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Suppose we start with the opposite view. That is, the square root of 5 is rational. This means that it can be expressed as a ratio of two integers. Let the square root of 5 in its simplest terms be a/b so that a and b are integers with no common factors and b > 0.
sqrt(5) = a/b so 5 = a2/b2
that is, 5b2= a2
5 is a factor of the left hand side (LHS) so 5 must be a factor of the right hand side (RHS).
Since 5 is a prime and a is an integer, 5 must be a factor of a. That is, a = 5c for some integer c and since a and b are coprime b and c must also be coprime.
But now,
5 = (5c)2/b2so that 5 = 25c2/b2and so b2= 5c2. As before, this implies that 5 must be a factor of b. But that contradicts the supposition that a and b are coprime.
The contradiction implies that the original assumption was incorrect. That is, sqrt(5) cannot be rational.
What else can I help you with?
Prove the square root of 2 not irrational?
This is impossible to prove, as the square root of 2 is irrational.
Is the square root of 6 rational?
No; you can prove the square root of any positive number that's not a perfect square is irrational, using a similar method to showing the square root of 2 is irrational.
Prove that root 2 is irrational number?
I linked a good resource that explains what you asked below.
Is the square root of five?
The square root of 5 is an irrational number
Is the square root of 5 an irrational number?
yes
How to prove that Assuming 5 is irrational how would you prove that 5 is irrational?
Root signs didn't show up
Prove the square root of 2 not irrational?
This is impossible to prove, as the square root of 2 is irrational.
Prove that square root of 2 is an irrational number?
The square root of 2 is 1.141..... is an irrational number
How can you prove that the square root of three is an irrational number?
Because 3 is a prime number and as such its square root is irrational
How do you prove that root 2 root 3 is an irrational number?
It is known that the square root of an integer is either an integer or irrational. If we square root2 root3 we get 6. The square root of 6 is irrational. Therefore, root2 root3 is irrational.
Is the square root of 25 rational or irrational?
They are +5 and -5, which are both rational.
Is 7 plus the square root of 5 irrational and why?
7 plus the square root of 5 is an irrational number because the square root of 5 is a never ending decimal number that can't be expressed as a fraction.
Is the square root of 125 an irrational number?
Yes, the square root of 125 is an irrational number. It can be simplified to ( 5\sqrt{5} ), where ( \sqrt{5} ) is an irrational number. Since the product of a rational number (5) and an irrational number ((\sqrt{5})) is irrational, ( \sqrt{125} ) is also irrational.
Is the square root of 6 rational?
No; you can prove the square root of any positive number that's not a perfect square is irrational, using a similar method to showing the square root of 2 is irrational.
How do you rationalize the square root of 5?
You cannot. The square root of 5 is irrational.
What are three irrational numbers?
This is an easy question : root 2, root 3, root 5. Any square root of an integer which is not integral itself is irrational.
Is the square root of 5 an irrational or a rational number?
A rational number can be expressed as the ratio of two integers (the number one does not count). An irrational number cannot. The square root of 5 cannot be expressed as the quotient of two integers. This assertion is by experience that the square root of a prime number is irrational. There are several methods to prove the irrationality -- look for them elsewhere. For the sake of curiosity, my calculator gives the following result for it: 2.2360679774997896964091736687312762354406183596115257242708972454105209256378048994144144083787822749695081761507737835042532677244470738635863601215334527088667781731918791658112766453226398565805357613504175337850034233924140644420864325390972525926272288762995174024406816117759089094984923713907297288984820886415426898940991316935770197486788844250897541329561831769214999774248015304341150359576683325124988151781394080005624208552435422355561063063428202340933319829339597463522712013417496142026359047379... You can say that it is irrational by looking at the inexact answer.
