Goal: To understand the mathematics that will be necessary for this course which you do not get in a math class Objectives: Learning how to use Significant Figures Learning how to work with Scientific Notation Learning about how to use Units/Directions Learning about the basics of Vectors

Significant Figures Significant Figures are the digits that you know their values. There is no guessing you know what it is. If you DON’T know a digit you have to put in a zero by default (called a place holder). The more significant digits you have the more accurate the number.

How to determine them For numbers greater than one: A) if there is no decimal place then all the zeros at the end are NOT significant. They are called place holders. So, 10400 meters has 3 significant figures (which I will hereafter call sig figs). B) If there IS a decimal then ALL of the digits are significant. So, 10.0203 has 6 sig figs

Less than 1 For numbers less than one the 0’s on the left side are placeholders. So, 0.0102 only has 3 sig figs

Some more complicated ones How many sig figs for each of the following #s: A) 2030.03020 B) 0.0020030 C) 4050700 D) 102

How many to use? Suppose I wanted to find the area of the white board, who would I do that? Find the area of the whiteboard.

What do these distances represent? 0.0000000000000007 meters 100000000000000000000000000 meters

Scientific Notation A * 10B A is greater than or equal to 1 but less than 10. A can have a decimal which means that ALL of the digits in A are significant (the 10B is the placeholder) B is an integer (which means 0, 4, -6, but not 2.3)

Math in scientific notation For addition or subtraction you almost have to treat the powers of 10 as a “Unit”. That is to say that to add or subtract – without using a calculator that is – you need to have everything have the same powers of 10. Once you have that you can just add and subtract the #s in front and leave the powers of 10.

For Example 5.0 * 104 + 2.4 * 104 = 7.4 * 104

Multiplication Suppose you have two numbers in scientific notation such as: A * 10B and C * 10D Multiplied you get A * C * 10(B+D) What is (3 * 104) * (2 * 103)?

Quick note about calculators: If I give you a number of 1*102 when you enter it into your calculator be sure to enter 1 power 2 and not 10 power 2. Go ahead and try this you should get 100. If you do 10 power 2 you will get 1000 because your calculator thinks you are trying to enter 10*102

One more note If you get the number in front to be more than 10 then you have to adjust it. To do so take off factors of ten off of the front number (i.e. move the decimal) Each time you withdraw a factor of 10 from the number in front you have to deposit that factor of 10 into the powers of ten (by adding 1 to the integer for each time you move the decimal)

Quick one I will do (3 * 104) * (7 * 103)

Division For division you divide the numbers in front and subtract the exponents in the denominator So when you divide A * 10B by C * 10D you get A / C * 10(B-D) You try, but no calculator for now: Find (4 * 105) / (2 * 103)

Calculator note Everyone use their calculators to find the answer to the following problem (even if you can do this one in your head): (2 * 6) / (3 * 4)

On the calculator Always put the denominator in brackets, i.e. ()’s. Otherwise your calculator won’t calculate the problem correctly.

One more note: If the number in front becomes less than 1 you have to adjust by adding factors of 10. That is you move the decimal place. Each time you add a factor of 10 you have to subtract a factor of 10 from the powers of 10 by decreasing the exponent by 1 for each power of 10 you add.

Using units/directions In physics values do not come usually as just a number. For example you don’t go to a store to buy 5. You don’t go to the store to buy 5 pounds. You may go to buy 5 pounds of apples. In this case pounds is a unit as is apples.

Units include Physics uses the mks system which stands for meters (m), kilograms (kg), and seconds (s) Combinations of these units can make up other units as well. For example velocity is m/s direction Direction is also a unit (so up = -down)

Very Important Units can be very important. Unit errors in real life have had disastrous effects. A units mistake was once claimed to cause the crash of a Mars probe. Some claimed that a unit error (million and billion can sometimes be considered a “unit”) was rumored to wipe a TRILLION dollars of value from the stock market.

Adding values When you add or subtract values you have to add values with the same unit. It would not make sense to add 5 apples to 5 pears.

Multiplying/dividing When multiplying or dividing you treat units like you would a variable in algebra. So, a meter times a meter is a square meter. A meter divided by a meter is 1 (i.e. they cancel)

Direction One special type of “unit” is direction. In this class it is VERY beneficial to think of direction as a unit and to include it for any value that requires it (which I will call vector values). Examples include and are not limited to: up, down, forwards, backwards, towards an object, away from an object, North, South, East, West. If you use graphing axis directions can be plus or minus x (called x hat) and plus or minus y (called y hat).

Adding Direction units Just like with other units you can only add 2 values if they have the same unit. So, you cannot add a North to a South much like you can’t add 5 cm to 2 m UNLESS you convert one of the two units first. The conversion is straightforward often times because North = - South However this also means you cannot add North to West as there is no conversion – i.e. these become two separate values as we will utilize in this course.

Vectors Vectors take advantage of the fact that each dimension (dimensions are separated by 90 degrees) is independent for the most part. Vectors in vector form have a component in each dimension (which for this class will be usually 2 dimensions). When you add vectors you have to add the same components together while separating out the components that are not in the same direction. If you want the hypotenuse of the vector, or the total value without sign or direction this is called a magnitude.

Conclusion We have learned the basics of math that we will need to succeed in this course. We have learned how to use significant figures and what they represent. We have learned how to use scientific notation even without a calculator. We have learned how to use units/directions and how they apply to vectors.