Zero

[clevedonboy]

As part of my degree course, I was taught at great lengths of the conceptual importance of 0, how the arabs had intoduced it to the west. that the fundamental calculation (+1) + (-1) = 0 rocked thinking, how 0!=1 is a contradion , 1/0 is infinity, Y to the power of 0 is 1 regardless of the value of Y. Without doubt for pure mathematicans 0 is cool. So how soul destroying was it to find this article about a Parrot who can uinderstand the concept of 0?

An African Grey Parrot at Brandeis University appears to understand a numerical concept akin to zero. In a series of counting experiments, Alex, a 28-year-old parrot who lives in a Brandeis lab run by comparative psychologist Irene Pepperberg, spontaneously and correctly used the label "none" during trials of his counting skills to describe an absence of colored blocks on a tray.

The findings, published in the current issue of The Journal of Comparative Psychology, add to a growing body of scientific evidence that the avian brain, though quite different from the brains of mammals, are capable of handling more complex processes than previously thought.

Generally, humans don't begin to understand the concept of zero until about three or four years old (longer in the case of some). Chimps and spider monkeys have been shown to have some rudimentary understanding of the concept of none. Grey parrots have brains the size of a walnut.

[bigdjiver]

Pi is irrational and so has no repeating sequences.

A normal number is one in whose decimal expansion all digits are equally likely to appear.

Is the claim about telephone numbers equivalent to being a normal number, or is irrationality sufficient?

The set of of telephone numbers is far from "normal". Just think how many begin with 0.

A vacuum technologist was introduced thius: "An expert has been defined as someone who gets to know more and more about less and less, until he knows everything about nothing. Mr .... knows practically everything about practically nothing."

[LMC]

I OFFICIALY declare this thread to be the Greatest discussion about absolutely NOTHING .... EVER.

Well, that saves me asking

damn you Gus...

[Clive Long]

And more than that, the sequences within that sequence (say you pick every other digit (or is it integer ) or every 500th or every squillionth) the resulting sequences are non-recurring.

References? I thought this was still unknown (same for fact that every digit sequence appears). Although the statistical evidence strongly implies Pi is normal, in which case both conclusions would follow.

You are very probably right - I'm really outta my league here.

All I can offer is:
"A real number is said to be irrational if it is not rational, i.e. it cannot be expressed as a quotient of integers. The decimal expansion is non-periodic for any irrational, but periodic for any rational number"

http://planetmath.org/encyclopedia/T...alNumbers.html

Which may be equivalent to what I wrote - but I might be well off.

Enough.

Clive

[bigdjiver]

References? I thought this was still unknown (same for fact that every digit sequence appears). Although the statistical evidence strongly implies Pi is normal, in which case both conclusions would follow.

There are normal, non-recurring sequences that do not contain every digit sequence, and have every sqillionth digit the same.

Take pi, and every time 34 appears replace it with 43. Swap every sqillionth digit with the nearest non-sqillionth 9.

[Feelingpink]

I OFFICIALY declare this thread to be the Greatest discussion about absolutely NOTHING .... EVER.

So aren't you pleased that YOU were the one who raised the issue in the first place!

[David Franklin]

All I can offer is:
"A real number is said to be irrational if it is not rational, i.e. it cannot be expressed as a quotient of integers. The decimal expansion is non-periodic for any irrational, but periodic for any rational number"

Which may be equivalent to what I wrote - but I might be well off.

No, it's not the same; again, the obvious counterexample is Liouville's number, which is non-periodic (and therefore irrational), but only contains 0's and 1's.

There are normal, non-recurring sequences that do not contain every digit sequence, and have every sqillionth digit the same.

That would be a direct contradiction of the definition of normal I'm using:

From http://mathworld.wolfram.com/NormalNumber.html:
for a normal decimal number, each digit 0-9 would be expected to occur 1/10 of the time, each pair of digits 00-99 would be expected to occur 1/100 of the time, etc

In fact every digit sequence has to occur infinitely often in a normal number according to that definition.

[bigdjiver]

Nice idea - but wrong.

Clive

Unhelpful - given that it is a simplification, how is it wrong?

[bigdjiver]

....That would be a direct contradiction of the definition of normal I'm using ... In fact every digit sequence has to occur infinitely often in a normal number according to that definition.

Apologies, I was using ESG's definition.

[Clive Long]

Unhelpful - given that it is a simplification, how is it wrong?

Good point.

I was being terse to the point of rudeness.

In fact I have been trying to justify what I wrote - I have go on a bit - I will PM you with my thoughts * (ramblings) rather than post them - as several people at Harrow tonight said the presence of this thread is sapping their will to live.

Clive

* and PM them to Dr Franklin for his review comments

[El Salsero Gringo]

Apologies, I was using ESG's definition.

I was being deliberately sloppy. It's the long run in-the-limit probablity of finding any digit sequence that has to equal out.

if you want a more "mathematical" definition of a 'normal' number, it looks like this: (from wikipedia):

Suppose b is an integer greater than 1 and x is a real number. Consider the digit sequence of x in the base b positional number system. If s is a finite string of digits in base b, we write N(s,n) for the number of times the string s occurs among the first n digits of x. The number x is called normal in base b if for every string s of length k:

[killingtime]

if you want a more "mathematical" definition of a 'normal' number, it looks like this: (from wikipedia):

Do you think we can get MathML support on this board so you don't need to attach images on the math involved ?