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January 15

Triangle inequality

I'm thinking of studying maths at uni so to see what it's like I've been looking at some textbooks. In the following textbook (http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF) on page 3 there is a proof of the triangle inequality which I don't understand. So can someone help me understand this proof? And also should I be put off studying maths as I'm already confused by the first proof in a textbook? --178.208.207.19 (talk) 18:25, 15 January 2017 (UTC)

Have you read our article Triangle inequality, and did it help? Rojomoke (talk) 18:45, 15 January 2017 (UTC)

Not really as I have read the Wikipedia article as well as other articles on the web dealing but they all deal with different forms of the inequality and different proofs which I don't understand much either. --178.208.207.19 (talk) 18:51, 15 January 2017 (UTC)

Which part of the proof confuses you ? One way of approaching a proof that you do not understand is to try to think of your own proof instead. The text splits the proof into four cases, but maybe it is simpler to think of two cases initially. If a and b have the same sign (both positive or both negative) then can you see that |a + b| = |a| + |b| ? If a and b have opposite signs then can you see that |a + b| < max(|a|, |b|) ? For completeness you also have to consider the cases where a or b (or both) is 0.

In answer to your second question I would say do not be put off. Reading a maths textbook for the first time is often confusing. You have to get used to the author's style and approach. In this text there seems to be very little motivation for proving the triangle inequality. You are not told why it is important or what would follow if the triangle inequality were not true. Try reading another real analysis text in parallel - topics that are obscure in one text may be clearer in another. Gandalf61 (talk) 18:54, 15 January 2017 (UTC)

Different parts confuse me for different reasons. I think my problem is that I'm trying to follow the authors thinking exactly. I can see for example why as Gandalf shows it works where a and b are positive but I don't see why it was important for the author to point out that "so a + b > 0". Then also assuming I understood cases a, b, c, and d they just seem to prove equality with an = not less than or equal to. --178.208.207.19 (talk) 19:05, 15 January 2017 (UTC)

The proof given isn't as clear as it could be. There are really six cases which the author is disguising as four:

1) a≥0, b≥0

2) a≤0, b≤0

3a) a≥0, b≤0, a≥−b

3b) a≥0, b≤0, a≤−b

4a) a≤0, b≥0, −a≥b

4b) a≤0, b≥0, −a≤b

Try working out the left and right sides of the inequality in each case.

One thing you learn if you try to teach yourself math is that a single book is usually not enough. Get access to another book or two on the same subject, that way if you're confused by one book you can try another one. There are many on-line resources now: course lectures on YouTube, free PDF's of textbooks, etc. It helps if you rewrite the proof in your own words, trying to guess what each next step will be before seeing it the text. There is a whole grammar of proofs that you learn at about college sophomore level, basically how to construct proofs of if-then's, either-or's, iff's, cases, induction, etc. If you've never been to exposed to it then it will be hard to follow a proof where the author is assuming you have. --RDBury (talk) 21:04, 15 January 2017 (UTC)

The reason the author points out that $a+b>0$ is to deduce that $|a+b|=a+b$ (by the definition of absolute value, if $x>0$ then $|x|=x$.)

In the author's cases a and b, he shows that $|a+b|$ is necessarily equal to $|a|+|b|$. In cases c and d he shows the (weak) inequality - $|a+b|$ is shown to be equal to something which itself is less than or equal to $|a|+|b|$.

Anyway, I don't think you should be put off by this. This is not a textbook designed for people who are just starting out, it assumes some background knowledge and mathematical maturity and skims over topics without much of an explanation. Try finding a more beginner-friendly textbook. -- Meni Rosenfeld (talk) 21:19, 15 January 2017 (UTC)

Actually, I would say that in cases c and d aren't really proved. The proof just asserts what |a+b| is equal to without showing why this is correct. If you are familiar with real numbers and absolute values then it should be obvious that these steps are valid on real numbers, but just using them without proof is like asserting that "these operations on any ordered field work the way they do on real numbers". Really they should be proved—and you might find it a valuable exercise to do that. --69.159.60.210 (talk) 21:33, 15 January 2017 (UTC)

I disagree, the proof is complete but you need to read it properly. The part with the black square should be applied to each of the cases c and d separately and then the proof follows directly from the definition (the one which appears just before the theorem, and says $|x|=x$ if $x\geq 0$ and $|x|=-x$ if $x\leq 0$.)

