math

Math.....

For some reason few school books show Archimedes' derivation of (pi), a strange number that doesn't repeat like normal numbers. Before looking at how pi, which is used to calculate the area and circumference of a circle, is derived, consider what it means to say that normal numbers are repeating decimals. As an example of a normal, or "rational" number (meaning it is a ratio of two numbers) and how it repeats, consider 3/7. To put this in decimal form what do we do?....We divide 7 into 3, which gives .4283714283714283714.... That's what we call a repeating decimal. Other examples 1/3 = .333333.... and 1/2 = .50000000.... But some numbers, including pi and the square root of two go on and on without repeating. Click the links shown below to see the original derivation of (pi) in steps. Close each frame to return to go on to the next step.

Archimedes' calculation of

(From the website Graphics for the Calculus Classroom. To go to site, click here.)

In the third century B.C., Archimedes calculated the value of to an accuracy of one part in a thousand. His technique was based on inscribing and circumscribing polygons in a circle, and is very much akin to the method of lower and upper sums used to define the Riemann integral. His approach is presented in the following sequence of slides.

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Here is an interesting math problem from Shaum's outline series on probability.

Problem: What are the odds that three points on a circle fall on a semicircle? (Any semicircle, not a fixed semicircle, for which the answer would be 1/2 x 1/2 that the points will land together in either half.) We would like to hear intuitive solutions to this problem, and look at how the calculus solution works. And we would like to consider algebra solutions. Note, a good question to ask, in approaching this problem is what about two points? How often do two points fall in any semicircle? Click here for an intuitive solution to this problem.

Problem: (This is a physics problem. We want to include non-complicated physics problems and concepts on this page)

A boat is in a swimming pool in which a large rock is immersed. The rock sits on the bottom of the pool, it does not float. The rock is lifted out of the water and placed in the boat. Is the resulting water level on the side of the pool higher than when the rock was in the water, or lower, or the same?

To the right is an example of an interesting SAT problem in math. The three triangles ABC, ADB, and BDC are similar triangles (they have the same angles and the sides are in proportion) which is why Y/3 = 3/(X + Y). One solution for the value of Y .....By the Pythagorean theorem 16 + 9 = (X + Y) squared, so X + Y = 5. Substituting for X + Y in the equation they give gives Y/3 = 3/5. Which gives Y = 9/5. We would like to collect problems that are basic, like this, but have a certain elegance to them, like this one, on our math page. To learn basic math a student has to know the Pythagorean theorem, and to do algebra at this level. If a student is not facile and quick with algebra, that's OK. But they should get what's going on and be able to work through problems like this........To show that the triangles are similar we can start by showing that angle DAB is the same as angle DBC in the right triangles ABC and ADB. Consider that line BC has perpendicular AB and line BD has perpendicular AD. It is taken as a fact (an axiom) in geometry that if two lines cross at an angle, then lines perpendicular two first to lines will cross at the same angle. Since the two triangles also have a right angle the third angle will be the same. (180 degrees minus the other two angles, which are the same.)

We would like to ask if there is an intuitive way derive the formula for a volume of a cone. The formula is 1/3 r squared times the height.

We would like to develop a folder of multivariable calculus with a lot of problems that are based on something physical like electric fields. Divergence, gradient, curl, line integrals, operators.

Calculus is often seen as something too sophisticated for most high school students to ponder, but that is ridiculous. Finding the area under a curve, or the slope at a point on a curve, whereby the power of calculus can be simply observed, are not difficult concepts, if one will put his or her mind to it. The trick, going from X squared to 1/3 X cubed, is a very simple manipulation, a nifty little tool. ...................But it is not difficult at the basic level, so why not show it to students? If sophisticated ideas come in simple packages, why not show the package? Click here for a discussion of elementary calculus.

The formula for the circumference of a circle integrates to the formula for the area of a circle -- 2 R integrates to R squared. This integration is just like going from a curve to the area under it, conceptually -- the formula for a curved line, when integrated, gives the formula for the area contained by that line. Likewise the surface area integrates to the volume....... 4 R squared becomes 4/3 R cubed. The concept is the same, the envelope integrates to what is inside it. (The circumference of a circle is to its contained area what the surface area of a sphere is to it's contained volume.) Is there a history of speculations by mathematicians about why the change in formula is the same? It is obviously the kind of thing mathematicians look at and say, why is this so?

Music is made of math in the sense that the twelve notes on the piano make chords according to math ratios, and all the world's people have the same scale and harmonies. If there are beings like us on other planets who breath an atmosphere, and if they hear via this atmosphere, and if they have music, then they may find the same scales and harmonies to be natural and enjoyable, and, quite arguably, may have similar instruments, like the flute.

Something that may be said about mathematics is that it is true, in an absolute sense, perhaps more than any belief of the mind. The whole world disagrees about who God is but almost everyone agrees that 2 + 2 = 4. Physics and chemistry cannot be prove that reality conforms to the descriptions science gives it, and even our perceptions of the world around us cannot be proven to be real, as philosophers have suggested. But a circle is a circle, and the angles of a triangle add up to half a circle, and pi does not repeat, no matter what anybody says about it.