# The Squared Circle

It is interesting how a simple quirk created by expressing one third in base ten can baffle so many people.

Probably blows their mind to see that 2/7 = .285714r and 5/7 = .714285r. Therefore 2/7 + 5/7 = .999999r

It is interesting how a simple quirk created by expressing one third in base ten can baffle so many people.

Probably blows their mind to see that 2/7 = .285714r and 5/7 = .714285r. Therefore 2/7 + 5/7 = .999999r

You need me to explain that to you too?

2/7 is 2 divided by 7.

7 goes into 2.0 .2 times (1.4) with a remainder of .6 (answer so far .2)
7 goes into .60 .08 times (.56) with a remainder of .04 (answer so far .28)
7 goes into .040 .005 times (.035) with a remainder of .005 (answer so far .285)
7 goes into .0050 .0007 times (.0049) with a remainder of .0001 (answer so far .2857)
7 goes into .00010 .00001 times (.00007) with a remainder of .00003 (answer so far .28571)
7 goes into .000030 .000004 times (.000028) with a remainder of .000002 (answer so far .285714)

So .2 + .08 + .005 + .0007 + .00001 + .000004 = .285714 with a remainder of .000002

Guess what the next number is gonna be?

We started with 7 going into 2.0
We now have a remainder of .000002

Now it is stuck in a loop of repeating reminders (in different decimal positions) of:

6, 4, 5, 1, 3, 2 - 6, 4, 5, 1, 3, 2 - 6, 4, 5, 1, 3, 2 .....

YOU CAN'T BREAK OUT OF THAT LOOP, so the answer will repeat .285714 285714 285714 285714 .........

The division CAN'T be completed! The remainder is never divided equally for the division to end, so it will continue INFINITELY!

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It is interesting how a simple quirk created by expressing one third in base ten can baffle so many people.

Probably blows their mind to see that 2/7 = .285714r and 5/7 = .714285r. Therefore 2/7 + 5/7 = .999999r

What is really interesting is that you swept a remainder of .000002 under the rug and claimed 2 divided by 7 was .285714r.

Let me clarify your mistake. That .285714 is with a remainder of .000002.

You did not complete the division of 2 divided by 7 because you never divided the remainder equally at ANY point in the division.

If the remainder would have been divided equally then the division would not continue.

So it appears YOU are the one BAFFLED by simple long division you learn in 2nd grade!

Here's an easy one for you:

1 divided by 2.

2 goes into 1.0 .5 times to equal 1.0 WITH NO REMAINDER.
1 is EQUALLY divided by 2, because 2 x .5 = 1.0. Because 50% +50% = 100% See how there are TWO 50%? See how 50% + 50% = 100%??

Here's one a little tougher for you, being that you are just learning long division:

1 divided by 9:

9 goes into 1.0 .1 time (9 x .1 = .9) WITH A REMAINDER OF .1

Now you have to divide the remainder of .1 by 9.

9 goes into .1 .01 times (9 x .01 = .09) WITH A REMAINDER of .01

So far .1 + .01 = .11, and there is a remainder of .01 left to be divided by 9.

CONTINUE INFINITELY, because the remainder will always be 1 in the next decimal position.

So your old practice of tossing the remainder and claiming 1 divided by 9 = .111... is BS! 1 divided by 9 is .111... with a remainder left over.

Check your work, .111... x 9 = .999... and that is less than what you started with, which is 1.0. What's missing? The REMAINDER that was never divided equally! If it did divide equally the division would be complete and the answer would not continue infinitely!

So, moral of the story: If the remainder can't be divided equally the division can't be completed. When you see a "..." it means "this is BS!"

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What is really interesting is that you swept a remainder of .000002 under the rug and claimed 2 divided by 7 was .285714r.

Let me clarify your mistake. That .285714 is with a remainder of .000002.

You did not complete the division of 2 divided by 7 because you never divided the remainder equally at ANY point in the division.

If the remainder would have been divided equally then the division would not continue.

So it appears YOU are the one BAFFLED by simple long division you learn in 2nd grade!
See thar little "r" at the end? That means repeating.

Seriously, are you that stupid or are you just a troll?

Seriously, are you that stupid or are you just a troll?
Obviously a retorical question

See thar little "r" at the end? That means repeating.

Seriously, are you that stupid or are you just a troll?