For example, part (c) says: If $a\geq 0$ and $b\leq 0$, then by definition $|a|=a$ and $|b|=-b$ so $a+b=|a|-|b|$. Now, either $|a|\geq |b|$ so $|a|-|b|\geq 0$ so by definition $||a|-|b||=|a|-|b|$ so $|a+b|=||a|-|b||=|a|-|b|$, or $|b|\geq |a|$ so $|a|-|b|\leq 0$ so by definition $||a|-|b||=-(|a|-|b|)=|b|-|a|$, so $|a+b|=||a|-|b||=|b|-|a|$. So $|a+b|$ is either $|a|-|b|$ or $|b|-|a|$, and in either case is $\leq |a|+|b|$ (since $|a|\geq 0$ and $|b|\geq 0$).

For part (d) the proof is the same.

The proof in the book says the exact same thing I did here, the only thing missing is pasting "by the definition of absolute value" everywhere, which is of course unneeded for a non-beginner text. -- Meni Rosenfeld (talk) 22:37, 15 January 2017 (UTC)

First, I didn't mean to suggest that it was defective; only that it skipped steps (as you say, the way a non-beginner text will). Beside the "by definition" step, there's also the bit where you wrote "$|a|\geq |b|$ so $|a|-|b|\geq 0$". That's true because in a field you can construct $-|b|$ and subtract it from both sides, but that's still a step in the full proof. --69.159.60.210 (talk) 08:08, 16 January 2017 (UTC)

Well, you can always take any human-readable proof and deconstruct it to smaller and smaller steps until you end up with a computer-verifiable proof. It doesn't detract from the original proof. $|a|\geq |b|\Rightarrow |a|-|b|\geq 0$ is as obvious as steps go. -- Meni Rosenfeld (talk) 11:43, 16 January 2017 (UTC)

January 16

Maths in school, maths in the real world and maths do by mathematicans.

What are the important similars and differents with maths in school, maths in the real world and maths do by mathematicans? --Curious Cat On Her Last Life (talk) 14:43, 16 January 2017 (UTC)

That's a fantastic question and I don't believe I can do it justice, but offhand I'd say that:

• Math in school is the worst because its defining feature is that it is easy to test. School is a grades factory, and if you can't test it and put a number on it, it's out, no matter how beautiful or useful it is. That's why it focuses on practicing techniques rather than true understanding - testing mastery of a technique is easier than testing understanding. But the techniques are not so important, the ideas are.
• Math in the real world is practical. Its only requirement is that it gets the job done. It can be very interesting and varied, because different jobs requires different kinds of math. There is always a breadth of low-hanging fruit that are fun to explore, because there are always new ideas for how to apply known mathematical concepts to solving the plethora of challenges that life has in store for us. Understanding is important because if you don't understand it, you won't be effective in getting the job done using it. But there comes a point where something seems to work and you use it without really understanding how it works, or even being absolutely sure that it's fundamentally true. Results are everything. And it's somewhat limited in depth, because the vast majority of mathematics has no known applicability.
• Math done by mathematicians is limitless in profoundness and beauty. Everything goes - an idea does not need to sully itself with the real world, as long as it is elegant, interesting and ties in to the great tapestry that is math. But it can be hard to find the motivation to explore abstract notions that are many inferential steps detached from what we are familiar with. Trying to actually contribute to the body of knowledge can be an arduous task, because so much has already been discovered, and the rabbit hole leading to the frontier of mathematical knowledge goes so deep. Mathematical research is the ultimate pursuit of truth, because a result must be proven rigorously to be accepted. This is a blessing to those who value correctness above all else, but to others it can seem like needless technical pedantry that detracts from focusing on the actual concepts.

