Wikipedia:Reference desk/Archives/Mathematics/2007 May 11

From Wikipedia, the free encyclopedia

Mathematics desk
< May 10 << Apr | May | Jun >> May 12 >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.


Contents


[edit] May 11

[edit] Additional Mathematics

Most secondary school students here take a subject called Additional Mathematics (A Maths for short), where we learn mathematics topics that are harder than those we learn in Elementary Mathematics (E Maths). For example, in E Maths trigonometry, we learn sin, cos and tan, but in A Maths trigonometry, we learn sec, cosec and cot. If you fail E Maths and hence don't take A Maths, you will learn all the A Maths topics, such as binomial and differentiation, when you go to JC.

Since Wikipedia doesn't have an article on Additional Mathematics, my question is: do other countries' education systems have Additional Mathematics, what is it called there and what mathematics topics do they learn?

--Kaypoh 08:11, 11 May 2007 (UTC)

In South Africa, Additional Mathematics is called Additional Mathematics (or Add Maths). I did it in 2001 and 2002 (Grade 11 and 12). The course covers Maths of Finance (interesst rates, annuities etc), Algebra, Co-ordinate geometry, Statistics (although you only needed 2 of the previous 4 to pass) and mainly calculus. The calculus section covered about 90% of the stuff we did in first-year at University.Zain Ebrahim 08:38, 11 May 2007 (UTC)

In the US it may vary from state to state and even from district to district. In my case, we had math in the following order:

Pre-Algebra
Algebra 1
Geometry
Algebra 2
Trigonometry
Pre-Calculus
Calculus

StuRat 08:56, 11 May 2007 (UTC)

The mathematics curriculum seems to have no international standard. Web searches may be your best research tool. For example, here is the core mathematics curriculum for the state of New York in the United States; and here is the mathematics framework suggested by the state of California, on the opposite coast. And the Asia Pacific Network of Education Knowledge Bank has a great deal of international information, covering many countries. You may also find some interesting reading here, especially with regard to mathematics teaching in France. --KSmrqT 10:39, 11 May 2007 (UTC)
In the UK, additional maths is called Further Maths. I'm not sure about what level you are talking about, though; Further Maths is Sixth Form (US Junior/Senior years), and tends to be very high level stuff (calculus of hyperbolic functions, the Argand diagram) Some school also run Pure Maths or Statistics course, which go beyond normal level maths but are far more focused on a single topic; since you say that additional maths covers cosec/cotan functions, I'm guessing it must be a lower level (at GCSE, the UK currently only has one maths course, although it is divided into Foundation, Intermediate and Higher, and there are plans to split it into core maths and further maths). Laïka 11:14, 11 May 2007 (UTC)
While the IB has four levels of mathematics, of which the highest is again Further Maths. Unlike the UK syllabus, this contains an element of genuine pure mathematics (rather than bucketloads of calculus), including some group theory, graph theory and analysis. Algebraist 11:30, 11 May 2007 (UTC)

Since, according to the Sixth Form article, students are usually 18 when they enter the sixth form and take their A Levels after that, this would make the sixth form equal to the second year of JC, after which we take our A Levels at the age of 18. Additional Mathematics is taken during the last two years of secondary education (at the age of 14-16), and is an O Level subject.

The topics you mentioned, such as calculus of hyperbolic functions, are only taught at H3 Maths in JC, which is reserved for the best students. Most students who take Additional Mathematics in secondary school will take H2 Maths in JC, while those who did not take Additional Mathematics will take H1 Maths in JC (the H1 Maths syllabus is roughly equal to the Additional Mathematics syllabus - only that the students learn it at JC instead of secondary school).

I should note that while Further Maths, and H3 Maths in JC, are reserved for the best students, most students here take Additional Mathematics. Only students who fail Elementary Mathematics do not take Additional Mathematics - although students who consistently fail Additional Mathematics will also be allowed to drop it. In fact, because the education system is so competitive, Additional Mathematics is compulsory in some schools.

Perhaps I should post the list of topics we learn in Additional Mathematics?

--Kaypoh 15:13, 11 May 2007 (UTC)

Students are 16 when they enter Sixth form; they graduate at 18 to go on to uni/vocational traing. Laïka 15:26, 11 May 2007 (UTC)

Then sixth form would be the equivalent of JC here. JC's a two-year course leading to the A Levels.

