Irrational Numbers

[ronthepon]

Originally Posted by Popular

LOL, Qfwfq.

so

You seem to want to have Qfwfq suffer. You will never find n and d, even if you refuse to give up until your mind goes into a enlightment.

[Farsight]

I was offering a proof!

If n² = d² x 9, n = d x 3, eg n = 9 and d = 3.
If n² = d² x 4, n = d x 2, eg n = 8 and d = 4.
If n² = d² x 2, n = d x , eg nuffin

..because root 2 is one of them irrational number jobbies.

[CraigD]

Originally Posted by ronthepon

Aah, CriagD, surds are not linear. (I hope I'm not drastically mistaken)

Surds are a kind of number, so can’t appropriately be called linear or non-linear.

"Linear equation" is usually a term for "linear polynomial equation", or "degree one equation", which is just any equation that has the form Y=aX + c, where Y and X are variable, a and c constant. Y, X, a, and c may be natural numbers, integers, rational numbers, real numbers (which include the irrationals), complex (which include the imaginary numbers), or numbers of any kind that are closed under addition and multiplication.

“Linear” can also refer to a function where and . Many functions with irrational values aren’t linear – eg: for the square root function, . However, a linear function (eg: ) is still linear when irrational values of are used – eg: .

The term “surd” is poorly defined, and archaic. As I’m most familiar with it, it’s a synonym for “constructable using compass and straigtedge”. Some old texts use it a synonym for irrational, either the mathematical or the common meaning. It’s likely best not to use the term, since it’s meaning is so poorly agreed on.

Not all irrational numbers are surds (constructable with straightedge and compass). and are, but is not.

For computer-age folk who don’t like old-fashioned “radical” notation, = m^(1/n), while radicals without the n are assumed to be “square roots”, = m^.5.

[CraigD]

In an effort to reserve the Math and Physics thread for more speculative discussion, this thread has been moved to Science Projects and Homework.

The idea is, if you want to discuss something you’re studying with the aim of learning its usual, academic meaning, SP&H is the place for it. If you want to discuss something new unconventional, or more advanced than would ever be considered “homework”, M&P is the place for it.

[Qfwfq]

More simply:

Originally Posted by Popular

so

if d is integer, will also be integer? IOW will n also be integer?

[ughaibu]

Irrational numbers are described along the lines of 'non-repeating, non-terminating decimals', doesn't this description suffer from a paradox?
As there are only ten numerals, individual numerals will be repeated in sequences of more than ten terms, likewise, pairs of numerals will be repeated in sequences of more than one hundred terms, etc, as irrational numbers are infinite sequences this means that all occuring finite sequences, no matter how long, will repeat. So, the only non-repeating sequences will be infinite and as the only infinite sequence is the irrational number itself, the description "non-repeating" has no meaning.
?

[CraigD]

Originally Posted by ughaibu

Irrational numbers are described along the lines of 'non-repeating, non-terminating decimals', doesn't this description suffer from a paradox?
As there are only ten numerals, individual numerals will be repeated in sequences of more than ten terms, likewise, pairs of numerals will be repeated in sequences of more than one hundred terms, etc, as irrational numbers are infinite sequences this means that all occuring finite sequences, no matter how long, will repeat. So, the only non-repeating sequences will be infinite and as the only infinite sequence is the irrational number itself, the description "non-repeating" has no meaning.
?

I believe ughaibu is confusing the idea of ”occurring” with what is meant by “repeating” in many of the usual definition of an irrational number.

It’s arguable - and, I’m fairly confident, provable - that and sequence of digits in the continuing decimal representation of an irrational number can be found in (“occurs”) another place in that expansion. For example, the first 4 digits of the decimal part of the irrational number , “1415”, appears again in positions 7701 - 7704.

This does not mean that has a 4 digit repeating decimal part – that is is not 3.1415 1415 1415 …

“Repeating” in the context of describing the decimal (or any other numeral base) expansion of an irrational number means that the sequence repeats with no intervening space. For example, the rational number 7/13 = 0.538461 538461 538461 …

You can continue expanding 7/13 forever, and the sequence “538461” will never fail to repeat with no intervening digits. You could find some number of consecutive (no intervening digits) occurrence of “538461” in an irrational number, but if you continue expanding it, eventually the next occurance will not be consecutive.

[ughaibu]

CraigD: Thanks for the reply. I understand that isolated sequences dont constitute repetition, if they did repetition would be established by a sequence of more than ten. Between any pair of repeating sequences there will exist a finite sequence, this intervening sequence will also inevitably repeat between the same terminal sequences and thus we will have a repeating sequence composed of our initial three sequences. This pair of repeating conglomerate sequences in turn will be seperated by a finite sequence, a finite sequence which will also repeat in the same situation.

[ughaibu]

My above post is badly worded.
An initial intervening sequence won't inevitably repeat in the same situation but to avoid it doing so the number of available intervening sequences will be reduced, thus increasing the frequency of repetition of an alternative intervening sequence.

[ughaibu]

I think that what I was trying to say can be expressed as follows:
1) any expansion of an irrational number consists of an infinite string of non-repeating digits, thus the cardinality of this string is aleph-zero
2) if such strings do not repeat, it seems plausible that they will be normal
3) an infinite normal non-repeating string will contain all possible strings of digits
4) it follows that such a string will be the power set of aleph-zero, ie aleph-one
5) a set of elements, such as an infinite string of non-repeating digits, cant have both the cardinalities of aleph-zero and aleph-one.