7
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Square root of 0<x<2
bit of a dated post but something I am curious about. I watched Terence Howard's first appearance on Joe Rogan and found it mostly funny but something he pointed out did pique my interest.
the root of his issues with basic math seem to stem from a fundamental misunderstanding (or dislike?) of 2 things in particular:
- that any number multiplied by a fraction results in a smaller number (basically he struggles a lot with logic of fractional math) and
- the fact that for x where 0<x<2 , that x2 < x*2 when he is apparently under the notion that x2 should always be bigger than x*2 and to him, the fact that that isn't the case for 0<x<2 is evidence of some big mystery or conspiracy
but it did make me wonder if there's a certain name or property given to the numbers where 0<x<2 to note the fact the fact that for those numbers, x2 < x*2?
Approaching the question from a different angle to the other, wonderful responses already present: I think this is illustrative of a flawed "intuition" of the "meaning" of multiplication, fractions, powers, etc. Unfortunately, I find this kind of misconception common amongst students I teach in my math classes.
To many math students, "multiplication" means to make numbers bigger, and "squaring" is taught to make numbers even bigger-er. And this is true for a while, because we work with positive integers. Limited to these, it's easy to dismiss 02 = 0⋅2, 12 < 1⋅2, and 22 = 2⋅2 as exceptions. But the second we start introducing rational numbers, things become much more complicated. And then there's negative numbers, where we start having to talk about absolute values, and so on...
Most students, unfortunately, tap out of understanding math far before then. As a high school teacher, many of my students arrive in my class having not understood multiple years of math classes before them, still with a calcified image of mathematics that fossilized sometime between elementary school and middle-school. So naturally, this more nuanced understanding of operations is very difficult for them to grapple with.
And while you do see counterexamples to this logic everyday in life, it takes mathematical understanding to interpret those real-life counterexamples. Yes, "multiplication by a fraction between 0 and 1" makes a number smaller, but to these kinds of students, they would not interpret it that way. They'd just see it as "something was divided, and then was multiplied" (or vice versa).
I really try to emphasize to my students that division and subtraction are special cases of multiplication and addition, precisely to help break these kinds of misconceptions, and to help them be able to 'reinterpret' expressions, which is (in my opinion) one of, if not THE, most important mathematical skills. I also try to incorporate a LOT of fractions into my curriculum, both to get them comfortable with working with fractions, as well as to help them realize the relative strengths and weaknesses of fractions and decimals (and why fractions are generally preferred in mathematics).
Separately from numeracy and procedural fluency, though, there's a broader conceptual understanding: helping them realize that much of their previous understanding of mathematics, in earlier classes, was in a much more restricted field of numbers and operations. That now, we're revisiting the same concepts, but they might not follow all the same intuitive rules and expectations we've built up so far. Unfortunately, many elementary school teachers actively disdain mathematics, or are uncomfortable teaching it. Too often, elementary school teachers teach outright incorrect math to their students, or apply broad statements, generalizations, and "handy shortcuts" that their students internalize, to their later detriment. ("Tricks" and "shortcuts" is a whole other rant that I'll hold back for now...)
I have to help my students realize that results like these aren't contradictions, as Terrence Howard claims, but instead a consequence of the fact that their horizons have now broadened. What seemed to be true was never actually true. I tell them the story of the black swan, and we play games that help them understand the role of conjectures in mathematics, and how they have to adapt and change in the face of seemingly-contradictory observations and evidence. I really hope this helps impart these important life skills, which are applicable far beyond the reach of mathematics, even.
To conclude: I mentioned before that the ability to rewrite and reinterpret expressions is one of the most fundamental skills of mathematics. And indeed, in the wonderful responses of @TangibleLight and @stu2b50, you can see they're using precisely those skills! It's a wonderful example of how we can use algebra to reaffirm why something makes sense, by rewriting a seemingly-unusual result into a much more understandable relation. And then, as @stu2b50 does, find a way to interpret the meaning of the relation in even more intuitive words: of course multiplying x by itself will be less than multiplying x by 2, if x is less than 2, because the bigger the multiplicand, the bigger the product! But it takes a significant amount of procedural fluency and conceptual understanding to do such rewriting, and unfortunately, many people try to do this, but will make some mistake in their algebraic manipulations at some point. You see this all the time with online viral stuff showing why 0 = 2 or whatever. @TangibleLight touches on this point as well. It takes a lot of mathematical proficiency to recognize false assumptions or errors in the algebraic manipulations, often because, again, they rely on issues caused by introducing non-natural numbers into the mix (often, it's 0 - damned 0).
