Should pi be replaced with tau in mathematics?

[2sin54]

Some mathematicians suggest, that it would be better, if instead of pi we would use 'tau', as pi very often occurs with a 2 nearby. So tau = 2*pi
What's your opinion about this?

http://www.dailymail.co.uk/sciencet...laced-alternate-value-tau.html#ixzz1QetdUCW7"

http://www.youtube.com/watch?v=IF1zcRoOVN0&feature=related"

 

[aim1732]

Haven't we been living with the 'positive' charge mistake?The conventional direction of current flow as opposed to the actual direction of electron flow?I don't see their point.

 

[Studiot]

Since we live in a 3D world \pi occurs more often multiplied by 4 so how would that help? And how would making the circumference of a circle equal to one half tau times the diameter help?

Further, tau is already well used for other purposes.

Perhaps these 'mathematicians' would be better employed performing some real mathematics, IMHO this is a really poor way to try to make a name for one's self.

 

[2sin54]

Video explains why it's better not to use diameter, but instead of that, the radius. So it is tau times radius.

 

[Studiot]

Video explains why it's better not to use diameter, but instead of that, the radius. So it is tau times radius.

You or they have obviously never measured the diameter of a round bar or ball with a pair of calipers or micrometer or ruler or whatever.

When you measure you obtain the diameter. It is not possible to directly obtain the radius of most round objects by direct measurement.

Further in order to obtain the radius you have to also know where the centre is. This is uneccessary to obtain the diameter.

 

[SpectraCat]

In any case, pi would not be changed by such a convention.

 

[disregardthat]

pi is not a convention, it is a constant. The whole discussion of a "paradigm shift" is ridiculous.

 

[blade123]

The reason why we use what we use is because it represents something.

Changing the way things are represented does nothing but create confusion.

Who CARES about notation? tau vs 2pi. They both mean the same thing.

Why create MORE confusion about math? Math is already a very difficult subject, why make it arbitrarily more difficult?

 

[SteveL27]

The reason why we use what we use is because it represents something.

Changing the way things are represented does nothing but create confusion.

Who CARES about notation? tau vs 2pi. They both mean the same thing.

Why create MORE confusion about math? Math is already a very difficult subject, why make it arbitrarily more difficult?

Well, using 2pi as the fundamental constant would have the property that tau/2 is 1/2 way around the unit circle. tau/4 is 1/4 of the unit circle, etc. No matter how familiar I am with the unit circle, I always have to hesitate a microsecond to remember that pi/2 is 1/4 of the circle, etc. That's why some people think 2pi is more natural.

Of course nobody's going to go rewrite all the textbooks. But I've often thought that 2pi should be the fundamental constant, just to make the fractions of the unit circle come out better.

 

[Samuelb88]

In the article, they make a point that a quarter of the circle corresponds to an angle of \pi/2, two quarters of a circle corresponds to an angle of \pi, etc... They conclude that this is confusing for children and adults alike that are trying to learn trigonometry. Referring to the point they're making, I don't think that the fact that a quarter of a circle corresponds to one fourth of pi is confusing at all. And anyone who has learned the subject learns to accept things like that. Therefore it should not be changed.

 

[SteveL27]

Referring to the point they're making, I don't think that the fact that a quarter of a circle corresponds to one fourth of pi is confusing at all.

Your accidental misstatement demonstrates the point. It IS very confusing to EVERYONE, even experienced practitioners. One quarter of the circle is pi/2; and you have to THINK about that every single time. That's the entire point of the tau discussion.

 

[Samuelb88]

Hehe, I meant to say half of pi.

At any rate... The fact that a quarter of a circle corresponds to an angle of pi/2 has never been confusing to me. Moreover I don't have to think about it every single time. It's become second nature. My point is after learning that a quarter of a circle corresponds to pi/2, it becomes second nature and therefore does not need to be changed.

 

[rollcast]

Thats a bit like saying the e series of perferred values fro electronics should be changed because its hard to remember.

