[Colin2B]

Doesn't the fact that Pi is not a clear-cut whole number, mean that the Universe is not 3-dimensional?

Not at all.
3 dimensional means defined by 3 dimensions (each having its own number). It has nothing to do with a single number that just happens to be close to 3.
You are confusing things which are not the same.

[Bored chemist]

Do you think there's something disturbingly wrong about Pi not being a whole number?

I mean, why isn't it a simple whole number, like 3.  If it was, that would make sense in a 3-dimensional Universe.

In such a Universe, the 3 dimensions - length, breadth, height  ( or equivalent terms for these properties )
would surely be capable of clear-cut representation by the numbers: 1, 2, 3.  There wouldn't be any "fuzziness"

Doesn't the fact that Pi is not a clear-cut whole number, mean that the Universe is not 3-dimensional?

If you calculate the ratio of the edge of a square to the diagonal you  will find it is also irrational. It's the square root of two, about 1.414......

[evan_au]

If you calculate ... the square root of two

The polygon method also assumes that you are able to calculate square roots accurately
- Unlike addition, multiplication and division, calculating square roots is not something they teach in school - you just press the square root button on the calculator.
- Or, in the bad old days, you looked it up in a book of mathematical tables, or worked it out with a slide rule.

I did not realise that one simple way of calculating square roots was known to the ancient Babylonians.
- And it has a built in error-correcting mechanism with upper & lower bounds
- Method to find SQRT(C) where C is a constant:
     1) Have an initial guess at the answer to SQRT(C); call it X
     2) Calculate the quotient Q=C/X; if guess X is a bit high (upper bound), Q will be a bit low (lower bound) - and vice-versa
     3) Average X (too high) & Q (too low) to give a better estimate X.
     4) Go back to step 2, if you think X and Q are still too far apart

I was surprised to see that this converges as quickly as Newton's method
- in fact it gives identical answers to Newton's method applied to find the zeroes of y=x²-C
- For example, to calculate SQRT(16), with an initial guess of 5:

Babylonians_vs_Newton.png (15.09 kB . 679x255 - viewed 2538 times)

Those Babylonians really knew their arithmetic (in base 60)!
- It requires fewer arithmetic operations than Newton's method
- It automatically give you upper and lower bounds for SQRT.

So why would anyone use Newton's method?
- I guess it is a very general method, that works for any continuous function for which you can calculate the derivative.

PS: I chose SQRT(16) as an example, because everyone knows SQRT(16) = exactly 4

See: https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method

[alancalverd]

Or, in the bad old days, you looked it up in a book of mathematical tables, or worked it out with a slide rule.

Hand caclulation was taught and examined in my schooldays. Slide rules were "only for engineers".

[syhprum]

My father had very little education but the one thing he was proud of was he could calculate square roots with pencil and paper (it is not very hard !)

[charles1948]

My father had very little education but the one thing he was proud of was he could calculate square roots with pencil and paper (it is not very hard !)

Yes, it can be done by iterations.  Long ago, back in the primitive 1970's, I bought a Sinclair "Executive" pocket calculator.  It wasn't much of a calculator, by modern standards.  Only had basic arithmetic functions: add, subtract, multiply, and divide.

But despite these deficiencies, it had a big selling point - it was very slim.  Only about a quarter of an inch thick, as I recall.  This made it look quite "futuristic".  When compared to other calculators of the time, which were much thicker.

Anyway, the Sinclair "Executive" came with an instruction booklet, which explained how it could be used to perform more complex arithmetical operations, such as deriving "Square Roots".

The method set out in the booklet, as far as I can remember, started like this:

1. Suppose you're asked to derive the square-root of 27.
2. Make a "reasonable guess" at what it might be.  Such as 5
3. Divide 5 into 27, then see what the result is, as displayed on the calculator's screen.

I can't remember what you were supposed to do after that.  Except that it might possibly have involved "doubling" the result, then dividing again by something, in repeated iterations, until the answer came up on the screen.

That was in the bad old days.  Thankfully, nowadays calculators are much more sophisticated.  They have "square root" keys.  You just enter "27", then press the "square root"  key, and the answer comes up on the screen instantly.

I suppose the calculator uses its internal micro-processor, to do the same kinds of things that your Dad did with his pencil and paper.  Only much faster?

[alancalverd]

www.freecodecamp.org/news/find-square-root-of-number-calculate-by-hand takes me back to the 1960s.

[charles1948]

www.freecodecamp.org/news/find-square-root-of-number-calculate-by-hand takes me back to the 1960s.