2 divided by 7 = .285714 with a remainder of .000002

7 x .285714 = 1.999998
1.999998 + .000002 = 2.0

7 x .285714285714 = 1.999999999998
1.999999999998 + .000000000002 = 2.0
Notice how the .000000000002 is not part of the 7 x .285714285714??
Are you just trolling or are you really that stupid?

Repeat it as many times as you wish, it will never divide equally so you can never finish the division.

You really should go back to 2nd grade and learn this stuff. Were you sick that day?

See thar little "r" at the end? That means repeating.

Seriously, are you that stupid or are you just a troll?
He's pretending he doesn't know that in maths the sum of an infinite series can converge to a finite value. It's rather like not understanding what an asymptote is, in a graph.

Now that we have established that 2/7 = (7 x .285714) + .000002 = 2.0, let's add the 5/7 to that:

5/7 = (7 x .714285) + .000005 = 5.0

So: 2/7 + 5/7 = ((7 x .285714) + .000002) + ((7 x .714285) + .000005) = 7.0 = 700%

700% divided by 7 = 100%, or 7 divided by 7 = 1.0 = 100%
7 x 100% = 700% = 7.0 (which is the 2.0 + 5.0 that you started with)

...and whodathunk that:
.714285714285714285 x 7 = 4.999999999999999995
4.999999999999999995 + .000000000000000005 = 5.0

Just luck I guess. (rolls eyes)

You know what you can do with your .999..., right?

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Moron,

Every decimal place to the right is 10 times smaller.

0.01 is 10 times smaller than 0.1

0.1 is 1 Tenth
0.01 is 1 Hundredth
0.001 is 1 Thousandth

0.9 is 9 Tenths
0.09 is 9 Hundredths
0.009 is 9 Thousandths

.99 is 99 Hundredths = 99/100
.999 is 999 Thousandths = 999/1,000
.9999 is 9,999 Ten Thousandths = 9,999/10,000
.99999 is 99,999 Hundred Thousandths = 99,999/100,000
.999999 is 999,999 Millionths = 999,999/1,000,000

They are ALL less than 1.0

The more 9's you place after the decimal point, the closer you are to 1.0, but you can never be equal to 1.0 by placing more 9's after a decimal point.

.999999999999999999 = 999,999,999,999,999,999/1,000,000,000,000,000,000

That does NOT equal 1.0

How could you be so F'n stupid?
I addressed this in post #140.

If Motor Daddy had bothered to actually read that post and attempt to understand it, we could perhaps have had a productive conversation.

Instead, all we get from Motor Daddy is another set of insults.

This is a pretty shitty note on which to end your time here, Motor Daddy. Surely you can't be proud of your behaviour? You've acted like a child who can't get his way throwing a temper tantrum.

What a pity you couldn't be a bigger man.

Goodbye.

Moderator note: Motor Daddy has been permanently banned from sciforums.

This is at his own request. However, given his constant insulting and flaming of other members and his trolling, this probably would have happened sooner or later, even without this request.

forum;

The 'square circle' title got my attention since I exchange posts with someone on another forum who proposes a square light clock.
The repeating decimal problem has appeared on most of the forums visited, with the same response, and is on my list of annoying topics.
Motor Daddy was correct some of the time and wrong some of the time.
Dividing by n depends on what is divided.
Examples:
If the task is to divide a collection of 10 objects into 3 equal parts, it's not possible.
You will get 3+3+3+1. You divide the collection, but not the parts. If the material is continuous like a fluid, you can divide it into n equal portions.
A circle (pie) can be divided into n equal parts if n is a factor of 360.
Division of geometrical shapes depends on symmetries.
Equal is understood to mean without defined significant differences.
----------------------------------
No one asks, why did the earlier mathematicians define 'limits' instead of declaring an equality? Primarily because of their experience with sequences.
a=sequence of terms a(n), either a term after n recursive applications of a function or a summation of n terms. In words, the limit of (a), as n (increases without limit), is L.
The phrase in parentheses is substituted for 'approaches infinity', a totally nonsensical term. You can't approach something that is unreachable (the horizon, a carrot on a stick, etc).
If infinity means without bounds, then there is no end. If there is no last term, then the sequence remains incomplete, and the current value is always an approximation. Many know and state 'infinity is not a number' then contradict themselves by using it in that sense.
If a result requires an 'infinite' number of clock cycles, it never happens.
In agreement with Motor Daddy, no one will ever see an exact value of pi, or an irrational number. They only exist as abstract entities in the world of pure math.

