Given a box of dimension $\sqrt{3}$ × $\sqrt{3}$ × $\sqrt{3}$ meters; Is the volume of this box "defined".

Question

Since $\sqrt{3}$ never terminates,there is no finite length ,breadth or height. By this I mean you never end measuring if you want to achieve extreme precision. Say you keep on measuring to get the correct length with a hypothetical instrument that is super precise. So is the box volume defined? (or length for that matter)

I am not confusing infinity with irrational here.

Infinity is like - i am trying to find the largest number and can never end my process of counting (just one example of infinity). Similarly , here I am trying to find the exact length by measuring unit by unit and I cant reach the end, because the number is irrational.

One more interesting question which is related to this is , is the area of a square paper with $\sqrt{2}$ definite.

Please correct me if I am getting the whole idea of infinity wrong.

Answer 1

Its all about how you define your units while measuring.

I can always plot √3 on a real number line . Let the length of this plot from 0 be defined as 1 unit. Then volume of the box is 1 unit^3.

This same volume when expressed in the units of centimeters, the volume is 3√3 cm^3.

Answer 2

If the real numbers are defined, then the volume is defined and it is $3^{3/2} = 3\sqrt{3}$.

Answer 3

As your second question asks "is the area of a square paper with $\sqrt2$ definite"

Consider:-

If you have a square of side 2 units , then what is the length of its diagonal? Is length of diagonal you got is precise(definite)?

By answering above questions you can answer your question yourself.

Answer 4

I believe you may be muddling physics with mathematics.

Let a cube have edge length $a$, a real number. Then its volume is $a^3$, irregardless of the rationality of $a$.

In the mathematical description, we're not concerned about the physicality of these objects. From a physical perspective, it is impossible to measure any length exactly. This is not a limitation of our tools, but a fundamental consequence related to the Heisenberg uncertainty principle.