The Additional Maths Syllabus:

  1. Sets
  2. Simultaneous Quadratic Equations
  3. Indices, Surds and Logarithms
  4. Quadratic Equations and Expressions (using b2 - 4ac to find nature of roots)
  5. Remainder and Factor Theorems
  6. Solving Cubic Equations
  7. Matrices
  8. Cooordinate Geometry
  9. Linear Law
  10. Functions
  11. Trigonometric Functions (sec, cosec, cot and the four quadrants)
  12. Simple Trigonometric Identities and Equations (tan = sin/cos, sin2 + cos2 = 1)
  13. Circular Measure (about radians)
  14. Permutations and Combinations
  15. Binomial Theorem
  16. Differentiation and Its Techniques
  17. Rates of Change
  18. Higher Derivatives (using dy/dx to find max/min points)
  19. Differentiation of Trigonometric Functions
  20. Differentiation of Exponential and Logarithmic Functions (we've learnt up to this point)
  21. Integration
  22. Applications of Integration
  23. Kinematics
  24. Vectors
  25. Relative Velocity

--Kaypoh 09:41, 12 May 2007 (UTC)

[edit] Limits for cosh and sinh

Is there an elegant limit of a sequence for cosh and sinh that is reminiscent of this limit for the exponential?

e^x = \lim_{n \to \infty} \left( 1 + {x \over n} \right)^n.

I could make use of say, cosh(x) = 1/2 exp(x) + 1/2 exp(-x) and substitute the expressions above, but I am looking for something else. Any ideas? --HappyCamper 17:57, 11 May 2007 (UTC)

I'm having some trouble accessing it, but I believe [1] has an answer to that. Black Carrot 23:14, 13 May 2007 (UTC)

[edit] Non-iterative PRNG

Is there a deterministic PRNG where one can calculate the nth value (for a very large and entirely arbitrary n) without calculating the preceding values? All the PRNGs I'm aware of are iterative, meaning generating the gagillionth number means generating all gagillion-1 numbers before it too. My intended use is for simulation, and I'm not terribly concerned with periodicity or cryptographicly-acceptable randomness (if a few hundred values in a row starting at some arbitrary point are random enough to look random to a human, that's sufficient). Thanks. -- Finlay McWalter | Talk 21:20, 11 May 2007 (UTC)

Do you consider the digits of pi to be random? There's a formula for calculating arbitrary digits of it. --Carnildo 22:01, 11 May 2007 (UTC)
You can modify any PRNG that accepts seeds, such as LFSR to do this by using successive seeds i.e. the first number is the one generated from the seed "1", the second from the seed "2" ect. You may want to iterate it a certain number of times in order to make it seem random if you use certain PRNGs. In some cases, just starting from a high seed would work. — Daniel 22:31, 11 May 2007 (UTC)
Another approach, which does yield crypto-quality randomness, is to use a block cipher in counter mode. (See also this page for further details.) —Ilmari Karonen (talk) 22:36, 11 May 2007 (UTC)
If you are satisfied with a linear congruential generator, then you can compute the nth successor of any value on the cycle in O(log n) steps. Although not the best (see the article), LCGs are "good enough" for many applications, including yours. The basic idea is to generalize the relationship (omitting the "mod M" everywhere – this is tacitly understood)
Vj+1 = AVj + B
to
Vj+2k = AkVj + Bk.
Clearly, you can use
A0 = A, B0 = B.
Further,
Vj+2k+1 = AkVj+2k + Bk
= Ak(AkVj + Bk) + Bk = (Ak)2Vj + (Ak+1)Bk,
so, moreover,
Ak+1 = (Ak)2, Bk+1 = (Ak+1)Bk.
You can compute and store the vectors A• and B• ahead of time for up to the largest possible value of log2(n) you might want to consider, or recompute them each time on the fly. To compute Vj+n from Vj, use a "binary" method similar to exponentiation by squaring. Let us know if you need more details.  --LambiamTalk 22:47, 11 May 2007 (UTC)
Why not just use 'n' as the starting value for a PRNG? I mean, use an iterative one, but pretend the last iteration's result was 'n'. A generator like (An+B)modC wouldn't be very good at this (subsequent values would always be AmodC apart), but you could start there and modify it somehow, like using a bitwise XOR to make these evenly spaced values look random, like if it was more like ((An)modC)xorB) it might be good enough for your purposes. - Rainwarrior 06:52, 12 May 2007 (UTC)
Absolutely. Most (all?) generators will let you provide the starting value, or seed, as it is called, so you need not make your own. You can do like this in C:
#include <stdlib.h>
#include <stdio.h>
#include <math.h>

int main() {
  for (int i=1; i<=10; i++){
    srand(i);
    printf("%u ", rand());
  }
  printf("\n");
 exit(0);
}
Each time we go through the loop, srand will give the PRNG a seed value and rand will give you the random number for that seed. —Bromskloss 08:13, 12 May 2007 (UTC)
This idea of using successive seeds was already presented by Daniel above. It is equivalent to the application of an (ideally strongly mixing) hash function.  --LambiamTalk 08:39, 12 May 2007 (UTC)