Here is a definition of 'limit' which I posted to mathforums, which they rejected (even after telling them it is from a lesson plan from the math dept. at MIT. It shows the degree of arrogance in their 'fraternity', and places them above physicsforums.

Here is the example for .9R.
Common practice truncates any sequence when it satisfies the precision requirements.

View attachment 4942

Here is the example for .9R.
Common practice truncates any sequence when it satisfies the precision requirements.
This is only true for a finite n.
If you are only considering a finite number of elements in such an infinite sequence, you will get close to, but not equal to, its limit.
Since the "..." and "R" denote "ad infinitum" / recurring etc, one must consider the entire infinite sum, not merely a finite number of elements. As such, 0.999... and 0.9R should be equated to the sum of the entire infinite sequence, not just the first n elements. And the sum of the entire infinite sequence really is exactly 1, not < 1.

If you disagree that 0.999... = 1 then surely you must think that there is a number between 0.999... and 1? Care to say what it is? For all A and B, where A and B are different numbers, there exists another number (A+B)/2 which lies between A and B. So what is it for 0.999... and 1?

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forum;

The 'square circle' title got my attention since I exchange posts with someone on another forum who proposes a square light clock.
The repeating decimal problem has appeared on most of the forums visited, with the same response, and is on my list of annoying topics.
Motor Daddy was correct some of the time and wrong some of the time.
Dividing by n depends on what is divided.
Examples:
If the task is to divide a collection of 10 objects into 3 equal parts, it's not possible.
You will get 3+3+3+1. You divide the collection, but not the parts. If the material is continuous like a fluid, you can divide it into n equal portions.
A circle (pie) can be divided into n equal parts if n is a factor of 360.
Division of geometrical shapes depends on symmetries.
Equal is understood to mean without defined significant differences.
----------------------------------
No one asks, why did the earlier mathematicians define 'limits' instead of declaring an equality? Primarily because of their experience with sequences.
a=sequence of terms a(n), either a term after n recursive applications of a function or a summation of n terms. In words, the limit of (a), as n (increases without limit), is L.
The phrase in parentheses is substituted for 'approaches infinity', a totally nonsensical term. You can't approach something that is unreachable (the horizon, a carrot on a stick, etc).
If infinity means without bounds, then there is no end. If there is no last term, then the sequence remains incomplete, and the current value is always an approximation. Many know and state 'infinity is not a number' then contradict themselves by using it in that sense.
If a result requires an 'infinite' number of clock cycles, it never happens.
In agreement with Motor Daddy, no one will ever see an exact value of pi, or an irrational number. They only exist as abstract entities in the world of pure math.

View attachment 4941

Here is a definition of 'limit' which I posted to mathforums, which they rejected (even after telling them it is from a lesson plan from the math dept. at MIT. It shows the degree of arrogance in their 'fraternity', and places them above physicsforums.
View attachment 4942

Here is the example for .9R.
Common practice truncates any sequence when it satisfies the precision requirements.
I don't understand your difficulty. Surely you are familiar with the idea of an asymptote, are you not?

A circle (pie) can be divided into n equal parts if n is a factor of 360.

But Motor Daddy told me that I can't cut one pizza into three equal parts, because one pizza is 100%, and 100 is not evenly divisible by 3, according to him. Now you are saying that one pizza can be thought of as 360 degrees of arc around the edge of the pizza, and since 360 is evenly divisible by 3, I can cut the pizza into 3 equal parts. Yet 360 degrees of arc around the edge is equal to 100% of the pizza, so you have inadvertently proved that 100 is evenly divisible by 3, contrary to MD's claims!!

Aren't 100% and 360 degrees just man-made conventions that have been chosen for our convenience? Couldn't we just as validly say that one pizza is equal to 99 Neddy units, and therefore cutting the pizza into 3 equal parts is possible because it is simply 33 Neddy units?

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Aren't 100% and 360 degrees just man-made conventions that have been chosen for our convenience? Couldn't we arbitrarily say that one pizza is equal to 99 Neddy units, and therefore 1/3 of the pizza is 33 Neddy units?

Of course a Pizza can be divided into 3 equal parts. It is a finite object not a mathematical abstraction

If you wish to get ultra ultra pedantic and count the number of atoms in the Pizza you might / could have a number not able to be divided equally into 3. But in such a case you are back, in effect to mathematics

Sarkus'

If you are only considering a finite number of elements in such an infinite sequence, you will get close to, but not equal to, its limit...