Anyway, that's a lot of rambling, but I hope it was interesting and/or helpful as an additional insight to this question.
You didn't directly call it out, but reading this, I think I did something in my response that is not fair to Terrance.
When I first started writing, I was going to show the converse: start with
x*x < x*37, and divide both sides byxto findx < 37. But it occurred to me that division with inequalities can be unintuitive, and although it's valid in this case I didn't want to deal with the rigor in this context. So instead I showed the direct statement using multiplication, which (at least to me) feels more intuitive than division even though it's subject to the same nuances.I didn't really think much about this at the time; while writing the comment I was just trying to think of the clearest justification for the relation. I only thought about that decision above for a maybe few seconds. But in some sense, I deliberately omitted details to make the thing feel more intuitive than it probably is, and that's unfair.
Reflecting on your comment has been a good exercise in empathy. I don't generally have much patience for cranks like Terrance Howard, and even less patience for those who provide a platform for such misinformation like Joe Rogan does. But it is important to remember that, almost every time, there's some nugget of a genuinely tricky concept at the heart of the bad mathematics. In this case it's that subtle ingenuity in how the rational numbers are defined, and how that relates to multiplication and division. That I learned it too long ago to remember the difficulty shouldn't invalidate others' confusion with those new ideas.
Thank you for the kind words, and I'm glad my rambling was able to be thought-provoking. : )
You're totally right that it becomes very difficult to remember what it's like to not have that intuition. It's why many math professors in my experience have struggled with pedagogy. They weren't trained in it, so they struggle to understand the perspective of students who lack the intuition that comes with practice, experience, repeated exposure, time to sit with concepts, experimenting with their niche cases and outliers and whatnot.
But that additional layer of not just understanding material, but also anticipating the common mistakes, misconceptions, the sore spots and tricky areas... that's one of my favorite things about teaching! Especially when I get to craft diabolical problems poking at each and every one of them weak spots. ; )
something I will say is that there are interesting patterns in math. speaking for myself, I find it really interesting how if you plot xn, you get a graph of lines which all start out as fractions, and then all converge on
(1,1)(no duh since 1any number is 1) and that they all pass the line of2xbetween1<x<2.it can look like art at a certain point.
!graph
tried to link to the graph but I don't think it worked :shrug:
Not really. I mean, the intuition is just that, if you limit X to be less than 2, of course x * x is less than x * 2. Because we just asserted that x < 2. So the right hand side of that multiplication is always greater.
If I may add to this; while there is no special name to the numbers 0 < x < 2, the technique of inferring bounds on the product x * x from this inequality is called an estimation.
More generally, for a given value of
x>0, you can satisfy the inequalityx^2 < Axby choosingA = x + εfor anyε>0.Consequently, for all
xsuch thatM > x > 0, the inequalityx^2 < Axwill be satisfied by pickingε = M - x(which is necessarily positive sincex<M).Are you just saying that x multiplied by a number bigger than x is bigger than x multiplied by x?
Yes, exactly. That's essentially what is being claimed when stating "
x^2 < 2xfor0 < x < 2". If you fix the constantA=2but reducex, then of course the right hand side will remain larger.As an analogy, given two algorithms that run in
O(t)andO(t^2), there is no guarantee that the former algorithm will always run faster, only asymptotically. The actual performance will depend on multiplicative constants.I don't think there's any special name for it but the definitions of inequality. There's nothing special about the number 2 here:
For all 0 < x < 37, x*x < x*37Notice on the left we have
0 < x < 37, so just multiply all parts byx:x*0 < x*x < x*37. Such insight!The pattern I notice with Terrence and the like is that they'll take some tautology like
0 < x < 2, therefore x < 2, then do some arbitrary algebraic manipulations to it till they find some counter-intuitive-looking relation, and claim it is some big problem. Not to mention if there are errors in any of the manipulations, or if they start out with false assumptions.I'm going to try to be upfront about my own prejudice here: I believe Joe Rogan is easily manipulated, and anyone who would be a guest on his show should be met with a large amount of scrutiny. Logical fallacies are highly likely.