-- Meni Rosenfeld (talk) 18:02, 16 January 2017 (UTC)

I think this you give a very nice answer to this question actually. Math in school is often considered boring and/or difficult, with lots of rules and hard work, but once you see the elegance of some of the derivations, you'll remember those. The problem is that most people don't, especially not in school where the grading is important, not the comprehension. Rmvandijk (talk) 12:43, 19 January 2017 (UTC)

You might want to break up "math used in the real world" to "math used by the average person", and "math used for jobs that require additional math skills". In the first case, I wouldn't expect much beyond compounded interest, logic, and basic geometry (say to calculate square footage of a house) is needed by the average person. But for those whose jobs require additional math, it can get quite involved, especially in the sciences, except that math proofs aren't often required outside of mathematics itself, but rather they would use existing, proven math techniques. StuRat (talk) 19:29, 16 January 2017 (UTC)

• In the general public (i.e. people who don't use advanced mathematics in their jobs), there are two "types" of mathematical ability that are considered important: numeracy and statistical literacy. Numeracy is the ability to perform mathematical calculations, and statistical literacy is the ability to understand and evaluate the results of calculations. Schools are generally good at teaching the former, but very bad at teaching the latter. This is why, for instance, politicians and marketers can often get away with making dodgy claims (eg these graphs), and why gamblers often incorrectly estimate their odds of winning. Here is the mathematical content of the English national curriculum, roughly in the order they're learned:

Basic arithmetic (counting, adding, subtracting, multiplying and dividing small numbers) - used in basically everything. If you can't do these, that's a serious problem (numeracy)

Number sense (knowing whether one number is bigger or smaller, and by how much). Again, a crucial life skill. (numeracy)

Measurement (reading scales, converting between units) - telling time, cooking, DIY. (numeracy)

Geometry (calculating areas, volumes and angles) - DIY, map reading. (numeracy)

Graphs (drawing and reading) - understanding data presented in articles and adverts (statistical literacy)

Advanced arithmetic (long multiplication, long division, working with complicated fractions) - this used to be important in a lot of everyday life, but calculators have mostly displaced it as a vital skill. (numeracy)

Ratio (proportion, interest) - cooking, DIY, finances (numeracy, but key to statistical literacy)

Probability - gambling, buying products like insurance (statistical literacy)

Statistics (different types of average, interpreting graphs) - understanding data presented in articles and adverts (statistical literacy)

Trigonometry - DIY (numeracy)

Algebra (rearranging formulas, solving for unknowns in simple equations) - in theory, useful for solving problems like "If my car does X miles to the gallon, do I have enough fuel for an hour on the highway", but most people generally don't use it - at least in a systematic way (numeracy)

There are also a few useful life skills that schools don't tend to teach - most don't teach much in the way of accounting, even though that's essential to planning a household budget, quick approximation techniques (for instance, if some politician says, "We spend $X billion on welfare, and that's [too much/too little]", it's very useful to have a rough idea of whether that's big or small compared to the government's budget and how much that is per household, without having to research the exact number), or how to look for signs of misleading data. Smurrayinchester 16:15, 18 January 2017 (UTC)

As an educator myself, I need to make a minor quibble with your description of numeracy. Numeracy is the mathematical analogue of literacy (for reading) and fluency (for speaking). Just as true literacy is more than being able to mechanically pronounce individual words as written on the page, numeracy is more than the mechanical act of being able to perform calculations. Literacy requires reading comprehension above all, the ability not just to understand the correspondence of written words to sounds, or even of being able to understand the definition of a word in isolation, but rather the ability to extract meaning from a passage of writing, and be able to understand the abstract connections between the written word and ideas. True numeracy must incorporate some form of number sense, not merely the ability to remember and perform a rote algorithm, but to understand the meaning of a number, and the meaning of the relationships between numbers and operations. Schools are good at teaching and testing for performing algorithms, but not at developing number sense; as noted, that's because it is complex to test for. Not that it cannot be tested for; just that it cannot be tested for on a scantron, and schools are not willing to put in the resources to assure it is assessed correctly at a systemic level. --Jayron32 13:09, 19 January 2017 (UTC)

