Simple question, but my son has a good argument: How do you answer his question? Suppose you had the line y=x and you are asked to express the interval in which x is positive. I would merely say without a doubt that 0 > x >= infinity since (a) the value of zero is neither positive nor negative and (b) such is the convention I was taught for the right side of this expression regarding infinity. Yet, my son argues, if you stop and think about this, the answer literally says x is greater than infinity which does not make sense. How do you answer that? Is it merely a convention he should accept, or does he have a valid point?
I’m confused. Is the question the interval on which the value of the function is positive? In that case, I’d express that interval as (0, ∞).
I don’t think the convention is to use an equal sign when referring to infinity.
To express it with inequalities, I’d write 0 < x < ∞
Your inequality signs are backwards - you wrote “0 > x” which says x is less than zero, and x >= ∞, which says x is greater than infinity. Both of those signs are backwards, and the one referencing infinity should not have an “or equal to” part.
Btw, he argues the interval should be: infinity => X > 0 to show that X is, at best, equal to infinity. But, like an optical illusion, this notation flip-flops about in my mind until I see it as expressing the very original thing he was trying to work around, only this would be an unconventional way of expressing his answer. [I WAG the paradox here is understanding the concept of infinity? …that infinity is not truly a tangible value, maybe?]
X cannot be equal to infinity. That’s the crux of the issue here. X can approach infinity, but it can’t be infinity.
Thanks!
“Infinity” is not a real number, by the way.
Certainly expressions like x ∈ (0, ∞) , x > 0, etc are all perfectly understandable, as is the phrase “let x be a positive real number.”
Or just 0 < x, or x > 0.
With inequalities, the “< ∞” part is redundant. But with interval notation, like (0, ∞), the ∞ symbol is required to show that the interval doesn’t have an endpoint: it just keeps going.
And yes, the convention is never to use “or equal to” with infinity, in anything I’ve ever seen.
Infinity is not an element of the real numbers, but there are other number systems, some of which look almost like the real numbers, that include infinity as a number. Or possibly +infinity and -infinity as two different numbers.
You might consider some sort of non-Archimedean real closed field, but then there will be lots of infinite numbers, not just “infinity”. Moreover, all those really big numbers will still be positive, and there will be no change to the formula x > 0 nor any need to mention infinity even in that case.
You can also extend the real numbers by adjoining a single point at infinity, but that can’t be what we are talking about since there is no order, or alternatively by adjoining both an ∞ and a −∞ , which also has its uses (though we might not say it looks so much like the real numbers after we find we cannot add ∞ − ∞…)
Is that really the question? Normally the question would be for what interval of x is y positive. Stating the interval of x for which x is positive is trivial (in the technical sense of the word).
Maybe I am remembering wrong, and I misguided him. In the meantime, we found the answer the teacher seeks uses a notation of which I am unfamiliar, for example (0, 9]. The parenthesis means “not included” and the square bracket means “included”. I was taught something different maybe not used anymore. Based on my very first example, it would read: {x: 0>x>infinity). I swear this is how we’d express it (trying to indicate x is between 0 and infinity), but now that I think about it, the signs should go the other way. Blame it on a rusty memory. smh
(Btw, shouldn’t the “reply” button below a post bring up a quote of that post? Has the SD changed this convention?)
Interval notation, which uses parentheses and brackets to indicate endpoints and whether they aren’t or are included, respectively is pretty common. (Not faulting you for not knowing it - I don’t know WHEN it became common.)
Your notation with the curly braces is also correct but a little more formal than the interval notation, which is why I suspect it gets used less - it’s also a bit more intimidating at first glance. And yes, it should be {x: 0 < x < ∞}
(And, for your other note, I would think the “reply” button would automatically quote the post in question, but it doesn’t - I think it’s just a function of the new board software or whatever. You can quote a post, though - once you hit reply and it brings up the edit window, the left-most icon (to the left of the icons for bold and italics) will quote the post you replied to.)
If you hit the Reply button below a post, your post is marked as a reply to that post, but the text is not automatically quoted. If you want to quote it, highlight the text you want to quote a Quote button will pop-up above your selection.
I learned it in 6th or 7th grade, around 1970.
You would use the infinity sign in interval notation, but not in inequality notation. {x: 0 < x < ∞} would be written as {x: 0 < x}. It’s not meaningful to specify x < ∞ because that’s true of every real number.