If you disagree that 0.999... = 1 then surely you must think that there is a number between 0.999... and 1? Care to say what it is? For all A and B, where A and B are different numbers, there exists another number (A+B)/2 which lies between A and B. So what is it for 0.999... and 1?

You gave the answer in the first line above. You can't consider the complete sequence, because it doesn't exist. Another instance of 'all things thinkable are not realizable'.
Performing the division above you get more sequences of 9's. The context is 'infinite sequence', without an end/boundary. There is no last '9'. There is no such thing as a 'thing without boundaries'. It's a contradiction of terms. Boundaries are what enables measurements including counting. You lose your perspective when expecting a number different from '9'.
If the sequence were an odometer, the only way it would roll over to 1 would be the addition of 1 at some position, but that is prohibited by its definition.
The small difference is never zero, so the last line states 'for all n'.
Most math people agree, there is no greatest integer n.
The root cause of this problem is the concept of 'continuum'.

exchemist;
Surely you are familiar with the idea of an asymptote, are you not?

Like the limit L of the hyperbola ya=sqrt(n^2+1)=1, which approximates the line yb=n for large n?
Notice, L=yb but not ya, i.e. ya NEVER equals 1.
'For all n' implies the function sqrt(n^2+1) is independent of time.
An appeal to long periods of time is irrelevant.

Neddy Bate;
But Motor Daddy told me that I can't cut one pizza into three equal parts, because one pizza is 100%, and 100 is not evenly divisible by 3, according to him.

Percent means 'parts per 100' (latin 'centum', roman numeral c), i.e. a ratio.
This is something that can be done with a compass. The circumference is magenta. At points 1, 2, 3, draw a gray circle of radius r. This divides the circle into 3 equal sectors of 120 deg. Maybe MD didn't study geometry.

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Percent means 'parts per 100' (latin 'centum', roman numeral c), i.e. a ratio.

Yes, and 1/3 is a ratio also. All we are trying to tell you is that 33.333...% (repeating decimal) is exactly equal to 1/3.

This is something that can be done with a compass. The circumference is magenta. At points 1, 2, 3, draw a gray circle of radius r. This divides the circle into 3 equal sectors of 120 deg. Maybe MD didn't study geometry.
View attachment 4945

So now that you have divided 100% of the pizza into 3 equal slices, the question is simply what percent does each slice represent? The answer is 33.333...% (repeating decimal). There is no little bit lost, as MD had claimed.

You gave the answer in the first line above. You can't consider the complete sequence, because it doesn't exist. Another instance of 'all things thinkable are not realizable'.
You can consider it, and it does exist. The sum of the sequence really is 1. Not "almost 1" but actually 1.
Next you'll be saying that you can't walk from A to B because first you need to walk to half way to B, then halfway between there and B, then halfway between there and B, etc. Yet, lo and behold, you can get there quite easily.
There are an infinite divisions on a ruler between 0 cm and 1 cm, as there are between 0 cm and 2 cm, or 1 cm and 2 cm etc. Yet we can run our finger across all of them rather easily, even by going half way, then half the remaining distance etc. We can do that by going twice as fast for each halved distance.
Performing the division above you get more sequences of 9's.
So what?
The context is 'infinite sequence', without an end/boundary. There is no last '9'. There is no such thing as a 'thing without boundaries'. It's a contradiction of terms. Boundaries are what enables measurements including counting. You lose your perspective when expecting a number different from '9'.
Yes, it is an infinite sequence, so stop being dishonest and only considering the first n finite elements of the sequence, rather than all of them.
If the sequence were an odometer, the only way it would roll over to 1 would be the addition of 1 at some position, but that is prohibited by its definition.
No, it wouldn't.
What is the number between 0.999... and 1.
Let's say A = 0.999... and B = 1.
What is (A+B)/2
You say A =/= B, so there must be a number between them that is different to both A and B. So, what is it?
The small difference is never zero, so the last line states 'for all n'.
It becomes zero. That is what it means for it to reach its limit at infinity.
If you continue to only look at the finite then you are not talking about the summation ad infinitum, and you are simply trolling by deliberately talking cross-purposes.
Most math people agree, there is no greatest integer n.
So what? We still have this concept of infinity. All you are doing is refusing to look at that and instead stick with a finitie number of elements as if that disproves the case for the infinite.
The root cause of this problem is the concept of 'continuum'.
There really is no problem, other than your misunderstanding of maths.
If you have a problem with the concept of "infinity", that's on you.