What Are the Fun Mathematical Equivalencies That Equal One?

[dkotschessaa]

Sort of a fun question, I think. Or perhaps silly. I was just thinking about all the different things in mathematics that happen to equal one. Such as

(sin x)^2 + (cos x)^2 = 1

in probability P(s) = 1

The radius of a unit circle: x^2 + y^2 = 1

Obviously I don't mean things like 3-2.

Since "Things which are equal to the same thing are equal to each other." any of these things, regardless of the branch of mathematics they are part of, would be equal to each other.

So then P(s) = (sin x)^2 + (cos x)^2

I have no idea what that would actually mean, of course. But it's interesting (to me) to think about.

What else?

-Dave K

 

[micromass]

So then P(s) = (sin x)^2 + (cos x)^2

I have no idea what that would actually mean, of course. But it's interesting (to me) to think about.

Check out https://www.amazon.com/dp/1860940013/?tag=pfamazon01-20 He actually talks about this!

 

[Ferramentarius]

In combinatorics {{n}\choose{n}} = {{n}\choose{0}} = 1, while \varphi(1) = \varphi(2) = 1 where \varphi is Euler's totient function, also F_1=F_2=1 for the Fibonacci sequence and famously e^{i\pi}+1 = 0 known as Euler's identity.

 

[Whovian]

1=1, 1-1+1=1, 1-1+1-1+1=1, ...

One could come up with an infinite list of things equal to 1. :)

 

[dkotschessaa]

1=1, 1-1+1=1, 1-1+1-1+1=1, ...

One could come up with an infinite list of things equal to 1. :)

Which is why I said I didn't mean stuff like that...

 

[Whovian]

How about the area of one quarter period of the sine function?

 

[dkotschessaa]

How about the area of one quarter period of the sine function?

ah, so integral from 0 to pi/2 of sinx

I got to start learning how to do Latex here.

 

[Whovian]

ah, so integral from 0 to pi/2 of sinx

I got to start learning how to do Latex here.

:) Not too hard. This would be, using fairly simple syntax, \int_0^{\frac{\pi}{2}}\sin(x)\ dx, surrounded with [ itex ] tags, or [ tex ] tags for better formatting but requiring their own line. (The other thing you should probably know is that spaces don't mean anything in LaTeX unless a \ is right before them.)

 

[dkotschessaa]

In combinatorics {{n}\choose{n}} = {{n}\choose{0}} = 1, while \varphi(1) = \varphi(2) = 1 where \varphi is Euler's totient function, also F_1=F_2=1 for the Fibonacci sequence and famously e^{i\pi}+1 = 0 known as Euler's identity.

Right, and if we use 2pi (or tau :) ) in place of pi we get

e^{i2\pi} = 1

 

[fortissimo]

There is a theory on the partition of unity actually. http://en.wikipedia.org/wiki/Partition_of_unity

 

[dkotschessaa]

:) Not too hard. This would be, using fairly simple syntax, \int_0^{\frac{\pi}{2}}\sin(x)\ dx, surrounded with [ itex ] tags, or [ tex ] tags for better formatting but requiring their own line. (The other thing you should probably know is that spaces don't mean anything in LaTeX unless a \ is right before them.)

Workin on it. Thanks. We should have a LaTex practice thread.

 

[bahamagreen]

(2i((1/2)!)))^2)/(-\pi)

 

[micromass]

(2i((1/2)!)))^2)/(-\pi)

This is going to be pedantic, but (1/2)! is not defined. The factorial (how you use it) is only defined for nonnegative natural numbers. You should use the Gamma function and not the factorial.

 

[bahamagreen]

opps!

 

[dkotschessaa]

Still, we can use the Gamma function. That's cool.

 

[micromass]

Still, we can use the Gamma function. That's cool.

Yeah, for sure. The Gamma function is one of the coolest functions in mathematics!

 

[Mute]

This is going to be pedantic, but (1/2)! is not defined. The factorial (how you use it) is only defined for nonnegative natural numbers. You should use the Gamma function and not the factorial.

The TI-83 I used in high school interpreted (-1/2)! to be $\sqrt{\pi}$. I remember being pretty surprised when I found out about that (this was before I knew about the Gamma function)! I think that may have been the only negative value for which the factorial function returned an actual answer, though, so it was probably specially programmed in.

