Show that these numbers are irrational

In summary, when proving that a number is irrational, we can use indirect proof or proof by contradiction. We can also use the contrapositive of a statement to prove the original statement. Additionally, a number can only be rational or irrational, not both.
  • #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 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.
 
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  • #2
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]?
 
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  • #3
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
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?
 
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  • #5
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?
 
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  • #6
Dick said:
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.
 
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  • #7
kaos said:
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.
 
  • #8
Dick said:
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
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
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
kaos said:
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
Thanks Dick and Hallsofivy.
 

FAQ: Show that these numbers are irrational

1. What does it mean for a number to be irrational?

A number is considered irrational if it cannot be expressed as a ratio of two integers. This means that the decimal representation of an irrational number is non-terminating and non-repeating.

2. How can you prove that a number is irrational?

There are a few methods for proving that a number is irrational. One common method is to assume that the number is rational and then lead to a contradiction, thus proving that it must be irrational. Another method is to use proof by contradiction, where you assume that the number is rational and then prove that it cannot be.

3. Can all numbers be proven to be irrational?

No, not all numbers can be proven to be irrational. Some numbers, such as pi and e, are known to be irrational but have not been proven to be so. Other numbers, such as square roots of non-perfect squares, can be proven to be irrational using the methods mentioned in question 2.

4. What are some examples of irrational numbers?

Some common examples of irrational numbers include pi (3.14159265359...), the square root of 2 (1.41421356237...), and Euler's number (2.71828182845...).

5. Why is it important to show that a number is irrational?

Showing that a number is irrational can help us understand and classify different types of numbers. It can also have practical applications, such as in cryptography, where irrational numbers are used for their unpredictability and security. Additionally, proving that a number is irrational can lead to new mathematical discoveries and advancements.

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