What's the point of radians?

[laughingboy]

@Stranger On A Train

Octol offers less “units” per digit (i.e. nn only covers 0-63, nnn only covers 0-511) and groups bytes into a rather awkward three digit binary pattern. Hex is certainly more widely used; the only place I can think of offhand is the use of octal in specifing user permissions with chmod, though I’m sure some Unix guru can come up with much more.

Octal is a holdover from the days of computers with a 36-bit word length for which an “awkward three digit binary pattern” is the bee’s knees.

[Derleth]

@Stranger On A Train

Hex is certainly more widely used; the only place I can think of offhand is the use of octal in specifing user permissions with chmod, though I’m sure some Unix guru can come up with much more. But hex is used all over the place to code instructions and represent data.

Octal is still part of the C programming language (Unix’s ‘mother tongue’, in a very real sense): A numeric constant that begins with a zero like ‘077’ is taken to be base-8 instead of base-10. One of many, many things about C that trips up newcomers to the language.

Unix was (first) developed on DEC machines in the very late 1960s-1970s, especially the PDP-11. And anyone who’s seen PDP-11 machine code knows that the standard notation is in octal, even though the opcodes are 16 bits wide. DEC seemed to love octal, using it in its 36-bit machines, where it made sense, and in its 16-bit machines, where it didn’t, until the advent of the hex-centric VAX. So to bring a geeky interlude to an end, Unix likes octal because DEC liked octal.

(On the PDP-11, the opcodes were six octal digits wide but the highest-order digit could only be 1 or 0. 3*5=15+1=16.)

These days, hex is used because there are eight bits to a byte and one hex digit is equivalent to four bits. Therefore, there’s always a whole number of hex digits in a machine word, whether it’s 16-bit, 32-bit, or 64-bit. I’m not sure why computer makers were so eager to move to the eight-bit byte.

[Ms2001]

@Stranger On A Train

Hex is certainly more widely used; the only place I can think of offhand is the use of octal in specifing user permissions with chmod, though I’m sure some Unix guru can come up with much more.

C uses octal numbers for escape characters: ‘\100’ is ASCII 64. Newer compilers also support hex escapes: ‘\x40’.

[dropzone]

@Mathochist

Welcome to integral calculus.

Sure would’ve been a faster introduction if I hadn’t had to invent it myself.

[Roches]

The reason why this works, by the way, is because sin(t) is very nearly t for small t – but only when t is expressed in radians. Which is another reason why radians are cool.

That one really was a trick to make using a slide rule easier; some slide rules had special scales for small-angle trignometry. If you use a calculator instead of trigonometric tables, you don’t need to use the shortcut.

OTOH, working in radians can make using a calculator difficult, because calculators display, say, 1.570796… rather than ‘π/2’, and because it’s more difficult to input angles in radians than it is for degrees. You can divide the result by π to get a more practical answer, though, and a calculator with a π key makes it easier to enter angles in radians.

Other than that, not liking radians is like not liking metric; radians are far more logical and less arbitrary than degrees, but degrees are what people use in everyday life.

[Stranger_On_A_Train]

@Roches

That one really was a trick to make using a slide rule easier; some slide rules had special scales for small-angle trignometry. If you use a calculator instead of trigonometric tables, you don’t need to use the shortcut.

The small angle approximation isn’t just useful for using a slide rule; it finds application in all sorts of scientific and engineering problems in which small angle oscillations are found. It allows you to reduce a complicated non-linear relation between oscillation and linear motion into a simple first order differential equation of the form

d[sup]2[/sup]θ/dt[sup]2[/sup] + 2ξω[sub]n[/sub] dθ/dt + ω[sub]n[/sub][sup]2[/sup]θ = 0

the general solution of which is

θ = e[sup]-ξω[sub]n[/sub]t[/sup] [size=5][C[sub]1[/sub]e[sup]ω[sub]n[/sub]√(ξ[sup]2[/sup]-1)t[/sup] + C[sub]2[/sub]e[sup]-ω[sub]n[/sub]√(ξ[sup]2[/sup]-1)t[/sup]][/size]

@Roches

OTOH, working in radians can make using a calculator difficult, because calculators display, say, 1.570796… rather than ‘π/2’, and because it’s more difficult to input angles in radians than it is for degrees. You can divide the result by π to get a more practical answer, though, and a calculator with a π key makes it easier to enter angles in radians.

[countenance=beaming proudly with geek pride]My calculator (HP-48G) displays answers with π as a constant.[/countenance]

Stranger

[CurtC]

@Stranger On A Train

θ = e[sup]-ξω[sub]n[/sub]t[/sup] [size=5][C[sub]1[/sub]e[sup]ω[sub]n[/sub]√(ξ[sup]2[/sup]-1)t[/sup] + C[sub]2[/sub]e[sup]-ω[sub]n[/sub]√(ξ[sup]2[/sup]-1)t[/sup]][/size]

I’m blown away - not by your math, but your ability to get the forum to display this equation.