Undergrad Irrational numbers aren't infinite. are they?

[CollinsArg]

Most than a question, I'd like to show you what I've got to understand and I want you to tell me what do you think about it. I'm not a math expert, I just beginning to study maths, and I'm reading Elements by Euclids, and I've been doing some research on immeasurable numbers.

My statement is that: Irrational numbers are not infinite number, but numbers that can not be measured and that we have only found infinite aporoximate numbers to real irrational numbers.

For example: The number 1.41421... is not √2 but an infinite aproximation of √2.

Using the pythagoras theorem we can get to √2 by trying to know the measurement of the hypotenuse of a right triangle whose legs are 1cm each.

x²+y²=z²
1²+1²=z²
2=z²
√2=z

then the square of the hypotenuse is equal of 2, but when you try to find one of its sides, you realize that (trying to approach the number which multiplied by itself is equal of 2 ):
1.4² = 1.96 (approximate n, but not 2) so you try 1.5² = 2.25 the number is bigger so we try to add decimals, then 1.41² = 1.9881 so you try 1.42² = 2.0164. So, you will keep trying adding decimals and you'll realize that you always get to an approximate number or a bigger number than 2, this way you create a number with infinite decimals which never get to approach the goal (multiplied by itself it's 2). Therefore 1.4121... is not exactly √2 but an infinite approximation to that number, therefore it is √2 immeasureble. And therefore 3.14... is neither π but an infinite aproximation of π. So irrational numbers can't be expressed in numbers. Am I right?

(English is not my first language, sorry if I made grammar mistakes).

 

[jedishrfu]

Your post is a little confusing.

Basically for irrational numbers, yes we use decimal approximations when using them for everyday calculations. The decimal approximations for these numbers are non-repeating strings of digits.

https://en.wikipedia.org/wiki/Irrational_number

There are an infinite number of irrational numbers just as there are an infinite number of integers, rational numbers and real numbers. However since reals are uncountable and rationals are countable then irrationals are uncountable meaning there are many more irrationals than rationals.

 

[mfb]

Therefore 1.41421... is not exactly √2

With the dots, it is exactly √2. The difference is zero.
See one of the many good explanations why 0.999...=1, same concept.

Irrational numbers cannot be expressed with a finite number of digits in the decimal system. So what. The decimal system is not special in any way. You can use base √2 instead of base 10 if you want, then √2=10 exactly.

 

[CollinsArg]

why 0.999...=1

Thank you! I didn't know about this