Exploring Equality: A Fun Look at Mathematical Equivalencies

  • Thread starter dkotschessaa
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In summary, there are many different things in mathematics that equal one, such as the Pythagorean theorem and Euler's identity. The use of LaTeX can also help in understanding these concepts. The Gamma function is a specific function that is defined in terms of an integral and can be used to find the value of 1 in certain cases.
  • #1
dkotschessaa
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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
 
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  • #2
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  • #3
  • #4
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. :)
 
  • #5
Whovian said:
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... :tongue:
 
  • #6
How about the area of one quarter period of the sine function?
 
  • #7
Whovian said:
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.
 
  • #8
dkotschessaa said:
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.)
 
  • #9
Ferramentarius said:
In combinatorics [itex]{{n}\choose{n}} = {{n}\choose{0}} = 1[/itex], while [itex]\varphi(1) = \varphi(2) = 1[/itex] where [itex]\varphi[/itex] is Euler's totient function, also [itex]F_1=F_2=1[/itex] for the Fibonacci sequence and famously [itex]e^{i\pi}+1 = 0[/itex] known as Euler's identity.

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

[itex]e^{i2\pi} = 1 [/itex]
 
  • #11
Whovian said:
:) 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.
 
  • #12
[itex](2i((1/2)!)))^2)/(-\pi)[/itex]
 
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  • #13
bahamagreen said:
[itex](2i((1/2)!)))^2)/(-\pi)[/itex]

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.
 
  • #14
opps!
 
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  • #15
Still, we can use the Gamma function. That's cool.
 
  • #16
dkotschessaa said:
Still, we can use the Gamma function. That's cool.

Yeah, for sure. The Gamma function is one of the coolest functions in mathematics!
 
  • #17
micromass said:
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.
 
  • #18
Mute said:
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)##
 
  • #19
bahamagreen said:
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##.
 
  • #20
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.
 
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  • #21
bahamagreen said:
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:

$$\frac{1}{z} = \frac{1}{z}\times 1 = \frac{1}{z} \frac{z^\ast}{z^\ast} = \frac{z^\ast}{|z|^2}.$$
 
  • #22
Mute said:
'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
 
  • #23
$$0.\bar{9}$$
$$\lim_{n \to \infty} \sqrt[n]{n}$$
More general, for every real a:
$$\lim_{n \to \infty} \sqrt[n]{n^a}
 
  • #24
-e^(i*pi*2k) where k is an integer
 
  • #25
JNeutron2186 said:
-e^(i*pi*2k) where k is an integer

Are you sure about that - ??
 
  • #26
Pretty sure its a more general euler's equation

micromass said:
Are you sure about that - ??
 
  • #27
JNeutron2186 said:
Pretty sure its a more general euler's equation


What happens if k=0?
 
  • #28
Ʃ(1/2^k) from k = 1 to infinity
 
  • #29
micromass said:
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.
 

1. What is the concept of "things that are equal to 1" in science?

The concept of "things that are equal to 1" refers to the idea that certain quantities or properties have a value of 1. This can include physical quantities such as mass, length, or time, as well as mathematical concepts such as fractions or ratios.

2. How is the concept of "things that are equal to 1" applied in scientific experiments?

In scientific experiments, the concept of "things that are equal to 1" is often used to standardize measurements and ensure accuracy. By setting certain variables or properties to a value of 1, scientists can compare and analyze other measurements in relation to this standard value.

3. Can "things that are equal to 1" have different units of measurement?

Yes, "things that are equal to 1" can have different units of measurement. For example, the length of 1 meter and the length of 100 centimeters are both equal to 1, but they have different units of measurement. In scientific experiments, it is important to convert measurements to a standardized unit for accurate comparisons.

4. Are there any exceptions to the concept of "things that are equal to 1" in science?

Yes, there can be exceptions to the concept of "things that are equal to 1" in science. In some cases, scientists may use a different value as a standard instead of 1, depending on the specific experiment or field of study. Additionally, certain properties or quantities may not have a value of 1 and cannot be compared in this way.

5. How does the concept of "things that are equal to 1" relate to scientific theories and laws?

The concept of "things that are equal to 1" is often used as a foundation for scientific theories and laws. By establishing a standard value of 1, scientists can make observations and predictions based on how other measurements or properties relate to this standard. This helps to create a uniform understanding of the natural world and inform scientific theories and laws.

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