Wikipedia:Reference desk/Archives/Mathematics/2014 December 3

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December 3

Graphing 1/log(x)

Hello! I am in grade 12 advanced functions, and one question I had was concerning the graph of f(x) = 1/log(x). There must be a vertical asymptote at x=1, as that is where y=0; however, why, when I graph the function on a graphing calculator like Desmos does the function exist at x=0 (i.e., the point [0,0]), considering log(0) = undefined? If someone could shed some light on this, that'd be much appreciated! 74.15.5.210 (talk) 00:02, 3 December 2014 (UTC)

Certainly $1/\log 0$ is undefined, but the limit (from the right) is 0. Also, IEEE 754 arithmetic specifies that $\log 0=-\infty$ and that $1/\pm \infty =\pm 0$ — based on the extended real numbers (and signed zero) — so some computer implementations will actually calculate 0 there. --Tardis (talk) 04:42, 3 December 2014 (UTC)

problem with the "Sunrise equation"

Madam/Sir: I was trying to calculate the time of Sunset using the "Sunrise" equation provided by Wikipedia. Unfortunately it gives me nonsens answers like 13:02:15 on Jan 1, 2000 in San Francisco, California. I am wondering if there is a missprint in one of the formulas. I would appreciate your help regarding this matter. With mybest regards, Laszlo Nadasdi — Preceding unsigned comment added by 98.248.180.82 (talk) 01:12, 3 December 2014 (UTC)

This is more an astronomy question than a math question so you might want to try the science desk. In any case, it would be a good idea to specify which formula(s) from which article and section you're trying to apply. The WP articles I saw assumed you already had a good grasp of technical terms like "solar declination" and "hour angle" which aren't really covered in a typical math syllabus (at least in this century). --RDBury (talk) 11:36, 3 December 2014 (UTC)

Conversion help

I'm trying to convert how much thiamethoxam (in ppm) is applied to one kernel of corn, but my estimates vary largely and I'm not sure which is correct, if any. [1] (p. 11 and 12) states not to allow more than "0.21 lb thiamethoxam per acre", with 75,000 kernels per acre assumed. 0.21 ÷ 75,000 = 0.0000028 lbs per kernel, 0.0000028 lbs to ounces equals 0.0000448. A corn kernel weighs 0.000697983522 lbs. Thus 0.0000448 ounces thiamethoxam to 0.000697983522 lbs corn, which equals 0.0641848963 ounces/pound. Using [2], 0.0641848963 ounces/pound equals 4011.556019 ppm (which seems really very high, especially when an experiment found mortality rates to be high for certain species at 4 ppm / kernel). Is this correct? How would I calculate Syngenta's recommended ppm / kernel? Seattle (talk) 02:32, 3 December 2014 (UTC)

This would probably be better at the science desk than the math one. That said, I don't think you are wrong that the number is large. However, I think the underlying concern might be the method of application. If you are spraying seed kernels prior to planting, then you can achieve an effective dose per kernel using relatively little thiamethoxam. On the other hand, if you are spraying a field after planting, then the required weight per acre is probably much larger. Ensuring that each planting site or seedling receives an effective dose after planting probably requires that a lot of the thiamethoxam will fall unproductively onto the ground. Dragons flight (talk) 05:19, 3 December 2014 (UTC)

This is seed corn, and the document you referenced does give lengthy seed bag labeling requirements, which include:

• Do not use for feed, food or oil purposes.

I also concur with your math, although your arithmetic path was convoluted. The conversion website you linked to is good when you are comparing two values in different units, but once you have the mass of both thiamethoxam and corn in the same units (lbs in this case), just divide! 0.0000028 lbs thiamethoxam / 0.000697983522 lbs corn = .004011556 = 4011.556 ppm. (Remember ppm is just parts-per-million, so multiply your fraction by 1,000,000 to get ppm.)

Alternately, you could work with larger numbers by noting that 75,000 × 0.000697983522 = 52.34876415 lbs corn, and 0.21 lb thiamethoxam / 52.34876415 lbs corn = .004011556 = 4011.556 ppm.

Where did you find that very precise value of 0.000697983522 lbs / kernel of corn? It is equivalent to (and thus, I assume, derived from) 0.3166 g / kernel, but that also seems overly precise. Your value is equivalent to 1432.7 kernels / lb or 3158.6 kernels / kg.

