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I have a question which comes from Rudin's Principles of Mathematical Analysis; specifically, from the introduction.

In example 1.1, the author clearly shows that no rational numbers satisfy the equation ##p^2 = 2##.

So, I am trying to imagine myself in a scenario in which I am in a time before the discovery of irrational numbers, which provide us with solutions to equations as ##p^2 = 2##. Furthermore, I am one of these pre-irrational-number mathematician who does a proof to demonstrate that ##p^2= 2## has no rational solutions.

Here is my question: What would make me think that there existed

that one of the irrational numbers—namely, ##\sqrt{2}##—would be one such solution to ##p^2=2##? In other words, does ##p^2 = 2## "get" solutions because we want it to have solutions?

Also, below this example the author attempts to demonstrate that the rational number system has gaps. But would this only be true if we wanted ##p^2 = 2## to have solutions, right? Two questions: (1) why does it matter if the rational numbers are incomplete, and why would we fill in the gaps with numbers and then refer to them as irrational numbers (Note: I am not questioning the name given to these sorts of numbers, but the numbers themselves); (2) by augmenting our number system by including these numbers which fill the gaps, we form the real number system, how do we know that ##\mathbb{R}## does not have any gaps?

In short, why should we want ##p^2 = 2## to have solutions?

In example 1.1, the author clearly shows that no rational numbers satisfy the equation ##p^2 = 2##.

So, I am trying to imagine myself in a scenario in which I am in a time before the discovery of irrational numbers, which provide us with solutions to equations as ##p^2 = 2##. Furthermore, I am one of these pre-irrational-number mathematician who does a proof to demonstrate that ##p^2= 2## has no rational solutions.

Here is my question: What would make me think that there existed

*solutions; and why would we think***any**that one of the irrational numbers—namely, ##\sqrt{2}##—would be one such solution to ##p^2=2##? In other words, does ##p^2 = 2## "get" solutions because we want it to have solutions?

Also, below this example the author attempts to demonstrate that the rational number system has gaps. But would this only be true if we wanted ##p^2 = 2## to have solutions, right? Two questions: (1) why does it matter if the rational numbers are incomplete, and why would we fill in the gaps with numbers and then refer to them as irrational numbers (Note: I am not questioning the name given to these sorts of numbers, but the numbers themselves); (2) by augmenting our number system by including these numbers which fill the gaps, we form the real number system, how do we know that ##\mathbb{R}## does not have any gaps?

In short, why should we want ##p^2 = 2## to have solutions?

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