Question: Which of the following statements is the most logical way to prove that root 5 is irrational?
- Root 5 is an integer.
- Root 5 is a rational number.
- Root 5 is an irrational number.
- Root 5 can be expressed as a fraction of two integers, where the denominator is not equal to 0.
Answer: √5 is an irrational number.
Prove that root 5 is Irrational Solution:
- A rational number is a number that can be expressed as a fraction of two integers, where the denominator is not equal to 0. An irrational number is a number that cannot be expressed as a fraction of two integers.
- To prove that root 5 is irrational, we can use a proof by contradiction. We start by assuming that root 5 is rational. We can then square both sides of this equation to get a^2 = 5b^2.
- Since 5 is prime, this means that 5 must divide a^2. But 5 also divides 5b^2. Therefore, 5 must divide both a and b. This contradicts our assumption that a and b are integers with no common factors other than 1.
So, we have √5 = p/q.
Now, let’s square both sides of the equation to eliminate the square root:
(√5)^2 = (p/q)^2 5 = p^2/q^2
Now, rearrange the equation:
5q^2 = p^2
From this equation, we can see that p^2 is a multiple of 5. This implies that p must also be a multiple of 5 since squaring an integer will result in a multiple of 5. So, we can write p as p = 5k, where k is an integer.
Substituting p = 5k back into our equation:
5q^2 = (5k)^2 5q^2 = 25k^2 q^2 = 5k^2
Now, we see that q^2 is also a multiple of 5, which means q must also be a multiple of 5.
However, this contradicts our initial assumption that p and q have no common factors other than 1 because both p and q are divisible by 5. This contradiction shows that our assumption that √5 is a rational number must be false.
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