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Three geometric construction problems from antiquity puzzled mathematicians for centuries: 1. The trisection of an angle (dividing a given angle into three equal angles). 2. Squaring the circle (constructing a square with the same area as a given circle). 3. Duplicating the cube (constructing a cube with twice the volume of a given cube). Are these constructions impossible? Click here to know... Impossible Geometrical Constructions

Division by zero is an operation for which you cannot find an answer, so it is disallowed. You can understand why if you think about how division and multiplication are related. 12 divided by 6 is 2 because 6 times 2 is 12 12 divided by 0 is x would mean that 0 times x = 12 But no value would work for x because 0 times any number is 0. So division by zero doesn't work. Another explanation Let's look at some examples of dividing other numbers. 10/2 = 5 This means that if you had ten blocks, you could separate them into five groups of two. 9/3 = 3 This means that if you had nine blocks, you could separate them into three groups of three. 5/1 = 5 Five blocks could be separated into five groups of one. 5/0 = ? Into how many groups of zero could you separate five blocks? It doesn't matter how many groups of zero you have, because they would never add up to five since 0+0+0+0+0+0 = 0. You could even have one million groups of zero blocks, and they w...

Pay attention to restrictions on formulas This is an error that is often committed by teachers too (I must admit this fact!). Teachers don’t give or make a big deal about restrictions on formulas. In some cases the teacher forget the restrictions, in others they seem to have the idea that the restrictions are so obvious that they don’t need to give them, and in other cases the teachers just don’t want to be bothered with explaining the restrictions so they don’t give them. For instance, in an algebra class you should have run across the following formula. The problem is there is a restriction on this formula and many instructors don’t bother with it and so students aren’t always aware of it. Even if instructors do give the restriction on this formula many students forget it as they are rarely faced with a case where the formula doesn’t work. Take a look at the following example to see what happens when the restriction is violated. So clearly we’ve got a problem here as we are well awar...

Powers of trig functions Remember that if n is a positive integer then The same holds for all the other trig functions as well of course. Also remember to keep the following straight. In the first case we taking the tangent then squaring result and in the second we are squaring the x then taking the tangent. The tan x^2 is actually not the best notation for this type of problem. We really should probably use tan( x^2) to make things clear. Another thing to always keep in mind is that -1 in the cos^-1(x) is NOT an exponent , it is there to denote the fact that we are dealing with an inverse trig function. ie. There is another notation for inverse trig functions that avoids this problem, but it is not always used.

A perfect number is a whole number, an integer greater than zero; and when you add up all of the factors less than that number, you get that number. For example: The factors of 6 are 1, 2, 3 and 6. 1 + 2 + 3 = 6 The factors of 28 are 1, 2, 4, 7, 14 and 28. 1 + 2 + 4 + 7 + 14 = 28 The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496. 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 The factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 and 8128. The first four perfect numbers were known over 2,000 years ago. Some ancient cultures gave mystic interpretations to numbers that they thought were magic. Using a computer program, eventually ran for a billion of time the following 5 perfect numbers can be found: 6 = 1+2+3 28 = 1+2+4+7+14 496 = 1+2+4+8+16+31+62+124+248 8,128 = 1+2+4+8+16+32+64+127+254+508+1016+2032+4064 All of the perfect numbers that have been found so far fit the formula 2^(n-1) * ( 2^n - 1 ) where "n" ...

By definition, pi is the ratio of the circumference of a circle to its diameter. Pi is always the same number, no matter which circle you use to compute it. For the sake of usefulness people often need to approximate pi. For many purposes you can use 3.14159, which is really pretty good, but if you want a better approximation you can use a computer to get it. Here's pi to many more digits: 3.14159265358979323846. A very brief history of pi: Pi is a very old number. We know that the Egyptians and the Babylonians knew about the existence of the constant ratio pi, although they didn't know its value nearly as well as we do today. They had figured out that it was a little bigger than 3; the Babylonians had an approximation of 3 1/8 (3.125), and the Egyptians had a somewhat worse approximation of 4*(8/9)^2 (about 3.160484), which is slightly less accurate and much harder to work with. The modern symbol for pi was first used in our modern sense in 1706 by William Jones. It(rat...

Degrees vs. Radians Most trig classes that I’ve seen taught tend to concentrate on doing things in degrees. I suppose that this is because it’s easier for the students to visualize, but the reality is that almost all of calculus is done in radians and students too often come out of a trig class ill prepared to deal with all the radians in a calculus class. You simply must get used to doing everything in radians in a calculus class. If you are asked to evaluate cos (x) at x =10 we are asking you to use 10 radians not 10 degrees! The answers are very, very different! Consider the following, cos (10) = - 0.839071529076 in radians cos (10) = 0.984807753012 in degrees You’ll notice that they aren’t even the same sign! So, be careful and make sure that you always use radians when dealing with trig functions in a trig class.