Is it possible to find a number x such that 5 < x < 1?

[ChiralSuperfields]

Homework Statement

I am trying to find a number x such that 5 < x < 1? I don't think it is possible.

Relevant Equations

No equations

I don't think it is possible, but can someone please explain why?

Many thanks!

 

[Frabjous]

Not possible because < is transitive. If x existed it would imply 5<1.

 

[fresh_42]

The answer depends heavily on the ordering, i.e. the role of the numbers and the meaning of $<\,.$ Let us assume we have natural numbers (including zero) and the natural order here. Then one way to see it is impossible is that the solutions $x$ have to be elements in
$$
\{x\in \mathbb{N}\,|\,5<x\}\cap \{x\in \mathbb{N}\,|\,x<1\}=\{6,7,8,\ldots\} \cap \{0\}= \emptyset
$$
which is empty, and without zero as a natural number, we would directly see that there is no solution to $x<1.$ This demonstrates that the domain we take the numbers from is essential.

Another way to see that there is no solution is by logic: Assume we had a solution $5<x_0<1.$ Then
$$
0< 4=5-1 < x_0-1 < 1-1 < 0
$$
but $0<0$ is impossible in our standard ordering.

I don't have a specific example in mind, but there could be a solution in domains other than the integers and with an ordering other than our normal ordering. It all depends on which meanings you attach to the symbols you use.

 

[YouAreAwesome]

In the real numbers $5 > 1$. But in this statement $5 < 1$. Therefore no such $x$ exists.

 

[SammyS]

Homework Statement: I am trying to find a number x such that 5 < x < 1? I don't think it is possible.
Relevant Equations: No equations

I don't think it is possible, but can someone please explain why?

Many thanks!

Why is it that you are looking for such a number?

 

[ChiralSuperfields]

Why is it that you are looking for such a number?

Thank you for your replies @Frabjous , @fresh_42 , @YouAreAwesome and @SammyS!

@SammyS, I starting wondering whether after I wrote a wrong domain restriction I think.

 

[WWGD]

This works, but needs rescaling to fit the OP:
Assume there are numbers S between 0 and 1. Then since the set is bounded, it has a least upper bound L, so that:
0<L<1
Now multiply thtough by L:
0< L^2<L
Then L^2 is a bound lower than L. Contradiction.

 

[Steve4Physics]

Homework Statement: I am trying to find a number x such that 5 < x < 1?
.
I don't think it is possible, but can someone please explain why?

As a non-mathematician, I’ll add my tuppence worth.

‘5 < x < 1’ is always wrong because it requires (as already noted by others) that 5<1, which is untrue.

No value of x can resolve this. ‘x’ is irrelevant. So trying to find x is a meaningless exercise. IMO.

 

[fresh_42]

As a non-mathematician, I’ll add my tuppence worth.

‘5 < x < 1’ is always wrong because it requires (as already noted by others) that 5<1, which is untrue.

No value of x can resolve this. ‘x’ is irrelevant. So trying to find x is a meaningless exercise. IMO.

You made a lot of assumptions here. Your "always" is plain wrong if we allow p-adic numbers. E.g. $|5|_5=\dfrac{1}{5} < 1=|1|_5.$

Yes, I agree that the OP presumably didn't have p-adic numbers in mind, but to claim "always" and underline it is voodoo and has no place in mathematics if there are counterexamples.

You could at least have listed all your assumptions!

 

[symbolipoint]

Homework Statement: I am trying to find a number x such that 5 < x < 1? I don't think it is possible.
Relevant Equations: No equations

I don't think it is possible, but can someone please explain why?

Many thanks!

WHY? The statement says effectively 5 is less than 1.

 

[Steve4Physics]

You made a lot of assumptions here. Your "always" is plain wrong if we allow p-adic numbers. E.g. $|5|_5=\dfrac{1}{5} < 1=|1|_5.$

Yes, I agree that the OP presumably didn't have p-adic numbers in mind, but to claim "always" and underline it is voodoo and has no place in mathematics if there are counterexamples.

You could at least have listed all your assumptions!

Fair comment.

I was assuming that '5 < x < 1' - with no ornaments or further qualifications - could only legitimately be interpreted as the comparison of 3 real (or rational or integer) numbers. I should have stated it explicitly.

But (in the context of 3 real numbers), I was trying to make the OP aware of the flaw in the original question. The underlining was meant to help the OP by emphasising a key point - in the same way we emphasise words in speech.

 

[fresh_42]

Fair comment.

I was assuming that '5 < x < 1' - with no ornaments or further qualifications - could only legitimately be interpreted as the comparison of 3 real (or rational or integer) numbers. I should have stated it explicitly.

But (in the context of 3 real numbers), I was trying to make the OP aware of the flaw in the original question. The underlining was meant to help the OP by emphasising a key point - in the same way we emphasise words in speech.

I am just annoyed by two facts here on PF, so sorry that it was your comment I complained about. It could have been any other in "mathematics" here.

a) We always pretend to know what the OP knows and what does not. I think they should speak for themselves.

b) We stick to school mathematics. This is embarrassing. Any post on the school level would be instantly destroyed in general relativity, but in mathematics, it seems to be ok to ignore everything above the high school level. That was why I said

It all depends on which meanings you attach to the symbols you use.