A really stupid idea, go do some proper work instead

 

[SteveL27]

Hehe, I meant to say half of pi.

At any rate... The fact that a quarter of a circle corresponds to an angle of pi/2 has never been confusing to me. Moreover I don't have to think about it every single time. It's become second nature. My point is after learning that a quarter of a circle corresponds to pi/2, it becomes second nature and therefore does not need to be changed.

Your own typo is the counterexample.

 

[Samuelb88]

It's a typo. Get over it.

 

[spamiam]

This has already been discussed before:
https://www.physicsforums.com/archive/index.php/t-480832.html

I think it would probably be a good idea to switch, especially for the nice consistency of quadratic forms:

But seriously, the section of the manifesto on quadratic forms was quite persuasive. It compared
\frac{1}{2}gt^2, \frac{1}{2}kx^2, and \frac{1}{2}mv^2, concluding with \frac{1}{2}\tau r^2 for the area of a circle. I was pretty impressed.

 

[Redbelly98]

There are two kinds of people in this world: those who are confused by math, and those who aren't.

I don't think switching over to using tau instead of pi would help the confused people understand math any better. And people who are not confused by math are able to use pi just fine.

 

[pwsnafu]

Further, tau is already well used for other purposes.

QFT. This more than anything else means it won't happen. We don't use pi as a variable so that the reader won't mistake it. But tau is too established in many fields of science. No chance.

It reminds me of using cis for the polar form rather than the complex exponential at high school. The argument for that is similar to the argument these guys are claiming.

 

[Jocko Homo]

It's a typo. Get over it.

What's there to get over?

You made the mistake for a reason but you're refusing (perhaps to yourself) to admit it. We know it wasn't a typo because you were spelling out your thoughts. You really did think that a "quarter" of a circle corresponds to a "fourth" of pi radians. Of course it doesn't... but it should, shouldn't it?


The tau manifesto makes a lot of sense. Circles are usually defined by their radius, rather than their circumference, which is why formulas usually look simpler with tau rather than pi. Even trigonometric and sinusoidal formulas look simpler because they're build on rotating a radius around a geometric origin. It's too bad we didn't do this from the start...

The real issue is whether this simplicity is worth overhauling a very well known convention. Is it? I think so. Convention isn't nearly as hard to break as one might think. Ask anyone from a metric using country. It certainly isn't easy but only one generation has to deal with it...

 

[rollcast]

Why should we change something that has been known about from the time of the egyptians and mesopotamians.

The metric system was changed in Britain mainly because it was already being used in science as SI units, and in business and technology.

 

[Jocko Homo]

Why should we change something that has been known about from the time of the egyptians and mesopotamians.

...because using tau will be simpler in most circumstances?

The metric system was changed in Britain mainly because it was already being used in science as SI units, and in business and technology.

...but why did science, business and technology change from the Imperial system to Metric? The change in convention was obviously hard but they did it because the metric system is simpler. Does this scenario sound familiar?

 

[Studiot]

Good morning, Jocko Homo.

You present both situations as though a change is a one way street in regard of benefits.

The reality is that there are benefits and disbenefits of both systems in both cases.

The main benefit of the metric system was decimalisation.
A serious disbenefit was the move from units that had arrived by humans taking a convenient sample of the quantity as a yardstick to base quantities that were imposed by arbitrary dictat, specifically designed to be different. As a result they are unwieldy.

On balance the ISO metric system is benficial.

As a matter of interest since you mention changes to metric, I note you avoid mentioning the change form one metric system to another vis cgs to MKS.

In the case of this proposed change the disbenefits would outweight the benefits.

I cannot agree that the use of tau would be simpler in most circumstances, what proof do you offer to back up this statement?

You have not addressed my comments in posts which indicate that we would be forever introducing another arithmetic step (=source of error) ie division by 2 when measuring round objects.

