Mathematics desk
< July 19	<< Jun \| July \| Aug >>	Current desk >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

July 20

Arrows on number line

The number line in wikipedia has arrows on both sides. Tha same is true in many American math textbooks for secondary school. However, in many mathematics textbooks of European countries, North Korea, Taiwan, and Singapore, the number line has arrow only on the right side. The number line represents the set of all real numbers, and the arrow on the right hand side indicates the positive direction. This means that the arrow on number line has an important mathematical meaning. So, the number line has to have only one arrow on the right hand side. It may be all right for the number line not to have any arrow at all. In this case we may understand that we are not considering the direction on the number line. It is remarkable that in wikipedia as well as in all the textbooks in most countries(including USA), the number lines in Cartesian coordinate system have arrow only on one side. It is quite confusing that in those books the authors claim that ‘the coordinate plane is formed by crossing two number lines orthogonally.’ However neither wikipedia nor ay textbook explains why the arrow on the left hand side has disappeared. It is desirable that in wikipedia number line has arrow only on the right hand side for mathematical meaning and internal consistence. — Preceding unsigned comment added by 210.93.99.240 (talk) 00:44, 20 July 2013 (UTC)[reply]

I think the diagram at number line is clear as it stands because it has integer points marked on it. In the absence of marked points, a single arrow can be used to indicate the positive direction, but the double arrow is used to emphasise that the line is infinite in both directions. Both conventions are common, and I don't think we need insist on either one or the other. Dbfirs 12:07, 20 July 2013 (UTC)[reply]

My understanding is that the arrow on the number line is not there for *orientation*, but is there to indicate *extension*. That is, "the arrow on the right hand side indicates the positive direction" is not really what it's doing. Instead, the arrow means "this keeps going, even though we haven't drawn it as such". When you see a number line (or a coordinate axis) with only one arrow, they invariably have an "origin point" - typically starting from zero. You wouldn't put an arrow on a number line of positive numbers, as the positive numbers don't keep continuing past zero, they only extend to the right. So if you had a number line of negative numbers, you'd have an arrow on the left, indicating the infinity of progressively more negative numbers, but none on the right, as negative numbers stop at zero. If you're listing both positive and negative numbers (as the linked article is doing), you have arrowheads on both sides, indicating that the set of all real numbers continues in both directions. -- 71.35.127.252 (talk) 17:31, 20 July 2013 (UTC)[reply]

One can come up with reasons for both, but the reasons for "arrowhead indicates orientation" are better.

When drawing the curve of a "really" continuous function on a subdomain–subrange "rectangle", you don't add an arrowhead to the curve to indicate that "the curve actually continues" (especially when the curve is cut due to the "top bound" of the rectangle). This adds inconsistency if an arrowhead is supposed to mean extension.
Indicating "extension" is less informative. The real line always extends infinitely to both directions. Use of, for example, the non-negative semi-line, is never required — it is formally sufficient to draw the function curve only where the function is defined.
Helper notation for extension vs non-extension already exists:
- If the drawn segment of the line ends at a point marker ("straw"), then the "line" doesn't extend (eg., |-----|---|----------| denotes an interval with some added point markers), otherwise the line extends (eg., ---|-----|---|----------|---> denotes the real line with some added point markers).
- A curve of a function, whose domain is not the full real line, should be accompanied by an empty or filled circle for each domain-subinterval-endpoint.

Except for a curve, just to confuse people, an arrowhead is sometimes used instead of an empty circle. – b_jonas 13:34, 25 July 2013 (UTC)[reply]

Using color as a fourth dimension in graphs

I've noticed that a 3-dimensional plot can sometimes use color as the third dimension, with the usual two spacial dimensions (especially in the complex graphs I've seen on Wikipedia). But what about using it for four-dimensional graphing? We could use the existing 3 spatial dimensions, and make the color of each point (at a certain spacing, in order to preserve visibility) the fourth dimension? It may be easier to picture curved space-time or quadruple integrals (as well as integrals over a 3-dimensional "supersurface" in 4-dimensional space, just like surface integrals over 2-dimensional surfaces in 3-dimensional space). Yet I haven't seen widespread use of this plotting method.--Jasper Deng (talk) 05:38, 20 July 2013 (UTC)[reply]

You can do this, and in some cases it may help visualize things; though, I don't think it really gives you that much useful information since you still don't see the "shape". Moreover, at the point that you'd be working with integrals of that nature, I'd imagine that you'd have gotten used to not visualizing things, or found another way to "see" it. In other words, I think the people who would be looking at the graphs, in general, wouldn't have the need; or wouldn't get much out of it. Just my two cents.Phoenixia1177 (talk) 06:50, 20 July 2013 (UTC)[reply]

I would imagine it would be most useful for plotting a scalar function of three variables, such as a density or temperature function of position. It would also be useful for investigating limits and gradients of such functions. However, yes, it's not very useful for shapes, and it doesn't give the same kind of perspective.--Jasper Deng (talk) 07:05, 20 July 2013 (UTC)[reply]

The rough part about making a four dimensional graph is that it would be difficult to see all four at once, even if only three are spatial. If you imagine your graph as some translucent blob in 3 dimensional space with varying color, I would personally have trouble telling the color at any internal position, being as I would be looking through the rest of the blob at the same time. Someguy1221 (talk) 07:08, 20 July 2013 (UTC)[reply]

4-d surfaces shown as transparent coloured volumes in 3d.

I think using tubes for the lines and making the width of the tube vary as the distance is far better for giving distance when showing 4 dimensional objects. Showing the surface of 4 dimensional objects is difficult as they become 3 dimensional volumes, you can use a light transparent colour for the volume so the nearest point which will be inside the 3d perspective projection still shows. If the object isn't made up of straight edges etc it certainly is much more difficult to represent a 4D object, I still think colour is the wrong way, you're better having something that ties in with your intuition of distance. Density of colour might work with more dense for closer parts but not changing the colour as the object turns. Dmcq (talk) 11:03, 20 July 2013 (UTC)[reply]

I've just attached a couple of pictures of tesseracts, one where the spheres and liness indicate the distance and one where the volumes are transparently coloured according to the sides. In both the hidden vertex has been eliminated. Note the nearest vertex is at the centre of the 3d volume, and there are 4 squashed cubes around that vertex representing the visible sides of th tesseract. Dmcq (talk) 11:36, 20 July 2013 (UTC)[reply]

Have a look at Scientific visualization there are a number of 3D plots using colour for a fourth dimension, for example Maximum intensity projection. The Volume rendering. There are a number of techniques used to represent complex maps C to C using colour.--Salix (talk): 17:35, 20 July 2013 (UTC)[reply]

Knight move degree

Chess Rook move at 0, 90, 180, 270 degrees and bishop at 45, 135, 225, 315. What are the degrees of the knight move? — Preceding unsigned comment added by 201.78.179.109 (talk) 13:15, 20 July 2013 (UTC)[reply]

tan^-1 0.5 is about 26.6 degrees so the knight's move angles will be about n x 90 +/- 26.6 degrees. Gandalf61 (talk) 14:51, 20 July 2013 (UTC)[reply]

But, of course, the knight is supposed to jump over the intervening pieces, so we need to add an angle of elevation as well. :-) StuRat (talk) 07:09, 23 July 2013 (UTC) [reply]

Does it jump? Or does it move through them? What about the nightrider? Double sharp (talk) 15:58, 28 July 2013 (UTC)[reply]