Penn State gives 1,450 kernels per pound, very close to your figure, but one Wisconsin supplier says that it varies with type, ranging between 1800 to 5000 seeds (kernels) per pound. Using the larger extreme brings your thiamethoxam concentration to 14,000 ppm.

This goes to show that there's more than one reason not to eat one's seed corn. -- ToE 02:51, 4 December 2014 (UTC)

@Thinking of England: Yes, I averaged three years' value from [3] for average corn kernel weight. 4,011.556 ppm seems high, especially considering when Arthur, Yue, and Wilde (2004) found that at one day's exposure to 4 ppm of thiamethoxam, the mortality rate for Sitophilus zeamais was ~90%, and that 40 CFR 180.565 limits ppm on corn forage, stover, etc. to between 0.02 and 0.10 ppm. The Syngenta label (p. 11) also suggests between 0.250–1.25 mg thiamethoxam per kernel for use against various pest insects. 0.250 mg equals 0.000000551156 lbs. 0.000000551156 lbs thiamethoxam / 0.000697983522 lbs per kernel equals 0.0007896404179. Multiplied by 1,000,000 equals 789.6404179 ppm. The corresponding value for 1.25 mg equals 3,948.20209 ppm. The diffusion from seed to forage, and later stover, must be quite high, but I can't find any relevant literature on absorption rate for thiamethoxam in corn. Perhaps its high GUS has something to do with it? Seattle (talk) 01:29, 6 December 2014 (UTC)

Ha! I was so close! If only I'd thought to Google "316.6 gram / 1000 kernel corn" I'd have had your reference on the first hit. I guess I'm not yet ready for the forensic numerical provenance investigator big-leagues. As for the rest, all I can say is that your math looks right. I assume that you are wondering why the concentration needs to be so high to keep the seeds safe from pests and are concerned about what remains at harvest. I can add nothing other than question how quickly thiamethoxam deteriorates when exposed to sun and weather. -- ToE 04:04, 6 December 2014 (UTC)

Now this really is straying into WP:RDS territory, but I see from thiamethoxam that it doesn't only protect the seed that it is applied to, but that "it is absorbed quickly by plants and transported to all of its parts, including pollen, where it acts to deter insect feeding". I had assumed that it was only there to protect the seed until germination, but for this use they would have to apply enough that it later protects the growing plant as well, and, evening assume that it is all incorporated into the plant, those overall concentrations would be much lower than for just the seed. How much heavier is the mature plant than the seed? Thiamethoxam#Toxicity indicates that it has a highly selective toxicity, affecting insects much more strongly than mammals. I see that this is the neonicotinoid that has been in the news recently as suspect in bee death. See Pesticide toxicity to bees. -- ToE 13:47, 6 December 2014 (UTC)

Zero and negative infinity

Zero is taken as an actual number; negative infinity is not. However, they have something in common:

To understand this, we must look at 2 mathematical worlds; the additive world and the multiplicative world. Note that every number in the additive world has a corresponding number in the multiplicative world; the conversion is to raise 2 to the power of the original number. For example, 3 in the additive world corresponds to 8 in the multiplicative world. Do you notice the properties of 0 in the multiplicative world?? The corresponding number in the additive world is negative infinity. What difference is there between 0 and negative infinity that makes it so that 0 (even in the multiplicative world) is a valid number, while negative infinity (which behaves the exact same way in the additive world) is not?? Georgia guy (talk) 17:40, 3 December 2014 (UTC)

I sort of get your point, but I don't think this is how it goes. I understand that one gets from your additive world to your multiplicative world by using exponents of 2 instead of plain numbers. As far as I remember, this works for every number except 0. You can't get 0 by raising any non-0 number to any power. You can't get any logarithm of 0. Negative infinity is a valid mathematical concept (although as far as I remember, it's not actually a number), but it's not something you get by taking a logarithm of 0. JIP | Talk 19:28, 3 December 2014 (UTC)

Positive and negative infinities are members of the extended real line, but they are not integers. Zero is an integer and a real number and a member of the extended real line. In math, what counts as a "number" depends on the problem at hand. Some people used to deny the existence of irrational numbers, and only believed that rational numbers truly existed. Sometimes it is useful to treat infinities the same as other numbers, sometimes it is not. SemanticMantis (talk) 20:08, 3 December 2014 (UTC)

You might also like to read about additive identity and multiplicative identity, these come up quite a bit in ring theory. SemanticMantis (talk) 20:11, 3 December 2014 (UTC) --p.s. additive identity is just a redirect to 1, so check out Unit_(ring_theory).