Thanks alan, for the link you very kindly provided.  I've studied it, but the method which it describes, such as dividing numbers into "pairs" and so on, is far more abstruse and complicated than what I remember from the Sinclair booklet.

The Sinclair method, as I attempted to describe earlier, was a simple iterative process, which boiled down to:

1. Take a reasonable guess at what the the square root might be;
2. Divide your guess into the number, and look at the result.
3. Do something with the result  (what you were supposed to do, I can't remember)
4. Keep repeating steps 2 and 3, until a square root of sufficient accuracy for your purposes, emerges.

It was just simple iteration.  It definitely didn't involve the stuff your link sets forth, which by the way slightly reminds me of a reputed old Russian method of multiplying two numbers, by writing down long columns of numbers, and crossing out various entries in the columns, following some arcane rules.

Thanks again for your reply, best wishes!

[evan_au]

...takes me back to the 1960s

That method is not much harder than long-division....

dividing numbers into "pairs"

What they appear to be doing here is making use of the fact that 100 is a perfect square = 10 x 10
- When they group numbers into pairs, they are effectively turning base 10 numbers into base 100 numbers
- So 20 25 = 20x100 + 25
      - as distinct from our Base 10 way of saying 2025 = 2x1000 + 2x10 + 5;
      - only 1000 & 10 are not perfect squares, and make the answer very messy!
- They proceed to find the √20, without mentioning that they are also multiplying it by √100 =10
- Anything "left over" from √20 must then be included as the leading digits when finding √25

[charles1948]

Alan when you mention "long division", this a difficult operation using decimal arithmetic. So difficult that I don't think it's even taught in schools these days.  There's some alternative procedure, I think, which is more on the lines of the Russian method I mentioned in a previous post.

Anyway, it leads me to this question:

Suppose we weren't using decimal, base-10 arithmetic.  But were operating in base-2 binary.

On this assumption, and going back to the original OP about PI,  when Pi is calculated using decimal base-10, the calculations seemingly go on for billions of decimal figures, without showing any obvious patterns in the figures.

Would this be the same, if we did the calculations in Binary.  Might not the masses of Binary figures when printed out,  in "1"'s and "0"'s , eventually reveal some kind of geometrical pattern.  Like a Circle?

[Colin2B]

Alan when you mention "long division", this a difficult operation using decimal arithmetic. So difficult that I don't think it's even taught in schools these days.  There's some alternative procedure, I think, which is more on the lines of the Russian method I mentioned in a previous post.

Long division is not difficult and is taught to 9-11yr olds in UK. In younger years they can be taught chunking which is a method using repeated subtraction which helps them understand the principle of division.

[charles1948]

Long division is not difficult and is taught to 9-11yr olds in UK. In younger years they can be taught chunking which is a method using repeated subtraction which helps them understand the principle of division.

Thanks Colin, I knew there was a word for the new procedure, which you've kindly supplied: "chunking".

If you don't mind me saying so,  I doubt that this "chunking"  business actually helps any of the kids understand long division in any respect whatsoever.

It reminds me of my years in Primary School.  Once, the headmaster made us kids get together in a room, and draw
 a chalk-marked diagram on the floor. 

Then we had to pick up bundles of straw-stalks, and keep putting them down, in varying orders, on the chalk marks.  This was meant to physically demonstrate the operations involved in long-division.

But it didn't achieve that at all. It just baffled and bemused  everyone except the well-meaning headmaster

[evan_au]

Might not the masses of Binary figures when printed out,  in "1"'s and "0"'s , eventually reveal some kind of geometrical pattern. 

People have searched for patterns in many bases, and not seen a pattern
- This is not surprising if you look at the infinite series for pi
- The first few terms produce a clear pattern
- But later terms come in and modify the pattern in later decimal places

some kind of geometrical pattern...Like a Circle?

This geometrical pattern is certainly present. Pi is the ratio of the Diameter to the Circumference of a circle.

[Colin2B]

If you don't mind me saying so,  I doubt that this "chunking"  business actually helps any of the kids understand long division in any respect whatsoever.

Don’t mind at all 
I think the more ways you have of visualising numbers and the various operations the more chance you have of understanding what is really happening. There are some useful ideas out there which help but, as you’ve experienced, quite a few that don’t - or maybe work for some.
I must confess that I really never got on with arithmetic, maths never made any sense to me until we got on to algebra and geometry.

[Bored chemist]

Would this be the same, if we did the calculations in Binary.

Yes.