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

Algebra.Com
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
The Out Campaign: Scarlet Letter of Atheism
RELATED QUESTIONS

Hi! Looking at my teachers notes, they do not make sense. I am also waiting on an emai... (answered by Glaviolette,jim_thompson5910,rothauserc,ikleyn)
My question is to describe two methods for adding and subtracting linear expressions, but (answered by MathLover1,josgarithmetic)
Hi, I had this as a bonus question on my Pre-Cal 12 Chapter 3 test. It's been bugging me... (answered by rothauserc)
If angle ABC is 39 degrees and angle CBD is 114 degrees, then what is angle ABD? I’ve (answered by Alan3354)
The test scores for a very large statistics class have a bell-shaped distribution with a... (answered by Theo)
Dear teaches, my math teacher gave me the following sequence: 1234321, 26121262,... (answered by greenestamps)
Could someone give me an example or two. All this talk about rational expressions has me... (answered by Edwin McCravy)
Ms.Ikelyn, for the solution you gave for the problem here:... (answered by math_tutor2020)
-2x + -^2x= 4 HOW does this make sense? Its on a pre test for the college entrance... (answered by Alan3354)