Physics 150

Basic Math

Algebra

The primary skill of algebra you will need for this course is to be able to solve for a desired variable in an equation or formula involving numbers and/or variables.

Notation

You will need to be familiar with the following notation:

Basic Manipulation

There are a few rules that you use in algebra to manipulate equations. We use these rules to go from one equation, which has a form we don't like, to another equation which has a form we want. When we use these rules to isolate a particular variable on one side of the equation, we call that "solving for" that variable. The rules of manipulation are:
  1. You can add or subtract zero from anything. So, x = 2y == x = 2y + (x - x) because x - x = 0.
  2. You can multiply or divide anything by 1. So, none because x/x = 1.
  3. You can perform any operation on an equation as long as you apply it to both sides of the equation. So, none, because we divided both sides of the equation by the same number (2), andx = 2y == x - y = y because we subtracted the same variable from both sides of the equation.
  4. When multiplying powers of a number, add the exponents, and when dividing powers subtract the exponents. Thus, none, and none.

Some Examples

  1. Solve for P in the equation none:
    1. First, multiply both sides by P^2. So, none.
    2. Next, divide both sides by k. So, none.
    3. Finally, take the square root of both sides, which leaves you with just P: none.
    4. Now, if you want to know a numerical value for P, you have to know numerical values for k and R. Let's suppose that R=3 and k=1.6875. Just plug those numbers into a scientific calculator, and you get none.
  2. Solve for x in the equation x^2 + 2 = y:
    1. Subtract 2 from both sides, so none
    2. Take the square root of both sides, so none. That's it!
    3. If you want a numerical value for x, you need to know y. Note that not every value of y will work....if none, then you'll be taking the square root of a negative number....something we don't know how to do in this class....

Scientific Notation

When dealing with very large or very small numbers, it is much more convenient (and takes a lot less typing) to use Scientific Notation. Scientific notation takes the form Ye+n, and the rule is quite simple. If n is positive, move the decimal point n places to the right. If n is negative, move the decimal point n places to the left. When typing, we often abbreviate scientific notation with an "e" instead of the none. Some examples:
  1. none = 1.234e+4
  2. none = 1.234e-4
  3. none = 1.234e+0
  4. none = 1.3e-2
  5. none = 1.3e+5

Geometry

A TriangleThe main fact that we will use in geometry is the Pythagorean Theorem. The Pythagorean Theorem is quite simple. Suppose you have a triangle like the one at the left. The Pythagorean Theorem says that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. So, a^2 + b^2 = c^2. Some examples:
    A Triangle
  1. Find the length of the unknown side of the triangle to the right. Since this is the hypotenuse, we just use a little algebra, and find that none.

    2. A Triangle
  2. Find the length of the unknown side of the triangle to the right. This time, it isn't the hypotenuse, but the algebra isn't much more difficult. With not much work, we find that none.
  3. Note that if the triangle isn't a right triangle (a right triangle has one angle exactly ninety degrees), we can still solve for the length of any side, given the lengths of the other two sides and the angles of the triangle. But, it is a lot more difficult, and we won't do that in this class....
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