Prove by contradiction

So the student is wrong because he assumed the N is positive because N could also be negative? Is this correct

No, they are wrong because they implicitly assumed that n-1 is a positive, rational number. When it could very well be negative. In the case it's negative, it sure as hell can't be called a new least 'positive' rational number.

Example: If 1/2 is the least +ve rational, then they're saying 1/2 - 1 = -1/2 is the new least positive rational. Wrong.

Reply 41

dasda

OP

15

Original post by RDKGames

No, they are wrong because they implicitly assumed that n-1 is a positive, rational number. When it could very well be negative. In the case it's negative, it sure as hell can't be called a new least 'positive' rational number.

Example: If 1/2 is the least +ve rational, then they're saying 1/2 - 1 = -1/2 is the new least positive rational. Wrong.

Ok i understand this part. The -1 part is wrong. How would you go about part b

Original post by dasda

Ok i understand this part. The -1 part is wrong. How would you go about part b

As I've said before, you can adapt that student's proof but fix it. Clearly, subtracting 1 was the problem there.

Is there any other way you can take n = a/b and obtain a SMALLER, and still positive rational number?

Reply 43

dasda

OP

15

I think i would use 2n but i dont know how i i would make ot smaller

Original post by dasda

I think i would use 2n but i dont know how i i would make ot smaller

2n wont help. It's bigger than n. Try the opposite.

Reply 45

dasda

OP

15

Original post by RDKGames

2n wont help. It's bigger than n. Try the opposite.

1/2n. Why cant we use a number like 1/3

Reply 46

dasda

OP

15

Would we do 1/2 of a/b. This will decrease it by a greater amount compared to sn3rd

Original post by RDKGames

2n wont help. It's bigger than n. Try the opposite.

Original post by dasda

1/2n. Why cant we use a number like 1/3

You can use a third if you want. Nobody is stopping you, but of course, simpler fractions are more favourable.

So clearly 1/2n is smaller than n. Can you now prove that it's also a rational number, and that it's also positive?

Reply 48

Sir Cumference2

12

Original post by dasda

1/2n. Why cant we use a number like 1/3

Both are fine. You can use any operation that makes the fraction smaller (and still rational) but still keeps it positive, thus proving that there is no smallest positive rational number.

Reply 49

dasda

OP

15

A and b are whole numbers so 0.5 of a whole number will stay positive

Reply 50

dasda

OP

15

Original post by RDKGames

You can use a third if you want. Nobody is stopping you, but of course, simpler fractions are more favourable.

So clearly 1/2n is smaller than n. Can you now prove that it's also a rational number, and that it's also positive?

Not sure how I prove that it is the smallest rational number

Reply 51

Sir Cumference2

12

Original post by dasda

A and b are whole numbers so 0.5 of a whole number will stay positive

That's not correct - you need to be really careful with your explanations and check they make sense.

When you multiply a fraction by 0.5, you don't multiply both the numerator and denominator by 0.5. I recommend sticking with algebra: if you have a/b and you multiply it by 1/2, what do you have now as an expression? Then conclude why it's smaller and also still positive.

Original post by dasda

A and b are whole numbers so 0.5 of a whole number will stay positive

0.5 of a whole number does not mean it will stay positive. You should be saying that 0.5 of a positive number will remain positive. But also, 0.5 of a whole number does not mean it will stay whole. 3 is whole but 0.5 of that is 1.5 which is not whole.

Anyway, I need to go now so I'll just finish this off.

\dfrac{1}{2}n < n

is good for us, we just now need to show that

\dfrac{1}{2}n

is rational, and positive.

To show it's rational, notice that

\dfrac{1}{2}n = \dfrac{a}{2b}

and since

a,b

are whole numbers, it means that

2b

is definitely a whole number. Hence

\dfrac{a}{2b}

is whole number over whole number, hence rational.

To show it's positive, well clearly since

n = \dfrac{a}{b}

is positive, it means a,b have the same sign (either both +ve or both -ve), in either case, we know that

2b

has the same sign as

b

, hence in

\dfrac{a}{2b}

we have that

a,2b

have the same sign therefore the fraction is +ve.

Hence, we found a new least positive rational number.

Bit more wordy than it needs to be, but you need to understand the specifics at play here in order to fully grasp what's happening.

Now just argue that there is no least positive rational number because we assumed there is one, but we found an even smaller one. Hence contradicting the assumption.

(edited 4 years ago)

Reply 53

dasda

OP

15

Original post by Sir Cumference

That's not correct - you need to be really careful with your explanations and check they make sense.

When you multiply a fraction by 0.5, you don't multiply both the numerator and denominator by 0.5. I recommend sticking with algebra: if you have a/b and you multiply it by 1/2, what do you have now as an expression? Then conclude why it's smaller and also still positive.