As for responding to any of the points raised: 2 x 0.5 doesn't result is a smaller number. You double the 0.5 into 1. If this individual would think about it that way, they are multiplying the "fraction" by the larger number, and their own way of looking at this would hold up. Ask yourself if you believe that they are truly unaware of this after the amount of effort they have put into selling their ideas to others.
I'm sorry if I come off as combative and dismissive. My issues are with Joe Rogan and the modern zeitgeist which he is a large part of. I'm not sure if these people are foolish, or taking advantage for profit.
At least when it comes to Terrence Howard, obviously nobody can be sure, but I've always assumed he's genuinely mentally ill. As far as I know, his "Terryology" doesn't seem to be in the service of a pyramid scheme, scam, or anything like that - I think he just genuinely has a paranoid suspicion of 'the establishment' which, in his mind view, has materialized as a 'new theory of mathematics'. I've unfortunately seen similar behavior in other, once-friends of mine. If I'm mistaken in any way about Terrence Howard, of course, totally willing to admit that - I'm not very familiar with him beyond the occasional reference to "Terryology".
Totally agreed with regards to Joe Rogan, however.
I've watched that episode multiple times, including the follow-up with Weinstein too. I don't think he's mentally ill. I think Neil deGrasse Tyson (and to some extent Weinstein) had it right on the money. He has a very active mind and has had some exposure to concepts which has made him over-confident, but something that Weinstein said that I hope Howard takes to heart is that Howard needs to have far more humility about the subjects he is dabbling in. I think it's just enthusiasm and lack of humility honestly, and something of a thin skin.
I mean, you know what I meant. the answer will be a smaller number than the one being times by the fraction :-P and if they are both fractions, then it'll be smaller than both of them
I do know what you meant, yes. Unfortunately don't think you know what I meant. The fallacy is representative of a misunderstanding of what multiplication is doing. It is easy to fall into, and the way we learned math in grade school makes it more likely to happen.
Maybe a more clear way of stating it is that you are counting x by x times. So if you have 1 * 2, you count 1, 2. If you have 1 * 0.5 you count 1 half a time, so 0.5. if you have 0.5 * 0.5 you are counting 0.5 half a time, so 0.25.
Others have mostly addressed the x<2 side of things, so I will tackle the other bit: why do we also need x>0? A student, approaching inequalities for the first time, will take x²<2x and divide by x, suggesting that x²<2x whenever x<2. Of course, we know better: if it happens that x=-1, dividing by x should also reverse the direction of the inequality. We don't like dealing with special cases, so a better approach is not to divide at all.
Instead, we can solve by factoring. Subtract 2x from both sides of x²<2x, we find x²-2x<0. Factor, and we find x(x-2)<0. The solution is the interval where the polynomial x(x-2) dips below the x-axis, which is precisely 0<x<2.
This strategy is more powerful than considering special cases. When I teach inequalities, I advise my students to never multiply or divide by anything which could possibly by negative -- there is always a better way using addition, subtraction, and factoring.
We can easily reverse this method to make up inequalities with whatever solutions we wish. For example, if I want an inequality which is true when 2<x<5, I can start at (x-2)(x-5)<0, multiply it out to get x²-7x+10<0, and rearrange to write x²+10<7x. Put into words, we could ask: for what numbers is ten plus the square less than seven times the number?
I hope that makes it a bit clearer why there is a special relationship between x²<2x and 0<x<2. It is the same as the relationship between x²+10<7x and 2<x<5. One expresses the interval directly, the other as a polynomial.
Generally you can say this is an issue of scale. Any greater order operation will at first be beaten out by a simpler one at very low values, but then overtake them in efficacy.
0.5 + 2 = 2.5
0.5 × 2 = 1
0.5^2 = 0.25
Here adding instead of multiplying or squaring is the most effective, but that reverses as soon as you get to the next whole number. It's like how points multipliers are most effective at the endgame when players already have big totals to multiply and comparatively wasted near the beginning.