Show that these numbers are irrational

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  • #1
kaos
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Greetings ,
Im taking an online course on mathematical thinking, and this question has me stumped.

r is irrational:

Show that r+3 is irrational

Show that 5r is irrational

Show that the square root of r is irrational.

Im sorry if i posted this in the wrong forum, but im not sure which category these questions fall under. I seriously dont even know where to start.

Btw these questions are under the proofs section of the course.
 

Answers and Replies

  • #2
HallsofIvy
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Use "indirect proof". If r+ 3= x, a rational number, what can you say about x- 3? If 5r= x is a rational number, what can you say about x/5? If [itex]\sqrt{r}= x[/itex] is a rational number, what can you say about [itex]x^2[/itex]?
 
  • #3
kaos
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Is it valid to say that a number is rational XOR irrational. My apologies as it over a decade since i attended high school. I also have only recently been doing proofs and so far only completed one proof in my life , so i am not very good at proofs.
 
  • #4
kaos
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Do you mean proof by contradiction ( supposing the negation of the statement , then disproving the negation) ?
We have just learned this type of proof in the last lecture

If so, would (there exists r)(for any p )(for any q)[(r not equals to p/q) ^
( p not equals to q)] , and ( theres exist x[x=r+3]) implies (for any x )(there exists a )(there exists b)[(x=a/b) ^
(p not equals to q) ]be a first step , then i negate the above
and try to disprove the negation, and thus prove the original statement?
 
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  • #5
Dick
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Yes, proof by contradiction. And yes, a number is either rational or irrational but not both - but that should be pretty clear from the definitions isn't it?
 
  • #6
kaos
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Yes, proof by contradiction. And yes, a number is either rational or irrational but not both - but that should be pretty clear from the definitions isn't it?

Well im not sure that there could be other types of number other than rational or irrational, which is why i asked. In other words im not sure that (~rational implies irrational). Is it valid to say that ~rational if and only if irrational and
~irrational if and only if rational? If i appear dense, guilty as charged lol as its been some time since high school.
 
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  • #7
Dick
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Well im not sure that there could be other types of number other than rational or irrational, which is why i asked. In other words im not sure that (~rational implies irrational).

A rational number is a number that can be expressed as the quotient of two integers. An irrational number is one that can't. I would say the definition of irrational is ~rational.
 
  • #8
kaos
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A rational number is a number that can be expressed as the quotient of two integers. An irrational number is one that can't. I would say the definition of irrational is ~rational.

Thanks Dick , your help is much appreciated.
 
  • #9
kaos
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Is this proof valid? I seen this proof from my study group in this course:

if r + 3 is rational, then r is rational
assume r + 3 is rational. then r + 3 can be expressed as follows
r+3=p/q
where p and q are integers
subtract 3 from both sides
r=p/q−3=(p−3q)/q
p - 3q (since q is an integer) as well as q is an integer, therefore r is rational
Contrapositive proof: if r is irrational, then r + 3 is irrational (~ (r+3 is rational) implies ~(r is rational)).
 
  • #10
kaos
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Is this proof valid?

Suppose 5r is rational
5r is p/q (p and q are intergers with no common factors)
r= p/5q
p and q are integers and 5 is an integer , and 5q is an integer since integers multiplied by intergers are always integers.
Since the r= integers/ integers , r is a rational number.

5r is rational implies r is rational.

Contrapositive of this is:

r is not rational implies 5r is not rational.
 
  • #11
Dick
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Is this proof valid?

Suppose 5r is rational
5r is p/q (p and q are intergers with no common factors)
r= p/5q
p and q are integers and 5 is an integer , and 5q is an integer since integers multiplied by intergers are always integers.
Since the r= integers/ integers , r is a rational number.

5r is rational implies r is rational.

Contrapositive of this is:

r is not rational implies 5r is not rational.

Yes, those look fine.
 
  • #12
kaos
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Thanks Dick and Hallsofivy.
 

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