That's a fair point. I was trying to make the difference between numeracy and statistical literacy clear, but I did oversimplify. Smurrayinchester 13:26, 19 January 2017 (UTC)

Comment. I know of one difference between math in school and math in the real world. In the real world, it's good to be accurate. In school, directions that say "don't be too smart" can occur; the most common is "use 3.14 for pi", meaning that you must answer problems as if pi were by definition the decimal 3.1400000, not 3.1415926.... Any other example of such a direction common in school math?? Georgia guy (talk) 17:51, 19 January 2017 (UTC)

On the contrary, in the real world we use answers to the correct level of precision, and do not introduce false precision. See significant figures. In school math, we assume every digit matters. In the real world, we know when they don't. --Jayron32 18:00, 19 January 2017 (UTC)

And the key fact is that schoolwork assigns the value 3.14 for pi with the assumption that it has an infinite number of significant figures. I want to know of another example of school math that has the direction "don't be too smart". Georgia guy (talk) 20:05, 19 January 2017 (UTC)

That probably goes back to Meni's "easy to test". The grader just has to look for one answer, with scribbles that look like they're probably right, and then can mark it correct.

On the other hand, it could be a valuable practical lesson — if you just need three sig figs in the answer, don't (usually) bother using seven sig figs in any of the inputs. It will just slow you down, to no benefit. --Trovatore (talk) 20:19, 19 January 2017 (UTC)

But the direction is still popular even though it's a remnant from before scientific calculators became common. Georgia guy (talk) 20:21, 19 January 2017 (UTC)

In the days before calculators, the usual instruction was: "take $\pi =$ ⁠22/7⁠" and the numbers were designed to cancel. Dbfirs 20:41, 19 January 2017 (UTC)

I remember it as ⁠22/7⁠ too! You could of course just leave pi in your answers until the very last step, but in practice I found that the cancellations were very useful: if they didn't happen, I had probably made a careless mistake somewhere. I must disagree with Georgia guy here: when areas of regions involving circles are involved, the problem is of a sort where in the real world, you would probably need that much precision: ⁠22/7⁠ is only about 0.04% off the right value, after all. Of course, in more purely mathematical problems, this would be nonsense. No one would ask for angles in radians where pi is approximated as ⁠22/7⁠ – or at least no one should. Double sharp (talk) 15:16, 20 January 2017 (UTC)

P.S. In the real world, it does not really matter that the expansion of pi never terminates. I know the first 50 decimal digits of pi (and I know some people who know even more), and I cannot think of a single real-world application when that accuracy would be insufficient. 51 orders of magnitude from 10⁰ to 10⁻⁵⁰ takes us from the estimated diameter of the observable universe way past the diameter of a proton, although it doesn't reach the Planck length yet. (This naturally excludes "unnatural" questions like tan(10¹⁰⁰) or tan(10¹⁰¹⁰⁰). Double sharp (talk) 15:22, 20 January 2017 (UTC)

January 19

Scalar triple product and the equation of a plane

I was answering a question on a third semester calculus exam when I realized a connection between the scalar triple product and the equation for a plane. It may be trivial, but I haven't seen it explained in the textbook exposition of the distance formula between a point and a plane. Okay, I crossed two vectors u and v to find a normal vector to a plane parallel to the two vectors. Then to find an equation that defines the plane, given a point P on the plane, I essentially found the dot product of the position vector of the point and normal vector to the plane (P dot (u cross v))--I didn't realize that was I was I doing. Then I was asked to determine whether a second point Q lies on the plane. So I realized that I just had to do same thing (Q dot (u cross v)). But this is just the scalar triple product, which defines the volume of a parallelepiped. What I did seems like can be interpreted as: given two non-zero vectors, the position vector of any point on a plane parallel to the two vectors will produce the same volume when forming a parallelepiped with the two vectors.

