 degrees of circumnavigation to one rotation of the hour hand, and 60 one-tenth degrees to one rotation of the second hand yielding a total of 3,600 one-tenth degrees to one hour.
Question
Given that all clock and watchmakers subdivide 360 degrees of a circle into a set number of equal lengths (Rational) parts, why is it that mathematicians and geometers cannot do the same?
Answer
You simply cannot subdivide a whole symmetrical unit, with an asymmetrical part.
Or put another way, you cannot subdivide a whole rational unit, with an irrational part.
Therefore as a circle is perfectly “Proportional - Rational - Symmetrical”, it is impossible to subdivide the length of a circle equally, using the number three + bit of a number.
Pi π is an outline that lies proximate to the symmetrical - rational edge of a circle that is subsequently irrational (asymmetrical) because it cannot divide a circle's edge into an equal number of parts.
The challenge to any and all mathematicians (Anyone): is to prove this simple arithmetic wrong, and if they cannot then they should take a watchmakers course and stop using Pi π.
CIRCLE
A circle's edge is perfectly rational having no beginning or ending, and it can be divided into any number of equal parts e.g. 360 degrees - 60 minutes - 3,600 seconds.
THE LENGTH OF A CIRCLES EDGE
Using a 120-centimetre length of diameter multiply this by 3
The circle's edge length is 360 cm long
The circle's edge has 360 degrees of subdivision
The circle's edge has 360 degrees and each degree is 1 centimetre long
THE SUMERIAN METHOD - CALCULATING THE AREA OF A CIRCLE
Using a 120-centimetre length of diameter multiply this by 3
The Circles Edge is 360 cm long 2. Multiply the 360 centimetres "Edge Length" by itself = 129, 600 square centimetres
Divide 129, 600 by 12 = 10, 800 Square Centimetres to the Area of the Circle
ARCHIMEDES: PROPOSITION
The area of any circle is equal to a right-angled triangle in which one of the sides about the triangle is equal to the radius, and the other to the circumference of the circle.
Archimedes Triangle
The Circle in question has a 120-centimetre Diameter length
The base right-angle is equal to the radius of 60 centimetres
The area of the circle is equal to the right-angle triangle, which has one side that is equal to the 60-centimetre radius, and the other to the 360-centimetre circumference of the circle
The 360-centimetre height of the right-angle is equal to 6 x the 60-centimetre radius length
(1r) 60 centimetres x (6r) 360 centimetres yields 21, 600 square centimetres the area of the rectangle
Half of the rectangle is 10, 800 square centimetres
The area of the triangle is half of the 1r x 6r rectangle
Half of the 1r x 6r rectangle is 1r x 3r (1r) 60 centimeters x (3r) 180 centimeters = 10, 800 square centimeters
THREE TIMES THE RADIUS SQUARED
The Diameter of the Circle is 120 centimetres
The diameter x 120 centimetres gives, 14, 400 square centimetres to the square of the diameter
The 60-centimetre radius x 60-centimetres yields 3, 600 square centimetres to the square of the radius
The square of the radius x 3 gives, 10, 800 square centimetres to the area of the Circle
SUMERIAN AREA: 10, 800 square centimetres
ARCHIMEDEAN AREA 10, 800 square centimetres
THREE TIMES THE RADIUS SQUARED AREA: 10, 800 square centimetres
FOUR QUADRANTS 10,800 square centimetres
Three Thousand Years Apart
And
Four Identical Results Is Not Coincidental
THE AREAS OF RINGS
Using An Eight Mile Diameter Right Angled Square (Or kilometres)
Begin by first finding the area of each circle Multiply the 2-mile diameter of the central yellow circle by itself = 4 square miles to the square of the diameter, divide by 4 = 1 square mile x 3 = 3 square miles to the central circle.
Multiply the 4-mile diameter of the red circle by itself = 16 square miles to the square of the diameter, divide by 4 = 4 square miles x 3 = 12 square miles to the red circle.
Multiply the 6-mile diameter of the blue circle by itself = 36 square miles to the square of the diameter, divide by 4 = 9 square miles x 3 = 27 square miles to the blue circle.
Multiply the 8-mile diameter of the green circle by itself = 64 square miles to the square of the diameter, divide by 4 = 16 square miles x 3 = 48 square miles to the green circle.
Deduct the 3 square mile area of the central yellow circle from the 12 square mile area of the red circle; = 9 square miles to the area of the red ring.
Deduct the 12 square mile area of the red circle from the 27 square mile area of the blue circle = 15 square miles to the area of the blue ring.
Deduct the 27 square mile area of the blue circle from the 48 square mile area of the green circle = 21 square miles to the area of the green ring.
Deducting the 48 square mile area of the green circle from the 64 square mile area of the overall pale blue square, = 16 square miles to the remaining area of the square, which is 1/4 of the area of the square of 64 square miles.
Check
Central Circle = 3 square miles
Red Ring = 9 square miles
Blue Ring = 15 square miles
Green Ring = 21 square miles
Pale Blue Area = 16 square miles
+
= 64 square miles
TIME
360 Degrees to one equatorial circumnavigation of the Earth’s surface around its fixed core (diameter) = One Day.
ONE DAY
360 degrees of one day divided by 24 = 15 degrees 15 degrees of one day multiplied by 24 = 360 degrees to one hour
360 degrees to one hour multiplied by 10 = 3, 600 seconds to one hour
3, 600 seconds of one hour divided by 60 = 60 seconds to one minute
CHECK
60 Seconds x 60 = 3, 600 seconds = one hour
One hour of 3, 600 seconds x 24 = 7, 200 seconds to one day
7, 200 seconds divide by 24 = one 15 degrees hour multiplied by 24 = 360 degrees to one day
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Sorry, but you have been programmed to see through the same rose coloured glasses that most of us have. Irrational numbers do not really exist, at least not in the way that is usually discussed. As we go through this series, you will see a more logical and careful approach to analysis; where rational numbers take center stage. What ``pi'' ends up being has yet to be seen--- a number like 22/7, or -14/1567 it certainly is not.