Can we just create an irrational number by a series of non zeros and non pattern numbers, if so how far must we go out to be declared irrational. And can there be a rational number that is not the ratio of two integers?

Sure.If you stop at any arbitrary decimal digit, then by definition you've written a rational number. Just move the decimal point all the way to the right of the last digit, make it the numerator of a fraction, and fill in a denominator with the same number of zeros as the numerator has digits. Those are the two integers whose quotient defines your number. Therefore, you have to continue to infinity to create an irrational number. Even then, the way you approach infinity probably matters. After all, at any given digit you can stop and you still have a rational number. You add one more digit and you've still got a rational number. You add any arbitrary number of extra digits and you've still got a rational number!No. That's the definition of a "rational" number.

> so how far must we go out to be declared irrational.

ask a psychiatrist

I was under the impression that an irrational number is a number with no repeating intergers or repeating patterns How does that fit into your definition of a rational number. If we strung out pi and arbitrarily assigned a decimal, would thayt make it rational?

No. Your definition of an irrational number is correct. Any number which has a repeating pattern can be written as the ratio of two numbers, and is hence rational. But since pi has an infinite number of digits without a repeating pattern, regardless of where you move the decimal point, it will still be irrational.

I dont get it. Then cant we create a number like pi (create an irrational number) by just selecting random numbers and string them out? and how far out do we have to go?

You don't seem to understand one of the basic properties of irrational numbers. They must go on forever, to infinity.

As soon as your number stops, meaning that all subsequent digits are zero, you've automatically got a rational number!

Let x = your original, hypothetically irrational, number.
Let n = the number of non-zero fractional digits in it before the infinite series of zeros starts.
Let b = 10^n
Let a = x*b (sorry, using Excel operator symbols here)

Notice that both a and b are integers.

Let c = a/b

It's an easy proof that c = x

In other words, x is a rational number.

Thanks for making that very clear. I appreciate that.

1) How do we prove that sum of Rational and Irrational no. is Irrational ?

2) Also how do we prove that sum of two Irrational nos. is Irrational ?

P.J.LAKHAPATE
pjlakhap@bechtel.com

Itseems to me that a rational number added to a cardical number is still irrational; multiplying an irrational number by a rational number is irrational. However can we make an irrational number rational by adding any infinite series of numbers to it that would make it have repeating patterns or zeros (theoretically of course) ?

sqrt(3) is irrational, as is 10-sqrt(3). However, their sum is integer.