Wikipedia:Reference desk/Archives/Mathematics/2014 November 19

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November 19

What's the point of ratios?

This is a two-part question. I have been wondering what the point of the concept of ratio is: It seems to me that nothing can be done with n:m that can't be done with n÷m. Is that correct?

If so, then maybe the mathematical justification for treating them separately is that the notation can be extended to more elements than two. n:m:k has properties beyond those of n÷m÷k. n:m:k can be expressed by an ordered set (n÷k,m÷k), but the set of 3-part ratios ℂ_:3 := {n:m:k | n,m,k∈ℂ} is not a vector space. (Using ℂ as the field, because I naively believe that it won't complicate matters over using ℝ.) Are there any mathematically interesting operations one can do with it? — Sebastian 03:56, 19 November 2014 (UTC)

You correct to some degree that ratios can be replaced by division. The ancient Greeks would disagree for two reasons though. First, in allowed them to incommensurable magnitudes, which correspond to what we call irrational numbers. The Greeks knew that the diagonal of a square to its side were incommensurable (we would say √2 is irrational), but their concept of number only included what we would call rationals. To them, having a number whose square is 2 would lead to a contradiction, so keeping ratios and numbers separate was necessary to keep mathematics consistent. Second, the Greeks regarded the operations of arithmetic as applying to numbers, not to things like line segments, or squares. To them you could no more divide one line segment by another than you could divide a dog by a cat. But they did consider the ratio of two line segments to be valid. From our viewpoint, several centuries after analytic geometry, we think of lengths and areas as numbers so naturally that we don't even think about it, but that's really the result of a couple of millennia of evolution of mathematical thought. In a sense, the Greeks have a point here in that all measurements is ratios; if I say a line segment is 12 cm long then what I'm really saying is that the ratio of the segment to a certain standard meter is 12:100. Also, there is a certain amount of tradition, both in the way that math is taught and in everyday usage, that keeps the idea of ratio alive and separate from division. It's possible that humans thought in terms of ratios before discovering division. ("My field is twice as big as yours so I'll harvest twice as much corn next Fall.") So you might say that we have ratios as well as division is for "historical reasons", not because of any mathematical need. After all, computers get along perfectly fine understanding only division and no one is beating a path to Microsoft demanding that the next version of Excel include a ratio operation. --RDBury (talk) 13:18, 19 November 2014 (UTC)

Yes, ratio is just a convenient alternative notation for fractions and for division, but manipulation of ratios is a skill worth learning for quick mental calculations. Dbfirs 23:24, 19 November 2014 (UTC)

Thank you for your replies. The historical aspect is certainly interesting. I'd say it's a common phenomenon in mathematics that things are initially seen more complicated than they are. Adam Ries, in his famous Arithmetic Book of 1574, presents not 4, but 6 elementary operations: addition, subtraction, duplication, mediation (i.e. division by 2), multiplication, and division. He didn't mention that some can be seen as special cases of others. As we now see it, it is easier to teach in accordance with Occam's razor, and reduce the number of operations.

However, the signs "÷" and ":" have been introduced long after Classical Greece, and even after Adam Rise's time. As I understand, they're even still being taught now. And they're special in that respect; we don't introduce different mathematical symbols for different readings, such as "♊a" for "double a" and "♎a" for "half of a". So I guess my first question can be reworded as: Why do math teachers violate Occam's razor here by introducing what amounts to a fifth elementary operation?

How about the second question? Is there anything mathematically worthwhile in ℂ_:3 := {n:m:k | n,m,k∈ℂ}? — Sebastian 20:10, 20 November 2014 (UTC)

I am not sure I completely understand what you want, but have you looked at homogeneous coordinates for a projective space? —Kusma (t·c) 20:30, 20 November 2014 (UTC)

Although their mathematics is the same, the notions of ratio and fraction are somewhat different in usage. Fractions typically connote portions of a single whole,

Two thirds of Americans enjoy consuming peanut butter,

whereas ratios typically connote comparisons of two quantities,

A typical flock of Monstrum spaghetii volante includes one male to every four females.

Of course one can argue that for every ratio there lurks a fraction ("The frequency of males is one fourth that of females"), but that illustrates only that discourse allows several similar ways of expressing things, each of which emphasizes slightly different features. After all, when we consider the portions of a whole, the fraction is not equal to the ratio:

In M. spaghetti volante, a flock is typically one fifth male.

