Proving 2/5*(2^0.5)-1/7 is Irrational: Elementary Math Proof

In summary, the given expression is proven to be irrational using the proof by contradiction method, assuming that the square root of 2 is irrational.
  • #1
lolo94
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0

Homework Statement


Proof 2/5*(2^0.5)-1/7 is irrational

Homework Equations

The Attempt at a Solution


I did this by splitting the expression and setting contradictions
2/5->rational
2^0.5->irrational

Proof first rational times irrational is irrational

Proof by contradiction

Assume the product is rational
let rational be x/y irrational s and the product u/t

rational*irrational=x/y*s=u/t
s=uy/tx

Contradiction s can't be rational

and then I do the same thing for irrational-rational

Is that the right approach?
 
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  • #2
Sure-- a proof is a proof, and that would prove it. The question I have is if you are allowed to take as given that the square root of 2 is irrational, but if you are, proof by contradiction is certainly the way to go.
 

What does it mean for a number to be irrational?

An irrational number is a real number that cannot be expressed as a ratio of two integers. This means it cannot be written as a fraction with a finite number of digits in the numerator and denominator. Examples of irrational numbers include pi and the square root of 2.

How is 2/5*(2^0.5)-1/7 written as a decimal?

The expression 2/5*(2^0.5)-1/7 is equal to approximately 0.3142857143 when written as a decimal. However, this decimal is non-terminating and non-repeating, which is a characteristic of irrational numbers.

What is the proof that 2/5*(2^0.5)-1/7 is irrational?

The proof of this statement involves assuming the opposite, that 2/5*(2^0.5)-1/7 is rational, and showing that it leads to a contradiction. This is done by expressing the number as a fraction and showing that the numerator and denominator have a common factor, which contradicts the initial assumption that the number is in its simplest form.

Why is this proof considered elementary math?

This proof is considered elementary because it only requires basic mathematical concepts such as fractions, exponents, and the definition of irrational numbers. It can be easily understood and solved by students in elementary or middle school.

Are there other ways to prove the irrationality of 2/5*(2^0.5)-1/7?

Yes, there are other ways to prove this statement. One example is using the proof by contradiction method, where you assume the number is rational and show that it leads to a contradiction. Another method is using the proof by contrapositive, where you prove the statement by showing that if the number is rational, then a certain condition must be true, and then showing that the condition is false. There are also more advanced mathematical techniques that can be used to prove the irrationality of this expression.

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