Advanced Math/infinite

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QUESTION: Hi,

I was reading about maths paradoxes. How could there be infinite number of points in between finite numbers. I dont find it rational. If there is an object with 1 cm lenght and 1 cm breadth, how could the diagonal be root of 2 which is infinite. It is inconceivable and irrational that a finite length is expressed as infinite. Please explain.

regards,

ANSWER: "How could there be infinite number of points in between finite numbers?"
Because there is no lower limit to the difference between numbers.

For example, find a point between numbers 0 and 8.
Halve the difference between 0 and 8 to find the midpoint:
0 < 4 < 8

Halve the difference between 0 and 4 to find the next midpoint:
0 < 2 < 4

Similarly,
0 < 1 < 2
0 < < 1
0 < <
0 < ⅛ <
...
0 < 1/2ⁿ⁺ < 1/2ⁿ
...
There is no limit to how small the difference can be. Since the difference can be halved infinitely many times, there are infinitely many numbers between 0 and 8.

"If there is an object with 1 cm length and 1 cm breadth, how could the diagonal be root of 2 which is infinite."
√2 is not infinite. It is a finite quantity that is represented by a non-terminating decimal (i.e., it has an infinite number of decimal digits).

As another example, recall that the decimal representation of ⅓ is 0.333...
⅓ is a finite quantity that is represented by a non-terminating decimal.


---------- FOLLOW-UP ----------

QUESTION: Thanks for your reply.

"√2 is not infinite. It is a finite quantity that is represented by a non-terminating decimal"

How could a finite quantity be expressed in non-terminating decimal number. If a number is not expressed with a terminating number, it means that the number is not finite..

It simply does not make sense to me.. If 1/3 is finite quantity, why the number is not expressed with terminating digits

ANSWER: Do you know how to convert a repeating decimal into its fraction equivalent? E.g., how to show that 0.3333... = ⅓ ?
=====
Both 1/3 and 1/10 are finite numbers. Expressed as decimals, 1/3 = 0.333... and 1/10 = 0.1 simply because we use are using base 10.

Suppose you used base 3 instead, then 1/3₁₀ would be 0.1₃, while 1/10₁₀ would be the non-terminating 0.00220022... in base 3. They are still both finite numbers, just expressed in a different form.


---------- FOLLOW-UP ----------

QUESTION: Do you know how to convert a repeating decimal into its fraction equivalent? E.g., how to show that 0.3333... = ⅓ ?

Thanks for your reply..I am not asking about conventional methods. I am asking for logical explanation.

Anyway, it looks like maths don't have proper answer.

if 1/3 is non-terminating, it is obvious that the number is not finite. if I say that object of 1cm length, 1cm height have 1.414cm as diagonal and you say that it is 1.41421cm, someone else would say something else. So this will never be ending. The whole thing is not logical in practical world.

Answer
"if 1/3 is non-terminating, it is obvious that the number is not finite."
Your conclusion is incorrect. As I have shown, the same number can be represented as either terminating or non-terminating, depending on the base used. The quantity itself does not change; just its representation.

As for the diagonal of your square: 1.414 cm and 1.41421 are approximations of its length, rounded to 3 and 5 decimal places, respectively. Its exact length is √2 cm.

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