—PaulTanenbaum (talk)

One difference is that a ratio is symmetric, in that it can be written either way round and makes sense. This is not true of a fraction when one of the numbers is zero. For example the equation of a line can be written

$ax+by+c=0$

The gradient is given by −a / b but it can't be calculated if b is zero. Using a ratio instead for the gradient avoids this problem. This is related to homogeneous coordinates mentioned earlier – that article mentions this particular form of the line equation and it can be thought of as an elementary example of it. It generalises into three and higher dimensions in various ways, and has practical applications in computer graphics with techniques such as Plücker coordinates. — Preceding unsigned comment added by JohnBlackburne (talk • contribs)

Thanks, guys. homogeneous coordinates is what I've been looking for; that answers my second question. As for the first, I guess it's largely been exhausted; what's remaining is not really a mathematical question, but one of didactic method and taste. Personally, I don't see why one can't simply allow a gradient of ±∞, and understand that either value means that the line is parallel to the y axis; that isn't perfect, but it seems better than teaching a new operation. But I'm not a teacher, nor do I have kids who are learning this, so we can just agree to disagree on this. Thanks! — Sebastian 01:19, 21 November 2014 (UTC)

Infinity is not a normal number, needs special rules when calculating with it, and so is not included in many number systems. Even when it is there's more than one way to do it; whether it's signed or not in particular gives you two or one infinities respectively.--JohnBlackburne^words_deeds 18:01, 21 November 2014 (UTC)

That is correct; that's why I wrote "that isn't perfect". My point was that one singularity normally isn't a reason to trow out the baby with the bathwater and define all ratios as non-numbers, which you can't use for any normal calculation. But I'm not on a war path for changing math education; I just am trying to understand and be understood. — Sebastian 01:37, 22 November 2014 (UTC)

Explanation of elliptic curve cryptography

RSA's operation is straightforward and it can be done by hand with small numbers. I don't understand ECC yet. Can someone explain it to me in a way that someone with third year calculus experience can understand and present a simple example to me? — Melab±1 ☎ 06:22, 19 November 2014 (UTC)

Since you understand RSA, I assume that you will have no trouble understanding the Diffie-Hellman key agreement protocol and the related ElGamal public key cipher. What both cryptographic schemes need is a finite group whose associated discrete logarithm problem is hard. What you'll want from the finite group in addition is that the discrete log problem becomes sufficiently hard without involving arithmetic of numbers that are too large. (The larger the numbers, the slower the arithmetic.) What elliptic curves give you is a way to construct finite groups that (are believed to) have such properties.

Elliptic curves are mathematical objects of certain forms defined over a field. (Basically an elliptic curve is a set of points, or coordinates, that satisfy an equation of a certain form.) The elliptic curves used in cryptography are defined over finite fields. Finite fields are of two kinds: prime fields and finite extensions of prime fields. A prime field is a finite field of prime order, an example of which is ${\mathbb {Z} }_{p}$, where $p$ is prime, together with modular addition and multiplication $\mod p$. An extension field is constructed from an underlying prime field (the base field) using an irreducible polynomial over the field. If the base field is of characteristic 2, e.g. $({\mathbb {Z} }_{2},+,\cdot )$, arithmetic over the extension field can be done very efficiently.

Going back to elliptic curves. Once you have an elliptic curve over a finite field, you can construct a finite group out of the the (finite number of) points on the elliptic curve. To do that, you add a point at infinity and you can define an operation, called point addition, that can serve as the group operation for the set of points on the elliptic curve. (See the article on elliptic curves for a geometric illustration of point addition.) Once you have point addition as the group operation, you can define an associated iterated operation, called point multiplication.

If we compare these operations on an elliptic curve with operations over the multiplicative group $({\mathbb {Z} }_{p},\cdot )$, point addition is the analog of multiplication modulo p; point multiplication is the analog of exponentiation modulo p.

To recap, you start with a finite field. From the finite field you construct an elliptic curve. From the elliptic curve you construct a finite group whose associated discrete log problem is (believed to be) hard. You now have a suitable finite group for use in discrete-log-based cryptographic algorithms, such as Diffie-Hellman.

--173.49.79.74 (talk) 05:17, 22 November 2014 (UTC)

inverse of sum of inverses (resistors in parallel)

hello, out of curiosity, does this formula: 1/(1/a+1/b...), have any significance in maths? Is it perhaps some kind of mean and does it have a name? Asmrulz (talk) 22:50, 19 November 2014 (UTC)

Yes, it's the harmonic mean except for a factor of n (the number of resistors). Dbfirs 23:12, 19 November 2014 (UTC)

cool, thanks! Asmrulz (talk) 23:53, 19 November 2014 (UTC)

Yes, it's called the parallel sum (my user space), and David Ellerman claims it is just as good as the serial sum --catslash (talk) 00:24, 20 November 2014 (UTC)

That sounds a good idea, but the term is already used in other senses. It would be the serial sum for capacitors, of course. Dbfirs 00:49, 20 November 2014 (UTC)

Many terms have different meanings in different contexts - and of course this leads to fights on WP as to which is the primary topic. As to capacitors, it depends whether you quantify them by their capacitance or their elastance - aside from convention there is no particular reason for preferring one or the other (which leads to the claim that there is no reason for preferring ordinary serial addition over parallel addition). Of course, if you choose capacitance, then you want the parallel sum for capacitors in series and the serial sum for capacitors in parallel. --catslash (talk) 01:56, 20 November 2014 (UTC)

... or a normal serial sum for resistors in parallel if you quantify them by their conductance. Dbfirs 09:22, 20 November 2014 (UTC)

Also, analagously, the sum of resistances to heat transfer in calculating the overall heat transfer coefficient. shoy (reactions) 13:53, 20 November 2014 (UTC)