This interpretation is not obvious to me and I hope someone can explain it to me. Thank you very much. 69.22.242.15 (talk) 21:17, 19 January 2017 (UTC)

The scalar triple product is a special case of a determinant. One of the properties of a determinant is that it is zero if, and only if, there is a linear relation among the rows. In the case of interest, u and v are two of the rows, which you've assumed are linearly independent (so they span a plane), a third vector x is in the plane spanned by u and v only if there is a linear relation among x, u, and v. This is the case only when det(x,u,v)=0 (i.e., their scalar triple product is zero). So det(x,u,v)=0 defines the equation of that plane. Sławomir Biały (talk) 21:46, 19 January 2017 (UTC)

But the scalar triple product, which I understand to be a 3x3 determinant, isn't 0 here (ax+by+cz=-d defines a plane, with <a,b,c> being a cross product). — Preceding unsigned comment added by 69.22.242.15 (talk) 00:26, 20 January 2017 (UTC)

The volume of the parallelepiped is just the base times the height (the distance from the plane). This height is just the length of the projection of the position vector to the unit normal vector, which is just their dot product. The dot product being zero is the same as the projection to the normal being zero is the same as the position vector lying on the plane is the same as the volume being zero. Hope this helps, too tired to write more clearly.John Z (talk) 11:05, 20 January 2017 (UTC)

I see it now, using the property P dot (u X v) = (P cross u) dot v. Very cool. 69.22.242.15 (talk) 14:03, 20 January 2017 (UTC)

January 20

How much of math is used/useful?

While reading this thread from a couple days ago, I was stopped in my tracks by something User:Meni Rosenfeld wrote: "[Mathematics in the real world] is somewhat limited in depth, because the vast majority of mathematics has no known applicability." This made me wonder, are there any even somewhat reasonable estimations of just how much of math is actually used to solve real-world problems? Is math anything like English, which has, I hear, over a million words now, but you can get along with just a couple thousand words in your vocabulary and be super-fluent with 20-30 thousand? I understand there are probably many obstacles to a precise answer to this type of question. I'm looking for more of a rule-of-thumb type of answer (and maybe a little reckless speculation, just for good measure :). Bobnorwal (talk) 04:26, 20 January 2017 (UTC)

Have to take the opposing position here. Basically all areas of math are applicable and applied these days. And frequently the stuff that people huff and puff the most to keep pure in one generation turns out to be the most useful and practical stuff to apply the next. <namedrop> I think Paul Dirac said something similar somewhere - so his answer / rule of thumb/ reckless speculation/reasonable estimation was - ultimately 100%. (I sat in front of him at a lecture once, when I was knee high to a grasshopper.)</namedrop>.John Z (talk) 10:39, 20 January 2017 (UTC)

I wasn't talking about areas (though arguably it's true with this as well, depending on what you call "area"). I was talking about actual specific results/theorems.

Also, it's true that many areas/results find an application eventually, but in any specific moment in time, most results have not yet been applied. -- Meni Rosenfeld (talk) 12:24, 20 January 2017 (UTC)

• It obviously depends on the quantification you take for "how much of math" (by amounts of characters written on a particular topic? by dollars invested in grants?), but also on what used/useful means. Technology readiness level jumps to mind: there are discoveries that everyone knows will be useful at some point in the future, but are not on the market yet; how do you count those? Tigraan^{Click here to contact me} 11:43, 20 January 2017 (UTC)

[ugh, I "posted" this hours ago, didn't notice there was an edit conflict, and left it as is... Posting for real now.]

I don't know the relevant numbers myself, but I'd say that yes, math is similar to English in this regard.

I'm weighing by specific results rather than general fields. For example, you could take the collection of all theorems that have been proven in peer-reviewed journal articles, and ask which of those have ever been applied. I bet the vast majority have not.

For example, I'm thinking about something like circle packing (Circle_packing_in_a_circle, [1]) (that's a relatively well-known problem, there are much more esoteric ones). In 1994 the optimal way to pack 11 circles inside a circle was found. This no doubt required a tremendous amount of work. Was this result ever applied? I doubt it. When we do want to pack circles, people will usually either truncate an optimal plane packing, or run a heuristic algorithm to find an approximate solution for the specific inner and outer radius given.

