SOLUTION: Hi, this is a question from a worksheet that I've been working on, it doesn't really make much sense to me. Here it is: Which number below is an element in the set of irrational nu

Click here to see ALL problems on real-numbers

Question 635202: Hi, this is a question from a worksheet that I've been working on, it doesn't really make much sense to me. Here it is: Which number below is an element in the set of irrational numbers? √4, 3.45, -8.7, √2
Please help, thanks.
Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
, so is rational

, so is rational.

Keep going. If you can represent the number using a fraction of integers, then it is rational. If not, then it's irrational.

Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

A rational number is a number that can be expressed as the quotient of two integers. An irrational number cannot be expressed as the quotient of two integers.

But:

1. Assume that is a rational number, meaning that there exists an integer and an integer such that .

2. Then can be written as an irreducible fraction such that and are coprime integers and .

3. It follows that and then .

4. Therefore is even because is even by definition of even.

5. It follows that must be even as (squares of odd integers are also odd, referring to ) or (only even numbers have even squares, referring to ).

6. Because is even, there exists an integer that fulfills: .

7. Substituting from (6) for in the second equation of (3): is equivalent to is equivalent to .

8. Because is divisible by two and therefore even, and because , it follows that is also even which means that is even.

9. By (5) and (8) and are both even, which contradicts that is irreducible as stated in (2).

Therefore the irreducible fraction does not exist.

John

My calculator said it, I believe it, that settles it