 

[bahamagreen]

The TI-83 I used in high school interpreted (-1/2)! to be $\sqrt{\pi}$. I remember being pretty surprised when I found out about that (this was before I knew about the Gamma function)! I think that may have been the only negative value for which the factorial function returned an actual answer, though, so it was probably specially programmed in.

You mean $\sqrt{\pi}/2$ ? Wolframalpha shows it as that... but it shows Gamma for (-1/3)! as $-\gamma (4/3)$

 

[Mute]

You mean $\sqrt{\pi}/2$ ? Wolframalpha shows it as that...

Nope, notice there's a minus sign: "(-1/2)!", if interpreted as $\Gamma(1-1/2)$, is equal to $\sqrt{\pi}$. "(+1/2)!", if interpreted as $\Gamma(1+1/2)$, is equal to $\sqrt{\pi}/2$.

 

[bahamagreen]

Hmm, I guess I don't understand Gamma...

The results of -(1/2)! and (-1/2)! are different as you indicate.

Gamma takes precedence in the order of operations?
I'll take a look at Gamma.

 

[Mute]

Hmm, I guess I don't understand Gamma...

The results of -(1/2)! and (-1/2)! are different as you indicate.

Gamma takes precedence in the order of operations?a
I'll take a look at Gamma.

'Gamma' is a function, so "$\Gamma(x)$" is the same sort of notation as "f(x)", except that f(x) is a general notation for a function while $\Gamma(x)$ generally refers to a specific function defined in terms of an integral (and the analytic continuation if we consider complex number inputs to the function).

It can be shown that for x = n, where n is an integer, $\Gamma(n+1) = n!$. The trick with the non-integer factorials comes from abusing this notation in the case where x is not an integer, i.e., writing $\Gamma(x+1) = x!$. From this it may be easier to see why (-x)! is different from -(x!).

Edit: to keep this post somewhat on the actual topic, one of the forms of 1 that I use often enough is introducing $1 = z^\ast/z^\ast$ when I want to rewrite a complex number $1/z$ in a more convenient form with the imaginary and real parts readily obvious:

$$\frac{1}{z} = \frac{1}{z}\times 1 = \frac{1}{z} \frac{z^\ast}{z^\ast} = \frac{z^\ast}{|z|^2}.$$

 

[dkotschessaa]

'Gamma' is a function, so "$\Gamma(x)$" is the same sort of notation as "f(x)", except that f(x) is a general notation for a function while $\Gamma(x)$ generally refers to a specific function defined in terms of an integral (and the analytic continuation if we consider complex number inputs to the function).

It can be shown that for x = n, where n is an integer, $\Gamma(n+1) = n!$. The trick with the non-integer factorials comes from abusing this notation in the case where x is not an integer, i.e., writing $\Gamma(x+1) = x!$. From this it may be easier to see why (-x)! is different from -(x!).

Edit: to keep this post somewhat on the actual topic, one of the forms of 1 that I use often enough is introducing $1 = z^\ast/z^\ast$ when I want to rewrite a complex number $1/z$ in a more convenient form with the imaginary and real parts readily obvious:

$$\frac{1}{z} = \frac{1}{z}\times 1 = \frac{1}{z} \frac{z^\ast}{z^\ast} = \frac{z^\ast}{|z|^2}.$$

Cool discussion, and thanks for humoring me. :)

-Dave K

 

[mfb]

$$0.\bar{9}$$
$$\lim_{n \to \infty} \sqrt[n]{n}$$
More general, for every real a:
$$\lim_{n \to \infty} \sqrt[n]{n^a}

 

[JNeutron2186]

-e^(i*pi*2k) where k is an integer

 

[micromass]

-e^(i*pi*2k) where k is an integer

Are you sure about that - ??

 

[JNeutron2186]

Pretty sure its a more general euler's equation

Are you sure about that - ??

 

[micromass]

Pretty sure its a more general euler's equation

What happens if k=0?

 

[Gackhammer]

Ʃ(1/2^k) from k = 1 to infinity

 

[Whovian]

What happens if k=0?

I'll agree, they made a minor fail, it should be $e^{2\cdot k\cdot\pi\cdot i}$, not the negative of that.