Wikipedia:Reference desk/Archives/Mathematics/2021 September 28

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September 28

Things appropriate for lower level math but not for higher level math

If the "higher level math" is calculus, there are 2 easy-to-understand answers to this question:

1. Degrees as the unit of angular measurement. Radians are very important in calculus. If we define angles in degrees the derivatives will all have $pi/180$ .
2. Radicals. In calculus exponents are played with a lot. For this reason, when doing calculus radicals should generally be avoided and fractional exponent should be used instead.

But, how about when the "higher level math" is topology or abstract algebra?? What answers are there now?? Georgia guy (talk) 23:43, 28 September 2021 (UTC)

I don't think angular measurements come up much in topology. Of course they do in geometry and analysis. In at least some parts of abstract algebra (viz. Galois theory), radicals are very important and fractional exponents aren't really used. Otherwise you'd want exponentiation to work with irrational exponents too, but $2^{\sqrt {2}}$ is not an algebraic number unless I'm confused. Exponentiation is more of an analytic concept rather than an algebraic one. 2601:648:8202:350:0:0:0:1598 (talk) 04:04, 29 September 2021 (UTC)

Are percentages appropriate for “higher level” math? In general – not only for exalted maths gymnastics but also for low-level maths involving some algebra beyond simple addition and subtraction – it is wise to begin one's calculations by converting percentage notation to plain numbers, and if the answer is desired to be in the form of a percentage finishing by converting back, using both times $P\%={\tfrac {1}{100}}P.$ Likewise, for angles, use $D^{\,\circ }={\tfrac {\pi }{180}}D.$, which gives one that

${\frac {d}{dx}}\sin x^{\circ }={\frac {d}{dx}}\sin({\tfrac {\pi }{180}}x)={\tfrac {\pi }{180}}\cos({\tfrac {\pi }{180}}x)={\tfrac {\pi }{180}}\cos x^{\circ }.$

This is not the derivative of the sine function, but of the function $\lambda x.\sin x^{\circ }.$ Allowing degrees does not necessitate learning any different or additional calculus rules; it merely requires the ability to convert notation. Rather than considering degrees or radical notation inappropriate, mathematicians will generally avoid having to introduce conversions, and therefore state rules in a form not requiring them. This is, as I see it, solely a matter of convenience. Another issue is that conventionally the argument in the notations $P\%$ and $D^{\,\circ }$ is a numeral; one is not likely to encounter $(2x+15)^{\circ }.$ Radical notation has a similar issue; while ${\sqrt[{3}]{x}}$ is fine,

${\sqrt {\frac {1+{\sqrt {x}}}{2}}}~~$ and $~~{\sqrt[{1+{\sqrt {x}}}]{x}}$

go from problematic to unacceptable. But that is purely a typographical issue, and the switch between radical notation and exponentiation may be made without any fuss. Mathematicians will also silently switch between $e^{x}$ and $\exp x,$ choosing whichever looks typographically best in any specific situation.  --Lambiam 06:58, 29 September 2021 (UTC)

Another one, eluded to in the post immediately above, is e. I mean, for lower level math you tend to use integral exponents and powers. If you encounter logarithms it's normally log to base 10, the common logarithm. But for calculus e is much better both in exponents and as a logarithmic base. The reasoning is the same as adopting radians for angles, it simplifies expressions as the factor log_e10 drops out.--2A00:23C8:4583:9F01:8C64:2FB8:E54E:2C6B (talk) 10:42, 1 October 2021 (UTC)

Percentages and degrees of arc should be taught; notations like 17.5% and 54°46′20″N, 1°34′31″W are in common use. Are common logarithms still taught in secondary education like they used to? They were useful when students had to use paper booklets of logarithm tables to compute the value of 4.56789 × 9.87654 (and in college when taught to use a slide rule). Basically everyone carries a powerful digital calculator in their pockets now; the only reason to pay special attention to base-10 logarithms is historical interest, just as for long division. It is good to know what common logarithms are and that an algorithm for long division exists, but drills in applying these is pointless.  --Lambiam 17:24, 1 October 2021 (UTC)

I disagree that the only use for the common logarithm is historic. When you e.g. use a logarithmic scale for graphing purposes, you almost always do so base 10. The only exception is base 2, as some things increase naturally in powers of 2, such as in computing and in music. But those are narrow, technical exceptions. When a log scale is used in finance, or to model general exponential growth, the base used is normally 10.--2A00:23C8:4583:9F01:ED56:12F5:54C2:E976 (talk) 12:14, 2 October 2021 (UTC)

The fact that the marking ticks on the y-axis of a log–linear plot are preferentially at powers of 10 does not mean the logarithms used to compute the plot are base 10; the plot can be read by someone who is not familiar with common logarithms but understands decimal notation. An ohmmeter usually has a scale that within a finite distance ranges from 0 to $\infty$ but is usually marked at 10, 100 and 1000 (or 1k). No logarithms are involved; the scale is related to the mapping R ↦ aR / (bR + 1). There may be a switch with settings × 1, × 10, × 100, ...; a user will not think of these settings as common logarithms.  --Lambiam 13:20, 2 October 2021 (UTC)

Certainly to read such a graph or scale you don't need, or even need to understand, logarithms. But to draw such a graph/scale you do need to use logarithms. More generally a good understanding of logs helps you choose when to use a log scale, whether to use a semi-log or log-log plot, what particular units to use so as to produce the clearest result. --2A00:23C8:4583:9F01:ED56:12F5:54C2:E976 (talk) 13:37, 2 October 2021 (UTC)

I did not say or suggest that teaching logarithms in general was pointless, but only drills in applying common logarithms.  --Lambiam 21:17, 2 October 2021 (UTC)

Units of decibels, based on common logarithms, are in common use in many fields. It's not necessary to do calculations with logarithms either by hand or using a slide rule, but it is important to understand the significance of reference points like 0 dB, 1 dB, 3 dB, 6 dB, and 10 dB. --Amble (talk) 20:11, 5 October 2021 (UTC)

Decibels are based on a logarithmic scale whose base is not 10, but the tenth root of 10, which is approximately 1.25892541179. Georgia guy (talk) 20:14, 5 October 2021 (UTC)