Does anyone know that the circumference of a circle was originally calculated by taking a large number of sides of a regular polygon and calculating the perimeter of the polygon. It was then divided by the distance of the segment through the center of the polygon. As the number of sides of the polygon increases, the distance of the side decreases and the polygon looks more and more like a circle and the value of Pi gets closer and closer to 3.1415916....  


No matter how large a circle is, the circumference divided by the diameter is equal to Pi.

A logarithm is an exponent. When we solve a problem  such as log 100 = y, first we assume that the base is 10. Then we must think of y as the exponent to which 10 is raised to get 100. Therefore y = 2.  The problem rewritten as an exponential equation is 10^y = 100.

Here's a few graphs of the conic sections. All of these are seen in my second book, which should be out before the summer.

Circle

Parabola

Ellipse

Hyperbola

When solving a problem such as 3^(2x + 1) = 9^(3x + 2), you can use logarithms or get the bases the same and set the exponents equal to each other. Getting the bases the same you have 3^(2x + 1) = 3^(6x + 4). Now set 2x + 1 = 6x + 4 and solve for x. When doing so, you get x = -3/4.

If we solve using logarithms, take the log of both sides to get

log (3^(2x+1)) = log(9^(3x + 2))

(2x + 1)log3 = (3x + 2)log9

(2x + 1)/(3x + 2) = log9/log3

(2x + 1)/(3x + 2) = 2

(2x + 1) = 2(3x + 2)

2x + 1 = 6x + 4

-4x = 3

x = -3/4

Staying on the topic of probability, suppose you want to figure out the probability of drawing a certain 5 card poker hand.  Sometimes we can list and the possible outcomes, but in this case it's virtually impossible to list all the 5 card hands that can be drawn from a standard 52 card deck.  So we use combinations to solve such a problem.  The number of possible 5 card hands that can be dealt from 52 cards if order is not important is 52!/(5! 47!)   Recall that 52! = 52 * 51 * 50 *..... *1. 

52!/(5! 47!) can be simplified to (52 * 51 * 50 * 49 * 48)/(5 * 4 * 3 * 2) = 2,598,960.  That's how many possible 5 card hands can be dealt.

Then we just figure out the number of ways the certain hand you are looking for can be dealt and divide that by 2,598,960 to obtain the probability of obtaining such a hand.

Ever think you have so much good luck that you could guess on a multiple choice test and achieve a perfect score? Well, think again.  If the test consists of 10 questions with 4 choices for each question, there are

4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 = 4¹⁰ = 1,048,576 ways to answer the 10 multiple choice questions. 

If each question has only 1 correct answer and you guess on every one, you have a 1 in 1,048,576 chance of getting a perfect score..

Better start studying!!

Remember when rationalizing a denominator involving a square root, multiply the numerator and denominator by whatever is under the radical in the denominator.

If the problem is 2/sqrt(3), multiply the numerator and denominator by sqrt(3)/sqrt(3) to get

2sqrt(3)/3.

If the problem involves a cube root, multiply the numerator and denominator by whatever will give you a perfect cube in the denominator.

For example, if the denominator is cube root(3), multiply numerator and denominator by cube root(9). This will make the denominator cube root(27), which equals 3.

Same idea for 4th root, 5th root and so on.