Custom and practice as a result of millenia of experience determines whether the radius or
diameter or other means is used to specifiy curvature. The most appropriate is used in most circumstances today.

 

[Ashwin_Kumar]

You people are crazy. Pi is very important in area of circle and volume of sphere and it is used in several equations without a two in front.
Only an idiot who is too lazy to right out two pi radius will get confused. Lazy scientists and their tau constant

 

[Ashwin_Kumar]

Pi is used so commonly without a 2 in front. Just look at the field equations, look at so many equations. Many of them involve pi squared. Now, if we used the darned tau sign, it will become one fourth tau square. Now, fractions are more confusing than whole numbers, and tau squared would just be a simple four pi square. Get used to it.

 

[agentredlum]

There are two kinds of people in this world: those who are confused by math, and those who aren't.

I don't think switching over to using tau instead of pi would help the confused people understand math any better. And people who are not confused by math are able to use pi just fine.

"The neck at the end of the rope is mine, I run the risks, so the next time I want more than half"

Even Tuco wants more than half of 2 pies.

 

[Jocko Homo]

You present both situations as though a change is a one way street in regard of benefits.

The reality is that there are benefits and disbenefits of both systems in both cases.

The main benefit of the metric system was decimalisation.
A serious disbenefit was the move from units that had arrived by humans taking a convenient sample of the quantity as a yardstick to base quantities that were imposed by arbitrary dictat, specifically designed to be different. As a result they are unwieldy.

On balance the ISO metric system is benficial.

As a matter of interest since you mention changes to metric, I note you avoid mentioning the change form one metric system to another vis cgs to MKS.

I was unaware of any further changes. MKS is all I've known. However, I should point out that these multiple changes show that change isn't so hard to do...

In the case of this proposed change the disbenefits would outweight the benefits.

I cannot agree that the use of tau would be simpler in most circumstances, what proof do you offer to back up this statement?

This thread is full of them. Did you look at the arguments presented by the OP? Did you see Samuelb88's "typo?" I assure you that he wasn't the first guy to do that and he won't be the last...

Fundamentally speaking, mathematicians define circles by their radius. I can't recall the last time the diameter of a circle was discussed...

You have not addressed my comments in posts which indicate that we would be forever introducing another arithmetic step (=source of error) ie division by 2 when measuring round objects.

I found this claim fascinating but otherwise I had nothing to say about it. It's a good point and perhaps engineers have reason to work with Pi rather than Tau. I'm not familiar, in detail, with what engineers deal with...

Custom and practice as a result of millenia of experience determines whether the radius or
diameter or other means is used to specifiy curvature. The most appropriate is used in most circumstances today.

I'm not apposed to teaching both constants and using one or the other where most appropriate, especially since Pi has great historical significance...

For example, do people doing signal processing ever need to know the diameter of anything? If not, working with the Tau equivalent of radians would make their work more natural and thus easier...

 

[disregardthat]

One important aspect of this is that sin(x) is measured in radians, and pi is 1 radian. sin(x) being measured in radians has the obvious analytical importance, and it would be awkward saying that very important functions such as sin(x) and cos(x) are measured in a unit for which tau/2 is 1 unit.

 

[pmsrw3]

One important aspect of this is that sin(x) is measured in radians, and pi is 1 radian. sin(x) being measured in radians has the obvious analytical importance, and it would be awkward saying that very important functions such as sin(x) and cos(x) are measured in a unit for which tau/2 is 1 unit.

Huh? What do you mean "pi is 1 radian"?

 

[Jocko Homo]

One important aspect of this is that sin(x) is measured in radians, and pi is 1 radian. sin(x) being measured in radians has the obvious analytical importance, and it would be awkward saying that very important functions such as sin(x) and cos(x) are measured in a unit for which tau/2 is 1 unit.

pmsrw3 is right, you're not making any sense saying "pi is 1 radian."

There are 2π radians in a revolution (a full circle). How weird is that? There's only a τ radians in that revolution. How nice is that?

