Show that these numbers are irrational

[kaos]

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 I am not sure which category these questions fall under. I seriously don't even know where to start.

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

 

[HallsofIvy]

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 $\sqrt{r}= x$ is a rational number, what can you say about $x^2$?

 

[kaos]

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.

 

[kaos]

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 ( there's 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?

 

[Dick]

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?

 

[kaos]

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 I am not sure that there could be other types of number other than rational or irrational, which is why i asked. In other words I am 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.

 

[Dick]

Well I am not sure that there could be other types of number other than rational or irrational, which is why i asked. In other words I am 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.

 

[kaos]

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.

 

[kaos]

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)).

 

[kaos]

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.

 

[Dick]

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.

 

[kaos]

Thanks Dick and Hallsofivy.