Show that square root of 3 is an irrational number

AI Thread Summary
The discussion demonstrates that the square root of 3 is irrational by negating the possibility of it being rational. It asserts that if √3 were rational, it could be expressed as a fraction of two integers a and b, leading to the equation a² = 3b². This results in a contradiction regarding the prime factorization of a² and b², as the presence of the factor 3 on the right side creates an odd count, while the left side maintains an even count. Therefore, the assumption that √3 is rational is proven false, confirming that √3 is indeed irrational. The argument effectively utilizes the fundamental theorem of arithmetic to support its conclusion.
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Homework Statement
Show that ##\sqrt{3}## is irrational
Relevant Equations
The fundamental theorem of arithmetic
##\sqrt{3}## is irrational. The negation of the statement is that ##\sqrt{3}## is rational.

##\sqrt{3}## is rational if there exist nonzero integers ##a## and ##b## such that ##\frac{a}{b}=\sqrt 3##. The fundamental theorem of arithmetic states that every integer is representable uniquely as a product of prime numbers, up to the order of the factors. So ##a## and ##b## are products of prime numbers.
$$\frac{a}{b}=\sqrt{3} \Rightarrow a^2=3b^2$$
Squaring ##a## and ##b## leads to the prime factors ## a^2## and ##b^2## existing in pairs. But, the right hand side is multiplied by ##3##, which means there is an odd number of ##3## in the right hand side. This leads to a contradiction because there is an even number of ##3##s on the left hand side.

So the negated statement is false, and hence the original statement is true.
 
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