31–40 of 53 posts
Pages like this will be so much better when we have MathML in the browser and can stop using fuzzy images for mathematical formulas.
Given how much harder MathML is to author, the jsMath solution is always going to be more popular among random math folks. Particularly since they're already used to writing TeX.
1- http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From
2- http://books.google.com/books?id=YXv6SEjTNKsC&pg=PA432&#...
Pi is Wrong: http://www.math.utah.edu/~palais/pi.pdf It should be e^(pi i) = 1, but pi was unfortunately defined at half the appropriate value in the 17th century.
Pi is defined as the circumference of a circle of unit diameter. This isn't somehow less correct than the alternative of making that circle one of unit radius. It causes some annoying extra factors (as the PDF points out), but the alternative would cause different complications. The area of a circle, for example, would be r^2*pi/2.
It was programming a lot that brought me to realize this. I got in the habit of always questioning the primitives I'm given, in the hope of simplifying the definitions built on them. It was a little boggling to notice that pi, something I'd grown up with, deserved the same kind of questioning.
Earlier quoted context omitted.
Pi is defined as the circumference of a circle of unit diameter. This isn't somehow less correct than the alternative of making that circle one of unit radius. It causes some annoying extra factors (as the PDF points out), but the alternative would cause different complications. The area of a circle, for example, would be r^2*pi/2.
I'm not convinced that redefining Pi as 2*Pi would cause an equal amount of complications (besides confusing everyone with the change). The radius is more fundamental to a circle. The diameter is just derived from the radius. The idea of a circle with a unit radius is so widely used that it is simply called a "unit circle" with the obvious implication that the radius is what the "unit" refers to. Every time I've done…
And yes, every time you've done math with radians, that problem has haunted you. That's because we decided to use radians instead of "diameter-ans", to keep with the above definition.
Imagine you're a complex number, which is just a type of 2-vector. Exponentiation is to do with growth at a speed which is a multiple of how big you are already. i is the multiplication which turns you through ninety degrees. If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude. Pi is how long it takes you to turn through a half circle. So if you grow at right…
Earlier quoted context omitted.
Thats neat. Only thing is that (to me anyway) this takes the idea of the complex plane as being very fundamental as opposed to just something convenient. I'm not sure how to convince someone that 1 + i is the same as the coordinate (1, 1) without saying "thats just how we define it because things work out."
http://betterexplained.com/articles/a-visual-intuitive-guide... http://betterexplained.com/articles/intuitive-arithmetic-wit... The first link was posted here a while ago, I think.
I really like Euler's Formula, but feel many explanations focus on the raw bits (showing the series are equal). The physics explanation "90-degree rotation" is a good one. Here's how I explained it to my mom:
"The formula is saying there's two ways to get to a place on a circle. One is to go across (cos) and up (sin), and the other is to go out (1.0) and rotate (x radians)."
This is a beautiful result which ties together radians, exponential growth, and complex numbers -- I have a post drafted that I want to share on this :).
Imagine you're a complex number, which is just a type of 2-vector. Exponentiation is to do with growth at a speed which is a multiple of how big you are already. i is the multiplication which turns you through ninety degrees. If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude. Pi is how long it takes you to turn through a half circle. So if you grow at right…
Imagine you're a complex number, which is just a type of 2-vector. Exponentiation is to do with growth at a speed which is a multiple of how big you are already. i is the multiplication which turns you through ninety degrees. If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude. Pi is how long it takes you to turn through a half circle. So if you grow at right…
In these terms, Euler's formula would be recast as
e^(i τ) = 1
That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.
Pi is Wrong: http://www.math.utah.edu/~palais/pi.pdf It should be e^(pi i) = 1, but pi was unfortunately defined at half the appropriate value in the 17th century.
Pi is defined as the circumference of a circle of unit diameter. This isn't somehow less correct than the alternative of making that circle one of unit radius. It causes some annoying extra factors (as the PDF points out), but the alternative would cause different complications. The area of a circle, for example, would be r^2*pi/2.
This is actually my favorite example of why pi is wrong—it's the "exception" that proves the rule. To see why, set τ = C/r = 2 pi, and then consider the following chart of common quadratic forms:
integral of u 1/2 u^2
kinetic energy 1/2 m v^2
distance fallen 1/2 g t^2
spring energy 1/2 k x^2
triangular area 1/2 b h
circular area 1/2 τ r^2
N.B. I made a previous comment along these lines, but put it in the wrong place. If you're still a "pi is wrong" skeptic, considering the improvement in radian angle measure may yet convince you:
The explanations mapping complex exponentiation to rotations are basically right. In this context, it's worth noting that the conventional choice of circle constant is off by two. We should be using tau = τ = C/r as the circle constant, rather than pi = π = C/D. Read τ as "turn" and all those radian angle measures suddenly make sense. Ninety degrees? Instead of the confusing π/2 we have 90° = τ/4 = one quarter turn. And so on: 60° = τ/6 = one sixth of a turn, 180° = τ/2 = one half turn, etc.
In these terms, Euler's formula would be recast as
e^(i τ) = 1
That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.