Assignment 88

"One cannot escape the feeling that these mathematical formulae have an independent existence and intelligence of their own, that they are wiser than we are, wiser even then their discoverers." - (Heinrich Hertz, 1857-1894)

Math History Tidbit:

Goldbach's Conjecture: In 1742, a mathematician named Goldbach came up with the following statement:

Every even integer larger than 2 can be written as the sum of two prime numbers.


16 = 5 + 11

28 = 11 + 19

56 = 3 + 53

114 = 17 + 97

Interestingly, no one has even been able to prove Goldbach's Conjecture. Since there are an infinite number of positive even integers, it is impossible to test all of them. If you can prove that Goldbach's Conjecture is true for all even numbers, you will be famous in the mathematical world.

Herkimer's Corner

Why did Herkimer refuse to throw his watch into the garbage?

Answer: He didn't want to waste time.

Herky wants to know:

As long as there are tests in schools, why shouldn't prayer be allowed?

Would a skydiver believe the saying "If at first you don;t succeed, try again?"

Does old age come at a bad time?


Reading: Section 13.4, pages 792-794.

Written: Page 795/18-31.

Items for reflection:

Mathematical defintion:
Transcendental number: A number that cannot be a root (solution) of any algebraic equation with rational coefficients. The numbers e and pi are probably the two most famous transcendental numbers.

If you use your calculator in degree mode, you can determine that sin(30o) = 0.5. Basically, sine is a function, and the point (30o,0.5) is a point on the graph of the function. Remember that the inverse of a function is obtained simply by switching the coordinates. If, on your calculator, you calculate sin-1(0.5), you will obtain a value of 30. Since the calculator is in degree mode, the 30 is interpreted as 30o.

There are, of course, many angles that have a sine value equal to 0.5. Any angle that is coterminal with 30o (30o + 360o = 390o, for instance) will have a sine value of 0.5. It's important to realize that your calculator returns only one of the angles. You have to use some MATH POWER to respond to requests like:

Find a negative angle with a sine equal to 0.5.

In the case, you could evaluate sin-1(0.5), and get 30o. But this can't be a correct response because a negative angle was requested. So, the calculator is useless, right? Of course not! We simply have to use some MATH POWER and realize that we can take the 30o, rotate the terminal side in a negative direction through 360o, and end up with a coterminal angle that is 30o - 360o = -330o. Ah, so -330o would be an acceptable response to the request. You realize, of course, that -690o would also be acceptable, as would many other negative angles.