Should Pi Be 2 x Radius? Analyzing Math Analysis

[waht]

We know that pi is circumference over the diameter. But how often do we talk about the diameter in any type of math analysis?

Radius is used 99% of the time so I think it should be more appropriate to define pi as circumference over the radius. It would only differ by a factor of two.

pi = 6.2831...

It just seems more right, and some equations might look even more elegant.

 

[Gokul43201]

It just seems more right, and some equations might look even more elegant.

Not the famous special case of the Euler Relation.

 

[Cyrus]

Engineers use Diameter, not Radius because were smarter.

 

[Integral]

Engineers use Diameter, not Radius because were smarter.

This is a classic.

I are college edumcated.

 

[Robokapp]

oh yea it would ruin Euler's Formula. Something so nice combining 5 best math elements in their simplest form is enough to keep Pi...Pi.

I never studied that formula, I have no idea what it's used for, but I have seen it and it's quite...impressive. I feel sorry for Euler's wife. :)

 

[Alkatran]

Oh please, Euler's formula would be arguably more elegant with a division by two in it. It adds another basic operation and important number to the mix.

Oh, what's that? Elegance isn't important enough to justify re-writing so much math it would take years?

 

[Robokapp]

Oh please, Euler's formula would be arguably more elegant with a division by two in it. It adds another basic operation and important number to the mix.

Oh, what's that? Elegance isn't important enough to justify re-writing so much math it would take years?

Well...you can treat that 2 as a square root...I mean look at this:

$$\sqrt{e^{i\pi}}+1=0$$ It looks a lot more...like something to scare those that haven't encountered it yet! high-tech .

but then...some smart guy will try to make this:

$$\sqrt{e^{i\pi}}+1=0 <=> \sqrt{e^{i\pi}}=-1 <=> e^{i\pi}=(-1)^{2} <=> e^{i\pi}=1 <=> e^{i}=1^{\frac{1} {\pi}} <=> e^{i}=1$$

Which raises my question...and I'd really love an answer to this one...

does $$1^{\pi}$$ really equal 1? I mean $$\pi$$ can't be written as a fraction made of natural numbers due to its irrationality...so how does 1 get raised to it?

 

[TD]

Which raises my question...and I'd really love an answer to this one...

does $$1^{\pi}$$ really equal 1? I mean $$\pi$$ can't be written as a fraction made of natural numbers due to its irrationality...so how does 1 get raised to it?

Yes, 1^x = 1 for all real x, not just for rational (or integer, natural) x.
a^b = e^(b*ln(a)) => 1^pi = e^(pi*ln(1)) = e^(pi*0) = e^0 = 1.

 

[Robokapp]

Yes, 1^x = 1 for all real x, not just for rational (or integer, natural) x.
a^b = e^(b*ln(a)) => 1^pi = e^(pi*ln(1)) = e^(pi*0) = e^0 = 1.

I never saw the $$e^{b*ln(a)}$$ part but it's defently handy! Thanks for the answer. So then you'd really have just $$e^{i}=1$$? if Pi would be replaced by a Pi/2 in Euler's orriginal formula?

 

[HallsofIvy]

Take a look at a tree trunk or column. Which is easier (for an ancient Greek, say) to measure, diameter or radius?

 

[Robokapp]

Take a look at a tree trunk or column. Which is easier (for an ancient Greek, say) to measure, diameter or radius?

Very good point. The bored guys in rainy days who were playing in the sand drawing circles didn't use "AutoCAD 2004" but a string and a piece of chalk or a hard object or...whatever. So they put one end of string in the desired center, let the string be the radius and rotated it 360 degrees around the point that they chose. They did not use diameter for anything except maybe to see if the apple basket fits or not in the back of the cart...

 

[waht]

Pi = 2Pi wouldn't ruin euler's formula at all.

instead e^(i *pi) = -1

we would have e^(i *pi) = 1

That's not the main point. Basically pi would be equavilan to 360 degrees.

so pi/2 would be 180 and so on.

When you get to polar or spherical coordinates and are faced with integration, it would go a lot easier if pi = 2 pi. Residue theorem is based on the radii. Even the definition of circle is based of radius,

"all points equal distant from a single point, we call that distance the radius"

Can anyone define circle in term of the diameter?

Not that pi = 2 pi wouldn't make any difference, it's still technincally correct.

 

[radou]

Radius is used 99% of the time so I think it should be more appropriate to define pi as circumference over the radius. It would only differ by a factor of two.

pi = 6.2831...

It just seems more right, and some equations might look even more elegant.

If it really seemed more right, someone would have thought of it already.

 

[Robokapp]

e^(i *pi) = 1

$$e^{i\pi}=1$$ can be rewritten as $$(e^{i})^{\pi}=1$$ still...and if you move the $$^\pi$$ it would just cancel in the 1. So you'd have two formulas...

$$e^{i}=1$$ and $$e^{i*\pi}=1$$ and $$(e^{i})^{\pi}=1$$ and then...well i can already see people trying to set them equal with each other and whining about how they know when to use one or another or why they're the same etc.

But the main point is, 0 is prbably one of biggest numerical achievements in mathematics. an equation combining things like e, i and $$\pi$$ simply...diserves to have a 0 in it in my oppinion...and -1 isn't that charming. 1 is an important number also...coefficients of 1, multiplying fractinos by 1...rewriting 1 in 100 different top/bottom ways...in trigonometry sin^2+cos^2...number 1 is pretty big also. -1 is not cool enough to make it in Euler's :D

 

[Moo Of Doom]

$$e^{i\pi}=1$$ can be rewritten as $$(e^{i})^{\pi}=1$$ still...and if you move the $$^\pi$$ it would just cancel in the 1. So you'd have two formulas...

$$e^{i}=1$$ and $$e^{i*\pi}=1$$ and $$(e^{i})^{\pi}=1$$

Ah, but you can't just move the $$\pi$$ over. It is not true in general that $$e^{zw}=(e^z)^w$$ for any complex numbers $$z$$ and $$w$$. Thus, $$e^{\pi i}=1$$ implies that $$(e^{\pi i})^{\frac{1}{\pi}}=1^{\frac{1}{\pi}}=1$$, but this does not imply that $$e^{\pi i \frac{1}{\pi}}=1$$. In fact, $$e^i \approx 0.540302 + 0.841471 i$$.