Wikipedia:Reference desk/Archives/Mathematics/2019 January 28

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January 28

Linear equations

Hey, I can use some homework help. Say you're given a set of coordinates, for instance (2,-1) and (-1,5). How do we figure the linear equation ("y=mx+b") that describes the line running through them? I figured out that m = (y2-y1/x2-x1), but what is b (the "y-intercept")? Thanks! Drmies (talk) 00:38, 28 January 2019 (UTC)

Your two data points give you the equations

(1) -1 = 2m + b and

(2) 5 = -1m + b

You need to remove one of the variables, so you can solve for the other, so you manipulate one of those equations so that their parameters (is that the right word?) match. For example, you can double eq 2

(3) 10 = -2m + 2b

Then add eq 1 and eq 3 together to get

9 = 3b

I'm sure you can take it from there. Rojomoke (talk) 00:53, 28 January 2019 (UTC)

Squank you, Rojomoke! Drmies (talk) 01:25, 28 January 2019 (UTC)

Another approach: You know that m = (y2-y1)/(x2-x1) for any two points* (x1,y1) and (x2,y2). Therefore m = (y-y1)/(x-x1) is another form of the equation for the line, based on point (x1,y1) instead of the y-intercept (0,b). Once you have m, you can write the equation in this form, then solve it for y in terms of x.

*Exception: if the line is vertical and therefore m is undefined. But y = mx + b doesn't work for vertical lines either.

--76.69.46.228 (talk) 04:01, 28 January 2019 (UTC)

Or another, perhaps simpler, way: Let your given points be A and B, and let the vertical intercept be C. We know C=(0, b). Then the slope from C to A must equal the slope m from A to B. This is one equation in the unknown b. Loraof (talk) 16:45, 28 January 2019 (UTC)

Mathematical analysis

Hello I am a newcomer to Wikipedia! Sorry for my bad grammar.. My textbook of Mathematical analysis told me to prove $\left\{n\sin {\frac {180^{\circ }}{n}}\right\}$ converge. Of course, it converges to π, then the proof says "Let $t={\frac {180^{\circ }}{n(n+1)}}$", then it started to use the monotone bounded sequence convergence theorem to prove it is convergent. But how did it get t?

Welcome to the Teahouse, Abel Sage Feynman. The Teahouse is for asking and answering questions about editing Wikipedia. Please ask your question at Wikipedia:Reference desk/Mathematics instead. Cullen³²⁸ Let's discuss it 02:01, 28 January 2019 (UTC)

This kind of question is tricky because every introductory analysis textbook has different ideas on what you're supposed to 'remember' from previous math courses and what you're supposed to 'forget' because it hasn't been proved rigorously in the context of the book. Apparently you're not supposed to 'remember' everything, since otherwise, as you pointed out, you could just say the limit is π by calculus. But it looks like you are supposed to 'remember' the trigonometric functions and their basic properties. (Another example of differences in context here is that what you call the "monotone bounded sequence convergence theorem" was an axiom in the book I used an an undergraduate, Protter & Morrey A First Course in Real Analysis. A later edition is available here.)

Without more context I can't be sure what the author of your text had in mind, but if you're going to show that the sequence {nsin π/n} is increasing then you need to examine sin(π/n - π/(n+1)) and it may simplify things a bit to call this sin t. My personal approach would be to first prove that if 0<b<a<π/2 then

${\frac {\sin a}{a}}<{\frac {\sin b}{b}}$

which can be derived from

$\cos \theta <{\frac {\sin \theta }{\theta }}<1$

which, in turn, is from Differentiation of trigonometric functions#Limit of sin(θ)/θ as θ tends to 0 and which I'm assuming is one of the facts your supposed to 'remember' from previous courses. Specifically,

$\sin a=\sin(b+(a-b))=\sin b\cos(a-b)+\cos b\sin(a-b)<\sin b+{\frac {\sin b}{b}}(a-b)={\frac {a}{b}}\sin b.$

It's easy to apply this with a = π/n and b = π/(n+1) to show the sequence is increasing. Anyway, I hope that at least helps with your question. As a bit of general advice, you might want to investigate other texts covering the same or similar material; many can be found on-line and if you get stuck on some point in one book you may find that a different author explains it better or that what you're stuck on isn't really that important. --RDBury (talk) 15:15, 28 January 2019 (UTC)

It actually converges to 180, not $\pi$. Ruslik_Zero 20:52, 28 January 2019 (UTC)

It converges to $180^{\circ }$, which is an old fashioned way to say $\pi$. (Recall $1^{\circ }={\frac {\pi }{180}}$). —Kusma (t·c) 21:05, 28 January 2019 (UTC)

ideal related

I and J are ideals of ring R and J⊆ I then I∩(J+K)=I+(J∩K ) proof — Preceding unsigned comment added by True path finder (talk • contribs) 12:09, 28 January 2019 (UTC)

Ok i have proved it which is actual I∩(J+K)=J+(I∩K ) — Preceding unsigned comment added by True path finder (talk • contribs) 12:45, 28 January 2019 (UTC)