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Well, I know that $\sqrt{2}$ is an irrational number and I am also familiar with the proof by contradiction method, but I'm confused by this notation as we can divide $\sqrt{2}$ by $1$ (as $\sqrt{2}$ is a real number and for a real number it is possible) and will get $\sqrt{2}$, so can't $\sqrt{2}$ be written as $\frac{\sqrt{2}}{1}$, and in that case $\sqrt{2}$ should be a rational number.

Amaan M
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senku ishigami
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    A rational number needs to be something you can write as $\frac{p}{q}$, where $p$ and $q$ are both integers, not real numbers, and $\sqrt{2}$ isn't an integer, so that doesn't qualify. – Amaan M May 06 '21 at 19:28
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    Rational numbers are those numbers who can be written as an integer divided by another integer. $\frac{\sqrt{2}}{1}$ does not satisfy this requirement since the numerator is not an integer. Whether or not a number can be written as a real divided by another real is irrelevant. – JMoravitz May 06 '21 at 19:28
  • And yes, we can of course divide $\sqrt{2}$ by $1$ and it be equal to the original number. That doesn't say anything about its rationality however. – JMoravitz May 06 '21 at 19:28
  • I think you should read about Dedekind cuts to see what irrational numbers can be. – Emil May 06 '21 at 19:30
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    @Emil You are suggesting learning about dedekind cuts to someone who asked a question of this level? Certainly not... that will only confuse them further. Let them learn math for another decade first. – JMoravitz May 06 '21 at 19:31
  • @JMorawitz: yes, I think diving in the deep end is good for you. I did not hear about it until several years after my engineering studies when I was autodidacting myself to make up for math I did not learn in uni. – Emil May 06 '21 at 19:33
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    @Emil There is a certain level of required "mathematical maturity" here. You don't try to teach a child about french baking techniques when they are confused by the concept that different powders are different ingredients and don't know how to boil water on their own. Letting them know that such techniques exist and that there are results, sure, but they aren't ready yet to understand or appreciate any of the details involved. – JMoravitz May 06 '21 at 19:40
  • @JMoravitz: Often mathematical ideas are really intuitive, I did not ask him to apply it for goodness sake :-) – Emil May 06 '21 at 19:50
  • Well, thnx for answering my question,and for an interesting convo b/w u both :) – senku ishigami May 08 '21 at 05:24
  • With this definition, EVERY real number would be rational. – Peter May 08 '21 at 08:49

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