what does irrational mean when it comes to numbers
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how many irrational number is ther between 1 and 6
In mathematics, an irrational number is any real number which cannot be expressed as a fraction a/b, where a and b are integers, with b non-zero, and is therefore not a rational number. Informally, this means that an irrational number cannot be represented as a simple fraction.
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between 1 and 6, there is infinity of irrational numbers
i would apreciate a new fan =D
To be more precise, in the standard model, there are exactly as many irrational numbers in any non-empty interval as there are real numbers all together. If you assume the continuum hypothesis, that is \[ℵ_1\]
The reason for this is, that there are only countably many rational numbers and if you remove a countable set from a non-countable set, the result remains non-countable.
wow, I can see we have a philosopher aboard
Yes...the infinity of irrational numbers in a set bound by rational numbers is "greater" than the set of all rational numbers.
The way I 'splain this to my kids is that there are alot of numbers on the number line; some are rational and can be actually found, they stay put we you look at them up close.
But then, there are these "irrational" numbers that are there, but no matter how close you look at them, they move a little.
We usually pinpoint this "irrational" little tykes by confining them to radicals, or constants like "pi" and "e". But in the end, you never quite know where they've been or where they are going :)
Also, I think rational means : ratioed... can be put in a ratio form.. 4/7 is a ratioed number.
Irrational numbers cannot be ratioed, ie, put in a ratio form...
Well, I wouldn't consider myself a philosopher. That was only a little set theory.
@amitstre64: your explanation might be ok for children, but relies implicitly on an understanding of real numbers through approximation by a decimal fraction, while at the same time summoning a geometrical notion. Though in the geometrical sense an irrational number is just as fixed as any other real number. You can easily see that by constructing \[\sqrt 2\] as the diagonal of a square with side length 1.
Of course from a practical standpoint you can always only observe any quantity up to a certain precision, but that's the case for anything and not only if its true value is irrational.