You're saying that negative infinity is not an integer, and this is a statement of the additive world. Does it have a corresponding statement in the multiplicative world?? Yes, this is that 0 is not a member of the set {...1/16, 1/8, 1/4, 1/2, 1, 2, 4, 8, 16...} Georgia guy (talk) 20:35, 3 December 2014 (UTC)

It will help if you use more conventional terminology. "additive world" and "multiplicative world" aren't really well-defined concepts. Your idea of raising 2^n seems like you are thinking of a logarithmic scale, but I'm not really sure. The set you denote is not a standard set of numbers that we commonly use in math, as it consists of powers of 2 and their multiplicative inverses. This set is indeed a group_(mathematics), where the group operation of "multiplication" is defined as exponentiation by 2. SemanticMantis (talk) 20:54, 3 December 2014 (UTC)

Georgia guy, what number in your additive world corresponds to -1 in your multiplicative world? -- ToE 00:46, 4 December 2014 (UTC)

That's a difficult question; I don't think there's any Wikipedia article revealing a good one. Georgia guy (talk) 00:500, 4 December 2014 (UTC)

Oh, but there is! The point is that your function, f(x) = 2^x, maps from all of the real numbers to the positive real numbers. If you want to extend your range (or, more correctly, your image) to include 0, you have to extend your domain beyond the real numbers to include -∞ from the extended reals. Likewise, if you wish to extend your image to include negative numbers, your domain must be expanded into the complex numbers. But that doesn't make -∞ part of the set we call the real numbers any more than it makes ln(2) π i (where i = √-1) part of the real numbers. Note that while the extended reals may have some niche usage, complex numbers are used much more extensively. -- ToE 03:18, 4 December 2014 (UTC) Many students find real analysis to be characterized by pathology, and complex analysis to be characterized by beauty.

The beauty of real analysis is in its pathologies. --Trovatore (talk) 06:18, 4 December 2014 (UTC)

Georgia guy, what do you mean by an "actual number"? Negative infinity is not a real number, but the "real" here is a term of art — it doesn't mean that negative infinity has any less objective reality than the reals do.

There seems to be some sort of meme about infinty "not being a number", but when analyzed, this claim is meaningless. --Trovatore (talk) 06:11, 4 December 2014 (UTC)

I tell my students all the time that infinity is not a number. What I mean is that it's not a real number, which isn't a meaningless statement. Staecker (talk) 19:55, 4 December 2014 (UTC)

Of course it's not a meaningless statement to say it's not a real number. But it'[s a different claim from saying it's not a number. Neither "infinity" nor "number" is well enough specified to make sense of the bare claim that "infinity is not a number".

Bottom line, in my opinion, you should stop telling your students that. It's a bad thing to tell them. Telling them it's not a real number is fine, provided you explain that "real" does not have its natural-language meaning here, but is a mathematical term of art. --Trovatore (talk) 07:32, 5 December 2014 (UTC)

I wonder if it's desirable, and feasible, to rename the real numbers to something less confusing. -- Meni Rosenfeld (talk) 09:47, 5 December 2014 (UTC)

Do people make this confusion, though, on the basis of name? Personally, most basic mathematics classes don't ever seem to move past the reals - save introducing the complex numbers as, what appears, a neat footnote (as opposed to deeply important) - and there is little, if any talk of abstract structure, etc. I would think that the natural assumption, from this, would be to assume that the reals were all that there was, or that anything else was kind of speculative - it also doesn't help that most k-12 (American) math seems to be taught as if one were learning magic than something logical - it feels more like the equations are neat little rituals that just work and need be memorized, the reasons are rarely offered (and proof is, certainly, not very central) - I mention this as most people seem very uncomfortable with the notion that "number" is rather fluid and contextual, and somewhat arbitrary, and instead seem to treat it as if it held some fundamental power of its own (if there were some decree that infinity were not a number, not in any form, it wouldn't change anything but what words needed to be used). tl;dr: I think it is less the name and more the treatment given in basic education.Phoenixia1177 (talk) 10:36, 5 December 2014 (UTC)