Provably optimal packings of squares inside a circle can be useful for the chip fabrication industry, but that is a separate, unique problem. There are tons of results in this overall type of problem ([2]) which, probably, have mostly remained unused. -- Meni Rosenfeld (talk) 12:21, 20 January 2017 (UTC)

I can imagine a lot of people would not get the 11 circle packing given a week to work on it heuristically! I wonder if anyone has ever turned chips to pack them on a wafer better, I think it would be possible to do accurately with the registration used in current step and repeat fabrication but it certainly sounds very worrysome. Dmcq (talk) 13:55, 20 January 2017 (UTC)

There's also a distinction between "useful now" and "useful when it was created". Prime numbers were for the most part trivia - it's useful to know about them for some everyday tasks (numbers with lots of prime factors are easier to divide arbitrarily, and gears with prime numbers of teeth are less susceptible to wear), and some people thought they had some spiritual value, but there was little value to knowing, say, Mersenne primes. Until that is, mathematical cryptography came along and suddenly being able to generate and study very large prime numbers was of critical importance for security reasons. Other examples of pure mathematics that graduated to applied are conic sections, which were just pretty curves until people realized they described orbits, non-Euclidean geometry and Riemannian geometry, which allowed the formulation of relativity, and the Hilbert space, which turns out to be very important in quantum physics. Lots more examples here. So even things that appear useless at the moment may have great applications in a few decades' time. Smurrayinchester 16:34, 20 January 2017 (UTC)

The integral test for convergence has a condition that the function be continuous

My professor says that this condition is unnecessary. I argue that if you have a function that's not defined at any x-value but the integers, the integral test would return a finite sum (namely 0), but I think he said something like the function having to have a value at every point, but isn't that just requiring a continuous function? Maybe I misunderstood his objection, but I think I'm right anyway. 69.22.242.15 (talk) 14:55, 20 January 2017 (UTC)

• "Continuous" is stronger than "defined". Consider for instance the Dirichlet function or Thomae's function, both of which are defined on all reals but not continuous on any nonzero interval.

Now, the integral test for convergence assumes the function to be monotonously decreasing and positive. There is a theorem out there that says such a function has a limit (0 in that case). And I think to remember another theorem that says that the set of discontinuities of a function that is monotonous (and bounded (not sure that is necessary?)) on an interval must be a discrete set. If so, that means the function is almost continuous. Tigraan^{Click here to contact me} 15:11, 20 January 2017 (UTC)

The integral test is always presented as being required to be eventually decreasing (for the region where it is not decreasing the sum is thus finite) and continuous, but I've not seen any proof that explains the continuity requirement. That's really what I'd like to know: a proof that you need a continuous function for the test. It definitely needs to be bounded (thus the non-negative requirement). 69.22.242.15 (talk) 20:41, 20 January 2017 (UTC)

Locus: Ellipse with Axes 1 and √3

About a couple of years ago, I was researching various geometric loci with interesting properties. (For instance, to give just one such example, I asked myself what is the set of triangles with two perpendicular medians; in this particular case, the locus described by the triangle point not belonging to the two perpendicular medians turned out to be a simple circle). Now, one of the several loci I've discovered at the time was an ellipse whose axes formed a ratio of 1 : √3. Unfortunately, I completely forgot what the locus was supposed to represent, and I cannot for the life of me remember what its defining property was. (Before anyone asks, this ellipse is indeed determined by two equilateral triangles sharing a common side, but that's not it). I know this is a long shot, but I was hoping that maybe one of you could help me. Thank you. — 79.113.193.22 (talk) 14:58, 20 January 2017 (UTC)

If the ratio between the ellipse's two axes were 1 : √2 instead of 1 : √3, for instance, then this locus would be related to the set of triangles whose angle bisector (also) bisects the segment determined by the foot of its median and the foot of its height. — 79.113.193.22 (talk) 18:04, 20 January 2017 (UTC)