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What subfield of mathematics? Surely if you do differential equations, your trigonometric functions will eat radians. But e.g. for geometry, where you do basic
by bmacho 4y ago
What subfield of mathematics? Surely if you do differential equations, your trigonometric functions will eat radians. But e.g. for geometry, where you do basic arithmetic operations on angles, turns are a bit more convenient than radians. (Radians are not that inconvenient if you denote 6.28 somehow, but then again, why not just use turns.)
- zarzavat 4y agoThe situation is similar to that of the logarithm and exponential functions[0]. There’s a log2 function and a log10 function and they are both useful. But when we talk about the log() function there can be no doubt that it is to base e. If you want to define a sinT() function that works in turns then that’s totally fine. But the sin() function is defined as taking an argument scaled in radians, because it is mathematically natural. Mathematically speaking, all trig functions are in radians. When you write sin(90°) the degree symbol ° is a conversion factor. I blame calculators for confusing high schoolers into believing that there is a separate set of functions that work in degrees. [0] unsurprisingly because Euler’s formula equates the trigonometric and exponential functions.
- bmacho 4y agoThe sine function, defined as x-x^3/3!+... doesn't take argument scaled in radians. It takes real numbers. It has nothing to do with radians, really. Or even with angles. The other sine function, defined using right triangles takes argument in Angles, also has nothing to do with the measuring unit. (Also I don't know who told you that log() denotes log_e. Maybe in your narrow environment, but definitely not universal between fields and languages. Personally I prefer ln, ld and lb for natural, decimal and binary logarithms.)
- lkuty 4y ago"Also I don't know who told you that log() denotes log_e. Maybe in your narrow environment," e.g. https://reference.wolfram.com/language/ref/Log.html https://reference.wolfram.com/language/ref/Log.html But I must confess that we had ln() in university courses and by default log used base 10. Now I use ln and a base for the log as a subscript like log_10, log_2, etc.
- bmacho 4y agoSrsly? Well, then according to https://en.wikipedia.org/wiki/Brainfuck https://en.wikipedia.org/wiki/Brainfuck the + symbol denotes cell increment and not addition. > But I must confess that we had ln() in university courses Same. I often wonder why would anyone denote the natural logarithm with log(), when ln is shorter, and easier to read (at least for the people that were thought to use it), also it is already somewhat established.
- lkuty 4y agoHaha the Brainfuck analogy was funny. Note that in programming languages it looks like Log denotes Ln. Anyway, Log_b(x) = Log_a(x) / Lob_a(b), so we're covered.
- tsimionescu 4y ago> The sine function, defined as x-x^3/3!+... doesn't take argument scaled in radians. It takes real numbers. It has nothing to do with radians, really. Or even with angles. The point is that a definition of the sine function where sin(pi/2) = 1 is equivalent to a sine function taking radians. You could also define sinT(x) such that sinT(1/4) = 1: sinT(x) = sin(2pi x) = sin (tau x) = 2pi * x - (8pi^3 * x^3) / 3! + [...]. Neither of these is more or less fundamental than the other, but one is more convenient in most (non-trig) calculations.
- jacobolus 4y ago> who told you that log() denotes log_e. Maybe in your narrow environment This is a nearly universal convention in modern mathematics (except a few niches like information theory and computational complexity theory where it means log base 2, which is usually clear from context). Engineering disciplines used to use "common" logarithms (i.e. base ten) all over the place back when most calculations were done with slide rules, lookups in paper tables, and pen-and-paper arithmetic, but with the advent of computers multiplication is just as cheap as addition, and expressing things on log scales is less necessary. Over time the mathematicians are winning the fight to define the symbol 'log'.
- bmacho 4y agohttps://www.google.com/search?q=log(10) https://www.google.com/search?q=log(10)
- jacobolus 4y agoYes, Google in this context is a glorified pocket calculator, and follows the convention from the slide rule era. But if you find log in a mathematics paper it almost surely means the natural log.
- Shorel 4y agoParent comment is way too dogmatic for my taste. It even mentions a mythological entity as an appeal to authority. In my opinion, mathematicians always choose the notation that's more convenient for them, at the moment, for a particular problem. If a given problem is easier using another form of sin/cos, etc., they will use it, and it will be used without hesitation. In that sense, mathematicians could not be more pragmatic. However, for many things, as long as the result is correct, they don't care about the operations' computability. Performance is an afterthought because for them (a*п)/п is exactly the same as 'a'. All operations are instantaneous. Taylor series for example are a perfectly fine final answer in calculus, but to a programmer they are an infinite set of partial approximations that can take any arbitrary time to execute. This is what makes computer science fascinating =)