Can Irrational Numbers be Defined Positively?

Question

Most definitions of irrationals that I've seen, basically boil down to merely stating that they're real, but not rational. This is a definition in negative terms: stating what irrational numbers are NOT. According to what little formal logic I've been told, formal, mathematically-logical, consistently well-behaved definitions are supposed to be stated in the positive. I 'd like to know whether irrationals can be defined positively: stating what they ARE, in general, universally-applicable terms; not just giving some specific, limited examples. If irrationals cannot be defined positively, how can we then define reals without referring to irrationals, i.e.: without saying that reals are the union of rationals and irrationals? ...And, furthermore, how can we then define complex numbers? How can we continue using Math, if such fundamental definitions cannot be stated in formal, mathematically-logical, consistently well-behaved terms?

Answer 1

The usual method is to define the reals. In fact, the usual method starts from the natural numbers (defined in terms of sets) and progressively defines the integers, the rationals, the real, and then the complexes.

There are two main ways of defining the reals: Dedekind cuts and equivalence classes of Cauchy sequences. Dedekind cuts are easier to explain, but Cauchy sequences allow easier proofs of most results. I'll explain the cut idea.

A real number is a Dedekind cut of the rational numbers: two sets, A and B whose union is the whole set of rationals and such that every element of A is less than every element of B. If B has a smallest element, we transfer it to A. Essentially, we identify real numbers as the 'holes' in the rationals. For a given real number x, we let A be the collection of rationals <=x and B the collection of rationals >x. Hence, the real numbers form the order completion of the rational numbers. it takes some work to define the operations of addition and multiplication, but it can be done.

The complex numbers are most simply defined as ordered pairs (x,y) of real numbers where (x,y)+(a,b)=(x+a,y+b) and (x,y)*(a,b)=(xy-ab,xb+ya).

Answer 2

Irrational numbers can be found by taking the limits at infinity of certain expressions. For example:

(1+[1/n])^(n)

For any specific integer n, this yields a rational number. However, the larger n gets, the closer your answer approaches 2.718281828... THis number has no end and no repetition (a fact that is very difficult to prove) and is known as e, the natural number. So, taking n all the way to infinity yields an irrational number.

I would say that, in general, irrational numbers can be found by using rational numbers in some infinite, convergent series.

ex:
.1 + .02 + .003 + .0004 + ....+.00000000010 + .000000000011 +.....=.1234567891011121314151617.... will be an irrational number.

The decimals go on forever and even though we see a pattern in the numbers, there is no repeated pattern. This irrational number comes from adding successively smaller rational numbers, though we have to do this to infinity for it to produce an irrational number. This would continue forever, to infinity and beyond (it's late, i'm tired, my family likes toy story).

Any irrational number can be constructed this way.

Hope this helps.

Answer 3

There is no need for definitions to be defined "positively", indeed it's not clear what you mean by this. I could say that an irrational number is any real number x with the property that for all integers p, q (q not equal to 0), |x - p/q| > 0. This probably sounds "positive". But it is exactly the same (just less clear) as saying that an irrational number is any real number that is not rational.

There's no problem defining irrationals as the reals minus the rationals (more formally, the complement of the rationals in the reals). Wheer you are going wrong is in thinking that the reals are defined as the union of the rationals and the irrationals. They aren't.

The real numbers can be defined from the rationals in a number of ways. One way is to define them in terms of Cauchy sequences of rationals - this means a sequence of rational numbers where for any epsilon > 0 you can pick a point in the sequence such that any two later points will be separated by less than epsilon. The real numbers can be defined effectively as the set of "convergence points" of these sequences. (It's clear to see that any real number can be represented as such a sequence: you take successive decimal approximations, e.g. for pi you take 3, 3.1, 3.14, 3.141, 3.1415, ... as your sequence.)

Note that I'm putting "convergence points" in quotes above in order to emphasise that that's only a way of thinking about them. Calling it a convergence point raises all sorts of questions (like "a point in what space?" and "how can we know that the sequence converges to that value?") - actually, they're defined in a more complicated way and we go on to prove these things later.

As I said, there are other ways of defining the reals, e.g. Dedekind cuts (look it up); all these ways can be shown to result in the same thing (i.e. the results are isomorphic to each other).

Answer 4

In mathematics, an irrational number is any real number that is not a rational number, i.e., that cannot be expressed by a quotient of two integers, i.e. it cannot be written as a fraction in which the numerator and denominator are integers.

That is any number that cannot be expressed in the form p/q integer / integer is irrational

when we go further it shall be non terminnating and non recurring.


for example sqrt(2) cannot be put as a/b where both and b are integers.

.3333,,,, can be put as 1/3 so it is rational

The fact that they cannot be expressed as rational means they are irrational. So -ve meaning may be there but meaning is well there.

Answer 5

You lose. There's actually a "positive definition" of irrational numbers, and that is, if expressed in decimal form (or any other base), the expansion is 1) infinite, and 2) is nowhere periodic. So, where are you?

Answer 6

um, prison? You are a number now ( number in prison) ….forever Doors define you With key( locked away behind prison doors) , define me So comforting, concrete so warm, ( the concrete cell) these gray bars ( the bars of the door) Number 23967, welcome ( the prisoner ID)

Answer 7

yes they can be defined positively.
irrational numbers are numbers with the decimal fractions non terminating and non repeating like 1.110111011110111110.....