It is almost impossible to increase the level in the mathematical forums without getting into trouble. The math forums suffer from too many teachers and too few mathematicians.

The question itself is either trivial, in which case every post after the second one is ridiculously obsolete (a), or the whole thread is substandard (b), or people could start to think about which premisses are actually behind the question and where 5 <1 does not apply (c).

We do apparently discuss trivial statements like riding a dead horse. But we ignore the science of mathematics completely.

 

[Vanadium 50]

While I understand, appreciate and at times share your annoyance, this was posted in a precalculus section. So it is not crazy to imagine the OP has the most straightforward interpretation.

 

[fresh_42]

While I understand, appreciate and at times share your annoyance, this was posted in a precalculus section.

Then it is even more important not less to strengthen:

It all depends on which meanings you attach to the symbols you use.

So it is not crazy to imagine the OP has the most straightforward interpretation.

So (a) or (b) apply and those additional answers (from #3 on) are only show-offs? Riding a dead horse?

Well, I agree. Not that this annoyed me less.

We have two different policies: mathematics = school, physics = science.

 

[malawi_glenn]

We stick to school mathematics. This is embarrassing.

There are plenty of fun math in the technical math sunforums. Like the recent lie algebra and representations thread.

This is precalc HW section.

 

[fresh_42]

There are plenty of fun math in the technical math forums. Like the recent lie algebra and representations thread

Yes, sometimes. I think of mathwonk's discussion with bhobba about hyperreals.

But in cases like this one here, either already my post #3 was far too long and the answer trivial, or we can discuss exceptions. I do not see how we reject the discussion about orders and still have more than one line as an answer. Job done after @Frabjous's post #2, maybe with the exception that someone could have - but did not - explain transitivity.

 

[malawi_glenn]

The math forums suffer from too many teachers and too few mathematicians

And too many physicsists ;)

 

[fresh_42]

And too many physicsists ;)

Yes. I barely dare to mention $\operatorname{char}(\mathbb{F})\neq 0.$

 

[fresh_42]

This is precalc HW section.

So? Doesn't this mean we should provide treats rather than self-censoring our answers?

There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.

My main criticism is that you get publically stoned if you answer beyond what some here define as the OP's horizon without even asking!

 

[Mark44]

I was assuming that '5 < x < 1' - with no ornaments or further qualifications - could only legitimately be interpreted as the comparison of 3 real (or rational or integer) numbers.

I was making exactly the same assumption.

a) We always pretend to know what the OP knows and what does not. I think they should speak for themselves.

I have responded to a large enough number of this OP's questions that I believe I have a reasonable idea of what the OP knows and what he doesn't.

This is precalc HW section.

Exactly, and my view is that we should tailor are answers with that in mind.

But in cases like this one here, either already my post #3 was far too long and the answer trivial,

Agree on both.

So? Doesn't this mean we should provide treats rather than self-censoring our answers?

If the so-called "treats" are so far above an OP's level, one can surmise that such treats are of no value. It's not about "self-censoring" -- it's about meeting a poster where he or she is. That's where a teacher has an advantage. When a response is way over the head of a poster, the expression "trying to drink from a firehose" comes to mind.

 

[WWGD]

Still, to keep some standards, we need to deal with Mathematical issues: What requirements must an ordering satisfy?

 

[FactChecker]

Homework Statement: I am trying to find a number x such that 5 < x < 1? I don't think it is possible.
Relevant Equations: No equations

I don't think it is possible, but can someone please explain why?

Many thanks!

Mathematically, you could say that the set, $\{x \in \mathbb{R} : x \lt 1 \} \cap \{x \in \mathbb{R} : x \gt 5 \} = \phi$.
That is, the set of x in the Reals such that x< 1 intersected with the set of x in the Reals such that x > 5 is the empty set.
You could also say several of the things that others have posted.

 

[WWGD]

So, the expression 5 < 1 by itself does not provide enough information to determine what this '<' means, nor how it works by itself.

 

[FactChecker]

So, the expression 5 < 1 by itself does not provide enough information to determine what this '<' means, nor how it works by itself.

As others have said, I think that we should restrict ourselves to precalculus scenarios. Otherwise, we could go off in many abstract directions.

 

[scottdave]

Just to throw something else into the mix: if we are dealing with ordinal numbers: First, Second, Third, Fourth, Fifth. One might say that First is "greater than" Fifth. If you are doing some calculations where these are predictor variables (or the result variable) it could make sense to view this way.

 

[malawi_glenn]

In the physics HW threads, we often assume things like motion is close to the surface of the Earth so that acceleration due to gravity is approx 9.8 m/s², and that air resistance can be neglected unless otherwise stated.

 

[WWGD]

Just to throw something else into the mix: if we are dealing with ordinal numbers: First, Second, Third, Fourth, Fifth. One might say that First is "greater than" Fifth. If you are doing some calculations where these are predictor variables (or the result variable) it could make sense to view this way.

You mean as in using Order Statistics? Say horse number five finished the race before horse number 1( of course, numbers were given before the race)?