Wikipedia:Reference desk/Archives/Mathematics/2020 December 30

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December 30

Finding Ratios when fraction are given.

A : B : C = 1/8 : 1/12 : 1/24

A : B: C = 3:2:1/24

Answer: A : B : C = 3 : 2 : 1

Why 24 is neglected in the final step? Rizosome (talk) 06:39, 30 December 2020 (UTC)

See Ratio#Proportions and percentage ratios:

If we multiply all quantities involved in a ratio by the same number, the ratio remains valid. For example, a ratio of 3∶2 is the same as 12∶8. It is usual either to reduce terms to the lowest common denominator, ...

Your second line looks a bit strange perhaps you meant:

A : B: C = 3/24 : 2/24 : 1/24

-- ToE 06:55, 30 December 2020 (UTC)

As ToE wrote, multiplying all quantities by the same (positive) number does not change the ratio; so 1⁄8 : 1⁄12 : 1⁄24 = m⁄8 : m⁄12 : m⁄24, whatever the value of m, provided that m > 0. This means that we can express the ratio in whole numbers by choosing m to be a common multiple of the denominators 8, 12 and 24. Their least common multiple equals 24. Using that as the value of m, , we find:

A : B : C = 1⁄8 : 1⁄12 : 1⁄24 = 24⁄8 : 24⁄12 : 24⁄24 = 3 : 2 : 1.

The first line of the question (the problem) is fine, the third line (the answer) is correct, but the line between these two, as written, is in error. Perhaps the intention is that (3 : 2 : 1) / 24 is a shorthand notation for 3⁄24 : 2⁄24 : 1⁄24, which is what you get if you express the fractions using their lowest common denominator. Then the final step is justified; it multiplies all quantities involved in the ratio by the same number 24.  --Lambiam 14:19, 30 December 2020 (UTC)

Which mathematician contributed more to Maths?

Which mathematician contributed more to Maths? Wiki says Paul Erdős was one of the most prolific mathematicians. Rizosome (talk) 07:16, 30 December 2020 (UTC)

Leonhard Euler says: "He is also widely considered to be the most prolific, as his collected works fill 92 volumes, more than anyone else in the field." PrimeHunter (talk) 08:17, 30 December 2020 (UTC)

I don't think you can really make any kind of objective assessment. It's like asking which author contributed the most to literature, or which composer contributed the most to music. Some are more prolific than others, and everyone has their favorites, and some names appear more often on lists of favorites than others, but in the end it comes down to a matter of opinion. --RDBury (talk) 12:04, 30 December 2020 (UTC)

Much of the activity of Erdős remained in a relatively small corner of the mathematical edifice, and I suspect that most mathematicians will not be able to mention even one result that is due to Erdős. In terms of output volume Euler may be the winner (perhaps), but I suspect most mathematicians will agree that Carl Friedrich Gauss, the Princeps mathematicorum, has been more influential. But indeed, how to compare for instance Isaac Newton and Henri Poincaré? The former has been more influential with regard to formulating the laws of physics using maths as the vehicle, but his contributions to maths per se, while impressive, do not quite reach the same heights.  --Lambiam 13:51, 30 December 2020 (UTC)

This is badly false with respect to Erdos. You think people don't know about the elementary proof of the prime number theorem, for example? --JBL (talk) 19:03, 1 January 2021 (UTC)

Yes, I may be wrong but I expect that most mathematicians, when prompted to mention a result due to Erdős, will not think of that, or any other result. (The outcome may be quite different if one restricts the question to the subset of number theorists and discrete mathematicians, which, however, most mathematicians are not.) One may also quibble whether a new proof of a known result counts by itself as a result (other than, as in this case, disproving a non-mathematical belief about the profundity of a theorem).  --Lambiam 21:24, 1 January 2021 (UTC)

I would probably come up with his work in transfinite Ramsey theory, for example the notion of an Erdős cardinal, but it's true I'm not sure what specific result I would name (I think introducing new concepts is often more important than quotable single results, though).

It's hard to argue with Gauss, though. His work touches so much of mathematics. He's also my great-to-the-ninth grand-advisor or something like that. A large fraction of mathematicians can say something similar. He's sort of the Genghis Khan of advisors. --Trovatore (talk) 21:39, 1 January 2021 (UTC)

John von Neumann is a more recent example who would likely have contributed a lot more if he hadn't died of cancer at the age of 53. Count Iblis (talk) 05:28, 2 January 2021 (UTC)

sideways root?

Is there a name for this operator?

8 ∘ 2 = 3

49 ∘ 7 = 2

243 ∘ 3 = 5

6561 ∘ 9 = 4

152399025 ∘ 12345 = 2

Steve Summit (talk) 17:06, 30 December 2020 (UTC)

That's the logarithm with base given by the second operand:

$\log _{2}8=3$

$\log _{7}49=2$

et cetera. --Wrongfilter (talk) 17:11, 30 December 2020 (UTC)

D'oh! Thank you, and pardon my ignorance. (I really should have been able to figure that out for myself!) —Steve Summit (talk) 17:22, 30 December 2020 (UTC)