Math Is Fun Forum

I think it’s called a stereoscopic Klein bottle.

Substitute x = 1 − my into the quadratic expression. If the line is tangent to the circle, this should be a perfect square. Complete the square (or otherwise) and solve for m.

Also try

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[*]Mathologer – https://www.youtube.com/channel/UC1_uAI … JjXWvastJg.[/*]
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The creator is a mathematics professor at Monash University, Melbourne; the channel is full of fun facts about various interesting everyday mathematical objects. I like the way he explains things step by step leading you from simple, easy-to-understand ideas to more complicated themes. In one video, he tries to explain why the infinite sum

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as derived by some string-theory physicists (who have absolutely no clue what they’re doing) is wrong. It turns out that what they are doing is evaluating the Riemann zeta function at the point −1 – this function being the complex analytic continuation of the Dirichlet series. Now, most people, including the proverbial man in the street, know what sums are, even infinite sums. But what is analytic continuation? Dirichlet series? Riemann zeta function? Follow the explanation given by the creator of this channel to find out.

The area between two curves is always the integral of the top curve minus the bottom curve. It doesn’t matter if part or all of this area is below the x-axis: as long as you take the top curve minus the bottom curve and integrate, the result is always positive.

Take these examples:

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They are plotted here. Find the area bounded by them and their points of interesection. Notice that this area is wholly below the x-axis. However the definite integral

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is positive; this is the area required.

In that case, the y-axis shouldn’t come into the question: it should just be area bounded by the two curves and the line x=3. I agree with Bob: the question is not very well stated.

Any time.

The dot product of two vectors depends (as far as angles are concerned) only on the cosine of the angle in between. Since

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it doesn’t matter whether we take the larger or smaller angle between them. Also, as

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it doesn’t matter whether we measure the angle clockwise or counterclockwise – only the angular magnitude is important. Hence, whether w lies between u and v (i.e. passes through the smaller angle between them) or not is not really important: we can always take theta to be the smaller angle between u and w, and phi the smaller angle between v and w.

In general, if the coefficients are rational and one root is an irrational surd, then the other root must be the conjugate surd. This is not true if the coefficients can be irrational as well as rational.

If the coefficients were rational, then the statement would be true. It’s false if you allow the coefficients to be irrational, as the replies below your post show.

If the coefficients are real and one of the roots is complex (and non-real), then the other root must be the complex conjugate. This statement also fails if you allow the coefficients to be complex.

Sorry, I made a mistake. See my revised post below.

If you can’t imagine, then don’t – just try the calculations and see for yourself!