Math Is Fun Forum

I've worked with infinity a lot so I'll chime in.  I also take issue with some of the conventional wisdom surrounding ∞ so the following is non-standard.

There are different uses of the term infinity in math.  Perhaps the most common is in reference to transcendental numbers.  Pi for example is an infinite decimal expression, but as a distance it's not ∞.  It exists somewhere between 3 and 4.  Pi is a finite distance from 0 (even if we can't quite pin down the exact distance).

When I refer to ∞ I mean the farthest possible distance from 0.

0 is the lower limit, ∞ is the upper limit. (think radial or scalar dimension)

0 * 0 = 0 (can't get any smaller)
∞ * ∞ = ∞ (can't get any larger)

Like 0, ∞ is non-polar.  +∞ and -∞ are the same thing.  If there is a way for +∞ and -∞ to be distinct, there also must be a way for +0 and -0 to be distinct (which there might well be but I'm not willing to get into that much detail just yet).

As measured from 0 to the tangent, ∞/∞ = 0 ± 1.  Details here, because words alone don't do it justice: [ http://www.perspectiveinfinity.com/root_grid.html#infdivinf ]
0 * ∞ offers a similar result but exists 90 degrees from ∞/∞ on the unit circle.  All points on the unit circle can be understood as distinct fractional representations of ±1.

In addition to 0 ± 1, ∞/∞ also = 0 ± every other natural number.  From an xy perspective when tangent is parallel with the x axis, ∞/∞ results in the entire y-axis.  Only by measuring to specific tangents can we limit the result, such as is the case with 0 ± 1.

0 is an origin.  When we count from 0, everything is relative to 0.  We don't preface every number with 0+ or 0-, even though that's exactly what we're doing.  ∞ can also be used as an origin.  Now instead of counting from 0 towards ∞ we can count from ∞ towards 0.  ∞ + 1 is a finite distance from ∞, but as far as 0 is concerned ∞ + 1 = ∞.  Different perspectives, different answers.

I refer to ∞ along with numbers relative to ∞ as "shadow numbers" since ∞ is not considered a "real number".

Also:
∞ + ∞ = ∞
∞ - ∞ = ∞
Remember, non-polar!  Like zero, infinity is timeless, non-dualistic.  Try splitting something timeless into finite parts and all we split is our perception of it.

Nothing is undefined.

Aww too bad the thread went off-topic.  Anyway I just stopped by to add that while people intuitively go from a circle to a sphere when thinking of 2D to 3D projections, this geometry is easier to project as a helix when going from 2D to 3D.

A sphere is a surface while a helix is still just a curve, so all that's required is to add a dimension and stretch the circle in the direction of the new dimension (in this case, z). This geometry matches that of a sine wave when viewed from a certain perspective: http://www.perspectiveinfinity.com/sinewave3d.html

Yeah lol I realized how that sounded after posting.

I had the two confused for a while.  Squared, square root... squirt, sqrt, sqrd... they blend together.  Thanks for clearing that up.

Umm, still nothing on my original question?

4/3 = 1.333
1.333 squared is 1.154

Thanks MathIsFun.  I've been using that resource frequently (I had come across it before on a search) but there's still so much to burn into my memory.

I think all these different uses of the word "inverse" have been throwing me off.  My first thought when I hear "inverse" is 1/something because that's the first way I learned it.  So invtan is the same thing as arctan?  Bloody hell...

Ok, so perhaps I should go with "sine/cosine swap" or even the more explicit "1/tan" to differentiate from tan-¹?  Sine flip is definitely out (even though I think it sounds catchy).

bobbyn: Thanks for the notation.  How is that text generated?  Is there a tool on this site?

You guys may or may not find it interesting that the only trig function I use in my program is arccosine, and only because I haven't yet found another way to obtain the angle.  I'm not using the unit circle or translating back and forth between radians and real numbers.  It's just basic addition, multiplication, and squaring.  Knowledge of pi isn't even required, except to use arcos of course.

All those seemingly arbitrary symbols and techno-babble is what turns me off from the more traditional approaches to learning math.  I tend to think in shapes, not in words and made up symbols that are used to represent ideas.  However, despite this I've been making an effort to learn what they all mean anyway.

bobbym: On my program I have tangent, sine, and cosine displayed along with other values.  The "sine flip" button swaps the values for sine and cosine, and inverts (1/n) tangent all at the same time.  Pretty simple right?  So what I'm wondering if there's perhaps a more common way of describing this operation?

Here's another question: Right now I'm describing "fractional roots" as this sequence: √1/2, √2/3, √3/4, √4/5... all the way to √∞/∞, along with their reciprocals: √2/1, √3/2, √4/3, √5/4... √∞/∞.  Such a sequence must have already been in use for a while and have a common name, but what is it?

gAr: Thanks.  My geometric calc is done in html5 canvas actually.  I used to write most apps in flash until a year or so ago.  I'm a graphic designer turned programmer.  Art and math go nicely together.

I made another, simpler version of the squarcle that perhaps lends better to it's name:
perspectiveinfinity.com/squarcle2.html

Note that when sine and cosine are matched (1/2sqrt) and tangent is 1, the guide vectors form a tetrahedron.  Cool stuff.

Hi bobbym,

Thanks for posting a link.

The squarcle is just a fun name for it. It's a sine wave projected into 3 dimensions.  As it rotates it looks like a circle from one perspective and a square from another.

A perhaps more "proper" version would use a circle in the middle for the odd numbers as well, but I didn't like having half circles on the sides.