Irrational Numbers

[Racoon]

OK, got alittle thing w/ these...
Can anyone help explain Irrational Numbers?

I haven't done serious Math in over a year or so, and I am going to be taking some heavy stuff this Fall, and want to start brushing up Now!

I did a search on this, and it seems CraigD had a pretty good thread going with Turtle a few months back..

so?:

Pi = irrational
Golden Ratio = irrational? --> Craig had stated (5^.5 + 1) / 2
Fibionachi Sequence = irrational?

Can we get an Irrational discussion going on?

thanks, and looking forward to comprehensible irrationality

[ronthepon]

Irrational numbers?

Numbers which are infinite decimals which do not repeat.

Example 1.101001000100001...

You can never use these numbers in linear equations properly without using approximation methods.

[Farsight]

You won't find me posting much on numbers. I find them somewhat artificial and struggle to maintain any interest. For example when it comes to irrational numbers, I think it's the measuring system that's irrational, not the number. Imagine, if I said measure the size of this third but only let you use a ruler divided into halves, and each half is divided into a half, etc.

Sorry.

[C1ay]

Irrational number

[Qfwfq]

The core of the matter:

Rational numbers are a ratio, which means a ratio of two of the previously defined integer numbers.

2/3, 7/5, 6/35

Since Pythagoras, we've been aware of the possiblity of defining numbers that can't be so expressed. A very simple case is any root of any prime number.

Popular, how would you change "the measuring system" in order to make the square root of 2 rational?

[Farsight]

I don't know, Qfwfq.

Use a saw-toothed ruler?

^^^^^^^^^^^^^^^

[Qfwfq]

Or the metric!

Actually, that would only mean the diagonal of the square ain't root two times the side...

Find integers n and d such that and I'll swim non-stop from Bering Strait to Southern Australia, all alone and in the nude.

[CraigD]

Originally Posted by Racoon

Can anyone help explain Irrational Numbers?

Several members have done a pretty good job already:

Originally Posted by ronthepon

…Numbers which are infinite decimals which do not repeat.

and

Originally Posted by Qfwfq

Rational numbers are a ratio, which means a ratio of two of the previously defined integer numbers.
2/3, 7/5, 6/35

Since any terminating decimal number can be written as a ratio (eg: 1.234 = 1234/1000), and, slightly less obviously, so can any infinitely repeating decimal number (eg: 0.333… = 1/3), these are in effect the same, correct definition of an irrational number.

Originally Posted by Racoon

Pi = irrational

Correct

Golden Ratio = irrational? --> Craig had stated (5^.5 + 1) / 2

Yes – is irrational, so the golden ratio is too

Fibionachi Sequence = irrational?

No, at least not usually. The ratio of terms of the 2 (or any) Fibonachi sequence is rational, because the terms of any ”standard” Fibonachi sequence (eg: 1,1,2,3,5,8,…) are integers.

You could create a weird Fibonachi sequence where the ration of terms is irrational so that this isn’t true, such as: 1,,,,,…

To head off any potential confusion, let me expand a bit on something in Ron’s post:

Originally Posted by ronthepon

You can never use these [irrational] numbers in linear equations properly without using approximation methods.

While it’s true that an irrational number can only be approximated by a rational number, irrational numbers can be used freely in equations without resorting to approximation. An expression involving Irrational numbers may even have a rational value, eg:

It’s important, I think, to note that there’s nothing necessarily unknown or inexact about most irrational numbers, just that they can’t be written as fractions of integers. They’re less convient, but this is due to coincidental qualities - such as most electronic calculators and computers storing numbers in some way using integers, and that it’s not as easy in most text editing/displaying software to displaying as it is 1.25 – not an inate quality of irrational numbers themselves.

[Farsight]

LOL, Qfwfq.

so

[ronthepon]

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