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basil32
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Homework Statement
Prove If x^2 is irrational then x is irrational. I can find for example π^2 which is irrational and then π is irrational but I don't know how to approach the proof. Any hint?
basil32 said:Homework Statement
Prove If x^2 is irrational then x is irrational. I can find for example π^2 which is irrational and then π is irrational but I don't know how to approach the proof. Any hint?
Curious3141 said:Try a proof by contradiction. Let's say you have such a rational [itex]x[/itex] where [itex]x^2[/itex] is irrational. Then let [itex]x = \frac{p}{q}[/itex] where p and q are coprime integers (meaning it's a reduced fraction). Now see what form [itex]x^2[/itex] takes. Can you arrive at a contradiction considering that this was supposed to be irrational by the first assumption?
Curious3141 said:Wouldn't it suffice to observe that [itex]\frac{p^2}{q^2}[/itex] is a reduced rational number since p and q are coprime? Which would imply that [itex]x^2[/itex] is rational as well, which contradicts the original assumption of the irrationality of [itex]x^2[/itex].
In other words, the negation of the proposition [itex]x^2 \notin \mathbb{Q} \Rightarrow x \notin \mathbb{Q}[/itex] leads to a contradiction. Hence the proposition is true.
basil32 said:yeah, the observation make sense but I need a lemma which proves that [itex]\frac{p^2}{q^2}[/itex] is reduced form whenever [itex]\frac{p}{q}[/itex] is reduced. How do you do that?
basil32 said:yeah, the observation make sense but I need a lemma which proves that [itex]\frac{p^2}{q^2}[/itex] is reduced form whenever [itex]\frac{p}{q}[/itex] is reduced. How do you do that?
basil32 said:yeah, the observation make sense but I need a lemma which proves that [itex]\frac{p^2}{q^2}[/itex] is reduced form whenever [itex]\frac{p}{q}[/itex] is reduced. How do you do that?
The statement is trying to prove that if the square of a number is irrational, then the number itself must also be irrational.
This statement can be proven using a proof by contradiction. This means assuming the opposite of what we want to prove, and then showing that it leads to a contradiction, thereby proving the original statement.
Sure, let's take the number √2 as an example. We know that √2 is irrational because it cannot be expressed as a ratio of two integers. Now, if we square √2, we get 2, which is also irrational. Therefore, based on the statement, we can say that if x^2 is irrational (in this case, 2), then x (in this case, √2) must also be irrational.
No, the statement is always true. It is a well-known mathematical fact and has been proven by many mathematicians.
This statement has important implications in the field of number theory and algebra. It helps us understand the relationship between irrational numbers and their squares, and also provides a powerful tool for proving other mathematical theorems.