Added By: Feedage Forager | |
Language: English | |
Tags: | |
case comment equilateral math natural numbers natural new number numbers org prime set term themathdoctors org triangle | |
Rate this Feed |
Comments (0) |
Feed Details and Statistics |
Last Build Date: Fri, 19 Jan 2018 19:49:10 +0000
Fri, 19 Jan 2018 19:49:10 +0000
I like it helps .me.on hard word problems(image) (image)Fri, 19 Jan 2018 17:17:55 +0000
Many thanks, Alan. There are so many ways of looking at this problem. Regards Gererd G.Fri, 19 Jan 2018 01:59:40 +0000
Congrats to Gerard for going the extra mile to get a proof for the scalene case. Like most other solvers I just assumed the triangle was equilateral and did it by partitioning the shaded area into two parts (one being 1/2 of 1/3 of the whole thing and the other being 1/6 of 1/4 for a total of 1/6+1/24=5/24). But Gerard's proof of the general case got me thinking! ...With two conclusions: First, the partitioning approach does still work in the scalene case, but one needs to use the facts that the lines joining feet of the medians are parallel to the sides of the triangle and so divide it into 4 congruent pieces, and that the areas of triangles from a point to equal bases on the same line are equal. Second, and maybe more amazing, is the fact that the general case (of this and many similar problems) actually follows from the equilateral case. The key is to note that any triangle can be projected onto an equilateral one. (For an informal "proof" just draw the triangle on a piece of paper and move around until it "looks" equilateral, or hold up a small cut-out equilateral triangle in between you and the paper and tilt it around until it exactly covers your view of the triangle you drew). But the projection just multiplies all areas by the same factor - and so preserves the ratios!Tue, 09 Jan 2018 00:00:14 +0000
Thanks for reaching out, Dave. I've amended my post. Good luck with your new venture!Mon, 08 Jan 2018 16:30:33 +0000
Thanks for your recommendation of Ask Dr. Math (mathforum.org/dr.math). It is important to us that, as you mentioned, we do not just give answers, but rather help students find their own. I want to let you know that, although that site is still accessible for viewing its archive of past answers, questions can no longer be submitted there. The Math Doctors have started a new and improved site at TheMathDoctors.org, where we now take questions as before, and also have a new blog discussing past answers. The following wording would now be appropriate for that entry: "Ask Dr. Math has been going since 1994 and has a wealth of answers to a broad range of readers' questions. It's searchable, as are all the recommendations below. New questions may be submitted at TheMathDoctors.orgFri, 05 Jan 2018 12:33:54 +0000
@Eric: This would be a good question for the IntMath Forum. Here's the appropriate section: Plane Analytic Geometry forum.Fri, 05 Jan 2018 11:41:47 +0000
Please Guys i will like to ask help on parabola dish Measurement i want a 40 feet dish Guys can you help me with the measurement ? i want to know the Depth of 40 feet dish Diameter = 40 feet Depth = Focal length =Thu, 04 Jan 2018 23:42:43 +0000
What is it about this question that has captured the imagination of so many people with such varying sophistication for such a long time? It is really quite fascinating and I cannot resist the temptation to weigh in just for fun. In Mathematics it often happens that the same term is used in different contexts to mean entirely different things. For example the word 'normal' can refer to a type of subgroup that is the kernel of a group homomorphism, to a vector that is orthogonal to a manifold or to a set of productions in a context-free grammar that is in a standard form (think Griebach or Chomsky normal form). The same is true of the term 'natural', we have the natural numbers, natural logarithms, natural transformations between functors and we have natural isomorphisms. However, the latter two concepts likely do not feel natural to the average adult, talk less of a seven year old. So, whether or not something is considered natural depends very much on your background. To be sure, the concept of zero may not be something you are born with but it doesn't take long before you become quite comfortable with it. (If it weren't for the potential confusion with the complex numbers, we could call {1, 2, 3, ...} the set of comfortable numbers . . .). However, it requires a bit more sophistication before the notion of a natural isomorphism actually feels natural. It is also true that the language of mathematics evolves over time, just as natural (oh dear) language does. Isaac Newton used the term 'fluxions' which was a precursor to the modern term 'limit' and abstract algebra used to be called modern algebra. Other examples abound. The language even seems to evolve during one's mathematical education. When we were in primary school, prime numbers were those that could only be divided by themselves and 1. Of course this includes 1. By the time we got to college the 1 was dropped so that the smallest prime number was now 2. What happened? Well for one thing 1 has some rather special properties that the other primes don't (it is a unit and an idempotent), but the main thing is that the prime factorization theorem would no longer be true as it is currently stated since the factorization would not be unique. The theorem is still true but its statement would become quite a bit more awkward, as would many other theorems about primes that would have to say things like "If p is a prime, where p > 1, then ..." instead of "If p is prime, then ..." Nowadays in mathematics when a new concept is discovered or invented and we want to call it something, we often seek some term from the vernacular that somehow gives us a bit of intuition about the concept. For example the notion of a sheaf of rings on a topological space is reminiscent of a sheaf of wheat - the common meaning element being something like 'a bunch of similar-looking things emanating from points on a surface'. Once the term is chosen something wonderful happens, the meaning of the term is purified and made precise by stripping the term of all its previous connotations and attaching the unique special meaning to it by way of a mathematical definition. This way, as distinct from natural language, the term becomes unambiguous. Or at least it should. Which brings us to the ambiguity in the term 'natural numbers' and how it should be resolved. It has been pointed out that number theorists like to start their natural numbers at 1, while set theorists like to start at 0. There is a good reason for this and it and it comes down to ease of expression. Set theorists find the set {0, 1, 2, ...} appears more frequently in the statements of theorems and proofs than the set {1, 2, 3, ...}, while number theorists find it the other way around. Thu[...]Sat, 30 Dec 2017 14:56:22 +0000
very good explanationFri, 29 Dec 2017 22:13:09 +0000
A log is used to make large numbers small to fit on a paper page. They are simply the power (exponent) on the base in question (usually 10 as in [for ex. base 10 to the 2.3423 power/exponent]) that will provide the large number in a less lengthy form thereby promulgating a snap shot visualization of graphical simplicity.