The subject here is understanding the difference between negative infinity in the additive world and 0 in the multiplicative world. Please make sure you don't mis-interpret it as the difference between negative infinity and 0 both in the additive world. The number line of the additive world has the integers on integer points. The number line of the multiplicative world has the numbers in the set {...1/16, 1/8, 1/4, 1/2, 1, 2, 4, 8, 16...} on integer points. In the additive world, negative infinity is an infinite distance to the left of any number. This belongs to 0 in the multiplicative world. Georgia guy (talk) 14:14, 4 December 2014 (UTC)

Your two "worlds" are analogous to some common mathematical notions (see above), but mainstream mathematics doesn't typically describe things in this way. It sounds like you're saying that 0 (along with all the negative numbers) doesn't even exist in the "multiplicative world". In this case 0 (multiplicative) is completely analogous to negative infinity (additive). Neither of them exist as "actual" numbers in their own "worlds". So there is no inconsistency at all. Staecker (talk) 19:55, 4 December 2014 (UTC)

Mathematicians use exponentiation and logarithms as a transformation where addition on one side corresponds to multiplication on the other. Hence the old saw about even adders being able to multiply on the a log table. This transformation is also exploited in the working of a slide rule. Mathematicians do not use your terms "additive world" and "multiplicative world" (with or without colors), but we understand what you are talking about. Similar transformations are used widely in mathematics. Quoting from Laplace transform#Properties and theorems:

The Laplace transform has a number of properties that make it useful for analyzing linear dynamical systems. The most significant advantage is that differentiation and integration become multiplication and division, respectively, by s (similarly to logarithms changing multiplication of numbers to addition of their logarithms). Because of this property, the Laplace variable s is also known as operator variable in the L domain: either derivative operator or (for s⁻¹) integration operator. The transform turns integral equations and differential equations to polynomial equations, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts to the time domain.

Your function f(x) = 2^x uses a base which might appeal to a computer scientist, while an engineer would prefer base 10 exponentiation, and a mathematician would more naturally use the natural logarithm base, but it is all really the same. To be most useful as a transformation, we like to express it as a bijection which maps from the Reals to the positive Reals, that is f:ℝ→ℝ⁺. Here there is no zero in the range (your multiplicative world) just as there is no -∞ in the domain (your additive world). If you want to expand the range (read image) to include zero, you must expand the domain into the extended real numbers to include -∞. While a few people might actually do this, in most fields it is not particularly useful, although everyone needs to understand the behavior from a limits point of view, that is $\lim _{x\to -\infty }f(x)=0$ and $\lim _{x\to 0}f^{-1}(x)=-\infty$.

You have good intuition here. There is something special about multiplication by zero that does not have a corresponding action with addition in the real numbers. You might enjoy reading Field (abstract algebra). One common example of a field is the real numbers under addition, with the additive identity 0, and multiplication, with the multiplicative identity 1. The field has two underlying groups, the real numbers under addition, and the real numbers without 0 under multiplication. 0 under multiplication has a special property that it yields itself when multiplied by any other number. (0 ⋅ x = 0) There is no such corresponding behavior for addition with the reals. If you do move to the extended real numbers you find a similar property that -∞ + x = -∞ as long as x ≠ ∞, but now it gets messy because we have elements that cannot be added (-∞ + ∞ is undefined), so we don't typically do algebra (as in abstract algebra) on the extend reals. See Extended real number line#Algebraic properties. -- ToE 20:33, 4 December 2014 (UTC)

You're saying that negative infinity plus infinity is undefined, and this statement from the additive world has a corresponding statement in the multiplicative world; this is that 0 times infinity is undefined. Georgia guy (talk) 20:49, 4 December 2014 (UTC)

Yes, that is true, but my point was that the real numbers under multiplication and addition form a nifty abstract algebraic structure we call a field, in which any two elements can be added or multiplied together, but that where the element zero under multiplication has a special property which is not shared by any element under addition. We can get that additive property by including infinity, but then we have to add special rules about who can play with whom (what multiplications and additions are undefined) and that breaks the simple rules of the algebraic structure and makes it less useful and interesting. By choosing to exclude infinities, in most cases we get better behaved constructions while still having the power to deal with infinite concepts through limits. That's what I was getting at. -- ToE 22:24, 4 December 2014 (UTC)