The use of radians is independent of which constant you wish to use. In the future, you might want to read the very source you reference!

 

[Char. Limit]

Here's a question. Wouldn't using tau just totally screw with the perihelion shift equation given by Paul Gerber?

Here's the equation, using pi:

\Psi = 24 \pi^3 \frac{a^2}{\tau^2 c^2 \left(1 - \epsilon^2\right)}

Now, here's the similar equation, using tau (marked as tau₁ to avoid confusion)

\Psi = 3 \tau_1^3 \frac{a^2}{\tau^2 c^2 \left(1 - \epsilon^2\right)}

Now, you may be asking "what exactly are you trying to say?" Well, this is just an example of how we already use tau in other contexts, and sometimes (in fact, I'd be guessing often) they also use pi. For example, don't proper time and torque use tau?

 

[Anonymous217]

If we're considering a change in denoting pi, consider a change in denoting the set of primes, ie. \displaystyle \prod_{p \text{ prime}}.

 

[Antiphon]

Tau is the wrong symbol. It should be pi-bar like h and h-bar in QM.

 

[SteveL27]

Tau is the wrong symbol. It should be pi-bar like h and h-bar in QM.

A mathematician, a physicist, and an engineer walk into the pi bar ...

 

[Studiot]

A mathematician, a physicist, and an engineer walk into the pi bar ...

So what 's the rest of it?

Go on do tell - it's after the 9 o'clock watershed.

 

[agentredlum]

Huh? What do you mean "pi is 1 radian"?

If the radius of the circle is pi, and you use an arc length of pi, then that arc length subtends an angle of 1 radian so pi can be 1 radian.

In that case the circumference woud be 2pi^2 and the diameter tau and in terms of tau the circumference woud be (tau)^2/2

 

[agentredlum]

Any arc length can subtend an angle of 1 radian. Just make the radius equal to the arc length.

If the arc length is e, and the radius is e, then the arc length subtends an angle of 1 radian. Diameter is then 2e, circumference is 2(pi)e, area is (pi)e^2.

This is fun! Insert your favorite constant please.

 

[SteveL27]

I just had a thought about how all this developed. In ancient times the diameter of a circle was more important than the radius, since they were measuring land. In modern times the radius is more important, because mathematicians care about the unit circle and its circumference. If aliens had math but not agriculture, they might take tau as being more fundamental than pi.

I hope SETI knows about this. It would be a shame if the aliens are transmitting 6.28... and we're missing it

 

[pmsrw3]

I hope SETI knows about this. It would be a shame if the aliens are transmitting 6.28... and we're missing it

As long as they transmit in binary we're fine.

 

[nickalh]

Thanks, guys. That last bit about SETI and binary was hilarious. I've been laughing through the next several several questions.

 

[thegreenlaser]

I think there's a lot more annoying conventions than pi. I never even thought about pi being annoying until reading this thread. However, when we learned about how current is opposite to the flow of electrons, because it's the flow of imaginary positive charge carriers (or the negative flow of negative charges) I immediately thought that was an irritating convention.

To be honest, I don't think scaling pi by a half will really help much of anything. I'm taking an electrical engineering signals course right now, so I'm doing a lot with periodic signals, which is one place people say that tau would make more sense. I do personally get confused when working with fundamental periods and frequencies and formulas like T=1/f = 1/(2*pi*omega). However, my confusion almost always comes from where the 2pi is supposed to go. That would be just as much of an issue if there was a switch to tau or pi bar or something representing 2 pi. I have no trouble remembering that 2pi is the period of a sinusoid.

 

[Twinbee]

thegreenlaser: see my earlier thread from here:
https://www.physicsforums.com/showthread.php?t=426341

We can cut out not just the "2", but also the "pi" completely by often measuring in revolutions (or cycles if you prefer). Since I'm making my own calculator, I'll be adding the new functions rsin, rcos, rtan etc. to represent this seemingly obvious function.

