Wikipedia:Reference desk/Archives/Mathematics/2008 October 27

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October 27

Accepted convention for combining radians and degrees

Is it accepted to combine radians and degrees i.e. π/2 + 360° As they can be converted to one another it seems like this is reasonable. Thanks. —Preceding unsigned comment added by Teopb (talk • contribs) 03:08, 27 October 2008 (UTC)

It's poor style; you generally wouldn't write "1 inch + 2 cm". What brought on the question? —Tamfang (talk) 03:29, 27 October 2008 (UTC)

Well if you define ° to be a constant equal to π/180, then it works. --71.106.183.17 (talk) 05:08, 27 October 2008 (UTC)

I used to annoy my teacher in high school by adopting the convention that % = 1/100. It makes the notation so much simpler, but it's not how the book says you're supposed to do it... —Ilmari Karonen (talk) 10:24, 27 October 2008 (UTC)

From a pedagogical point of view I find it’s difficult enough to get students to write math correctly, that viewing % as any other unit so that 1%=0.01 just as 12 inches = 1 foot makes life much easier – and I don’t see a good reason not to do so. GromXXVII (talk) 10:47, 27 October 2008 (UTC)

It could create a few problems. "During these centuries, the population increased by 1200%" = "During these centuries, the population increased by 1200 * 1/100" = "During these centuries, the population increased by 12". But they will be few. -- Jao (talk) 11:18, 27 October 2008 (UTC)

I think those problems will be many. If you are going to use that notation you need to explicitly say "1200% of the population at the beginning". --Tango (talk) 11:28, 27 October 2008 (UTC)

And don't forget that "of" used here, of course, means multiplication. And multiplication (of real numbers) is commutative. So that "using ethanol caused 40% of the difference" = "using ethanol caused the difference of 40%". (Abusing English is fun....) Eric. 131.215.45.193 (talk) 00:17, 28 October 2008 (UTC)

Writing "population increased by 1200%" is silly anyway; what's wrong with "population increased 13-fold"? Inserting "00%" is false precision and makes the reader/listener do extra mental work. —Tamfang (talk) 03:15, 1 September 2023 (UTC)

The notation is unambiguous, so I see no harm in it. Mathematicians virtually never use degrees, though. Angles are actually dimensionless quantities so it's pretty meaningless to have a different unit for them. --Tango (talk) 11:28, 27 October 2008 (UTC)

It is common to omit the unit name radian. The unit s⁻¹ can mean radian per second or perhaps cycles per second = hertz. Radians are for scientists, but are not common in practice. Mixing one unit omitted and the other unit named is not advisable: π/2 + 360° = 1.57 + 360°. The symbol π is considered a number while ° is considered a unit. This is purely conventional. π = 3.14159 and ° = 0.0174533, so 2π = 360° = 6.28319. It was earlier common to mix units, omitting the plus sign, in 57°17'45", and in 3 feet 3 inches, and in 5 shillings 4 pence. Numbers were thought of as whole numbers. Today decimal fractions are common. Bo Jacoby (talk) 12:25, 27 October 2008 (UTC).

People tend to assume if there is a π in the measurement it is in radians, you hardly ever see a pure radian measurement. Mixing different systems of measurements though is very strange, why would one want to do such a thing? Twit, messer is what I'd think if I saw it. The business about yards feet and inches is more like a mixed number base system than anything to do with different units. Dmcq (talk) 22:55, 27 October 2008 (UTC)

Any angle measurement that isn't a rational multiple of pi is going to be an irrational proportion of a complete circle, which isn't likely to come up very often, so there almost always will be a pi in there. There is a far easier way to tell if its radians or degrees, though - degrees should always have the symbol, a number written on its own will be radians (since radians are, in fact, just numbers since angles are dimensionless). --Tango (talk) 23:35, 27 October 2008 (UTC)

Angles are dimensionless by convention, but that's only because mathematicians aren't fond of units. Logarithms to different bases are related by constant ratios, just like physical quantities expressed in different units. We could abolish ln and subscripted log and define log without a subscript to return a unitful logarithm taken to no particular base. We already have names for the units! Instead of log₂ 32 = 5 and ln 32 ≈ 3.5 and log₁₀ 32 ≈ 1.5, we would have log 32 = 5 bit ≈ 3.5 nat ≈ 1.5 ban. One radian equals one nat if you define the trig functions from Euler's formula in the usual way (you have to write it with exp(iθ) rather than e^iθ, where exp = log⁻¹). Sure, keeping track of units is sometimes a hassle, but, dang it, "nat" and "degree" and "bit" are units; we all treat them as units, and I think we'd be better off officially recognizing them as such. People who prefer unitless log can always "work in natural units" as physicists often do. -- BenRG (talk) 14:17, 28 October 2008 (UTC)

I have missed those names for the units of logarithms! I haven't heard about the nat before but I like it better than the neper, which I haven't heard used in practice. Especially we need something like centi-nat (sounding better than centi-neper) in places where financial people use percent, which courses endless confusion. I believe that i radian (rather than 1 radian) equals 1 nat, (according to Euler's formula). The order of magnitude is sometimes used for your 'ban'. Bo Jacoby (talk) 14:12, 29 October 2008 (UTC).

Whether or not a value is dimensionless depends on how it is defined. I define an angle to be the ratio of the length of an arc which subtends that angle and its radius, that gives a dimensionless quantity. How do you define an angle? --Tango (talk) 15:23, 28 October 2008 (UTC)

The question comes up because the example I gave was an answer on a multiple choice question that asked for an angle terminal at π/2. —Preceding unsigned comment added by Teopb (talk • contribs) 00:49, 28 October 2008 (UTC)

I'm not sure what you mean by "angle terminal", but regardless of whether it is acceptable to mix units like that is certainly isn't ideal and no multiple choice answer should do so. --Tango (talk) 01:00, 28 October 2008 (UTC)

Is this even possible?

I must have done something wrong.. at the end of the problem I have something like Integral(x * e^x) or Integral(x^2 * e^x)

Don't solve it for me or anything, but I don't think these are even possible o_O —Preceding unsigned comment added by 71.176.156.86 (talk) 22:21, 27 October 2008 (UTC)

See integration by parts. --Tango (talk) 22:25, 27 October 2008 (UTC)

Differentiate (x*(e^x))-(e^x)+C for some constant C.

Topology Expert (talk) 04:14, 31 October 2008 (UTC)