Equality of irrational numbers and the need for a proof

Question

This seems obvious to me, but do we need a formal proof to establish this is always true?

Let $x$ and $y$ be irrational numbers with decimal representations

$x = 0.x_1x_2x_3 \dots$

$y = 0.y_1y_2y_3 \dots$

If $x_n = y_n$ for $n \in \mathbb{N}$

Can we then infer that $x = y$ ?

Answer 1

Perhaps it seems obvious to you because the symbols would be identical. But a real number is not a symbol, it is an abstract mathematical quantity.

So if you are defining the real numbers by a construction (Dedekind cuts; or equivalence classes of Cauchy sequences of rational numbers), or if on the other hand you are starting from axioms (the axioms for a complete ordered field), there is indeed something to prove.

What a decimal number really is, is just a shortand for a certain infinite series: $$0.x_1x_2x_3\cdots = \sum_{n=1}^\infty \frac{x_n}{10^n} $$ So you will need at some point to prove a theorem which says that if an infinite series converges then the number it converges to is unique.

Answer 2

Disclaimer: if you want to prove something basic like this, it means that you allow yourself only very elementary tools, that are so elementary that they are in fact extremely abstract and make the proof not so easy to grasp.

A real number is an equivalence class of Cauchy sequences of rational numbers, under the equivalence relation $$(u_n)\sim (v_n)\Longleftrightarrow \lim_{n\to \infty} (u_n-v_n)=0$$

In your case by definition $x$ is the class of the Cauchy sequence of rationals $(u_n=0.x_1x_2\ldots x_n)$ and $y$ is the class of the same sequence, therefore the limit of the difference is $0$ and $x=y$.

Answer 3

It seems to me that if we accept the axiom of extensionality then we have the proof.

Then the only question is "can we convert a decimal expansion into a set?"

Let Set $A = \{ \frac{x_1}{10},\frac{x_2}{100},\frac{x_3}{1000} \dots \} $

and Set $B = \{ \frac{y_1}{10},\frac{y_2}{100},\frac{y_3}{1000} \dots \} $

Then by the axiom of extensionality https://en.wikipedia.org/wiki/Axiom_of_extensionality

we can say the two sets are the same and the two decimal terms are also the same.