E.g.:
rsin(1/8) = 0.7071... (1/8th of a turn or circle)
ratan(1÷1) = 1/8... (1/8th of a turn or circle)

 

[Studiot]

Since I'm making my own calculator

In which case you may be interested in a paper I wrote some years back published in the
Empire Survey Review and entitled

The use of the 5th quadrant.

go well

 

[Twinbee]

Studiot: What was it about? Any link to the paper?

 

[Studiot]

Twas not really a full blown paper, more of a simple article and appeared in June 1986.

A simple computation that causes much confusion in navigation, surveying and engineering is the calculation of partial coordinates by distance and bearing.

Calculators and computers are programmed to provide angular functions according to the mathematical definition.
That is the defining rotating arm rotates anticlockwise from the horizontal or x axis.

Distance and bearing coordinates are defined by an arm that rotates clockwise from the y or vertical or north axis, resulting in cumbersome 'sign rules' to be applied or up to three computer tests if programmed.

It is possible to remedy both these issues by using the fact that calculators will accept input angles greater than 360 degrees and modifying the traditional formulae to suit.

A suitable input angle may be obtained by noting that as a result of the 90^o shift the whole circle bearing (wcb) and the mathematical angle(ma) may be related by the equation

ma + wcb = 450^o

This takes us into the fifth quadrant.

In order to cope with the differing directions of rotation the usual projections onto the x and y axes must be inverted so

\DeltaY = Rsin(450-wcb) = Rsin(ma)

\DeltaX = Rcos(450-wcb) = Rcos(ma)


These formulae are automatically correct for sign rules, unlike the traditional
ones.

go well

 

[dimension10]

He makes use of Euler's identity to strengthen his claim but that doesn't make any sense.

It works with pi too.

{e}^{i\pi}=0-1

To change it to the form of Euler's identity, it would be

{e}^{i\tau}-1=0

Tau is also very annoying as writings involving tau would be very confusing. He is not even taking trigonometry and area into consideration.

 

[Char. Limit]

He makes use of Euler's identity to strengthen his claim but that doesn't make any sense.

It works with pi too.

{e}^{i\pi}=0-1

To change it to the form of Euler's identity, it would be

{e}^{i\tau}-1=0

Tau is also very annoying as writings involving tau would be very confusing. He is not even taking trigonometry and area into consideration.

Not to mention, the latter equation tells us LESS. Try taking the square root of e^{i \tau} and 1. You'll get e^{i \pi} and... ±1. So which one is e^{i \pi}? The latter equation doesn't tell us.

 

[dimension10]

Not to mention, the latter equation tells us LESS. Try taking the square root of e^{i \tau} and 1. You'll get e^{i \pi} and... ±1. So which one is e^{i \pi}? The latter equation doesn't tell us.

Not too sure if you could say that. Even with pi there is a problem. It doesn't tell us \exp(i \frac{pi}{2})

\sqrt{\exp(i \pi)}= \pm i .

But the correct one is i sin (pi/2)+cos(pi/2)=i+0=i not -i.

So the best is just \exp(ix)=i \sin x +\cos x.

 

[disregardthat]

Not too sure if you could say that. Even with pi there is a problem. It doesn't tell us \exp(i \frac{pi}{2})

The point is that the equation corresponding to tau can be algebraically derived from the equation correponding to pi. So Char.limit is correct, essentially the equation for tau gives less information, because it cannot derive the equation for pi algebraically.

 

[dimension10]

The point is that the equation corresponding to tau can be algebraically derived from the equation correponding to pi. So Char.limit is correct, essentially the equation for tau gives less information, because it cannot derive the equation for pi algebraically.

And that for pi cannot derive that for pi/2 algebraically so there is not best equation whatsoever.

 

[disregardthat]

And that for pi cannot derive that for pi/2 algebraically so there is not best equation whatsoever.

You can't derive Fermat's last theorem either, so what?