Scientific Notation and Factor Label Method Lesson 2 January 20th, 2011 Scientific Notation and Factor Label Method

Scientific Notation Dealing with larger numbers can be difficult. We like to deal with small numbers. Ex 0.000 000 895 kg = 0.895mg If we cannot change the units we can use scientific notation. It allows us to manage large numbers without changing the units

Scientific Notation Scientific notation allows us to write numbers in the form of a x 10n Where 1 < |a| <10 and all the digits in the coefficient are significant Example 124.5 million kilometres = 124 500 000 km Or = 1.245 x 108 km

Scientific Notation How do I find n? Count the decimal places 230 000 000 000 units If 2 is the number out in the front, the rest of the numbers must be behind the decimal place. Count from the right how many times the decimal place would have to move to make it 2.3 n= 11 2.3 x 1011 units

Scientific Notation Bring out your calculators Look for a key that says EXP Or EE This key allows you to just type in the start and it fills in the “x 10”. Then you just add the exponent. Try it out. Type in 1.5 EXP 5 = It should say 150 000

Scientific Notation Practice Decimal Notation Scientific Notation 127 1.27 x 102 0.0907 9.07 x 10 – 2 0.000506 5.06 x 10 – 4 2 300 000 000 000 2.3 x 1012 Decimal Notation Scientific Notation 127 1.27 x 102 0.0907 9.07 x 10 – 2 0.000506 5.06 x 10 – 4 2.3 x 1012 Decimal Notation Scientific Notation 127 1.27 x 102 0.0907 5.06 x 10 – 4 2.3 x 1012 Decimal Notation Scientific Notation 127 0.0907 5.06 x 10 – 4 2.3 x 1012 Decimal Notation Scientific Notation 127 1.27 x 102 0.0907 9.07 x 10 – 2 5.06 x 10 – 4 2.3 x 1012

Factor Label Method

The factor label method Used to convert between units km to miles, m to km, mol to g, g to mol, etc. To use this we need: Desired quantity, Given quantity, Conversion factors Conversion factors are valid relationships or equities expressed as a fraction

E.g. for 1 km=0.6 miles the conversion factor is

Q. write conversion factors for 1 foot =12 inches Q. what conversion factors can you think of that involve meters?

Conversion factors Conversion factors for 1 ft = 12 in There are almost an infinite number of conversion factors that include meters:

Conversion Factor Steps 1.Write down the desired quantity/units 2.Equate the desired quantity to given quantity 3.Determine what conversion factors you can use (both universal and question specific) 4.Multiply given quantity by the appropriate conversion factors to eliminate units you don’t want and leave units you do want 5.Complete the math

First write down the desired quantity example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km First write down the desired quantity

Next, equate desired quantity to the given quantity Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi Next, equate desired quantity to the given quantity

Now we have to choose a conversion factor Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi Now we have to choose a conversion factor

What conversion factors are possible? Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi 1 km 0.621 mi 0.621 mi 1 km What conversion factors are possible?

Pick the one that will allow you to cancel out miles Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi 1 km 0.621 mi 0.621 mi 1 km Pick the one that will allow you to cancel out miles

Pick the one that will allow you to cancel out miles Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi 1 km 0.621 mi 0.621 mi 1 km Pick the one that will allow you to cancel out miles

Multiply given quantity by chosen conversion factor Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi 1 km 0.621 mi 0.621 mi 1 km Multiply given quantity by chosen conversion factor

Multiply given quantity by chosen conversion factor Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) x 1 km 0.621 mi # km = 47 mi Multiply given quantity by chosen conversion factor

Cross out common factors Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) x 1 km 0.621 mi # km = 47 mi Cross out common factors

Cross out common factors Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) x 1 km 0.621 # km = 47 Cross out common factors

Are the units now correct? Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) x 1 km 0.621 # km = 47 Are the units now correct?

Yes. Both sides have km as units. Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) x 1 km 0.621 # km = 47 Yes. Both sides have km as units.

Yes. Both sides have km as units. Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) x 1 km 0.621 # km = 47 Yes. Both sides have km as units.

Now finish the math. x 1 km 0.621 = 75.7 km # km = 47 Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) x 1 km 0.621 = 75.7 km # km = 47 Now finish the math.

The final answer is 75.7 km x 1 km 0.621 = 75.7 km # km = 47 Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) x 1 km 0.621 = 75.7 km # km = 47 The final answer is 75.7 km

Summary The previous problem was not that hard In other words, you probably could have done it faster using a different method However, for harder problems the factor label method is easiest

Rearranging Equations

Rules to rearranging equations RULE #1: you can add, subtract, multiply and divide by anything, as long as you do the same thing to both sides of the equals sign.  In an equation, the equals sign acts like the fulcrum of a balance: if you add 5 of something to one side of the balance, you have to add the same amount to the other side to keep the balance steady

Rule 1 example Find b in the equation y = mx + b Subract mx from each side of the equation Y – mx = mx – mx + b Y – mx = b

RULE #2: to move or cancel a quantity or variable on one side of the equation, perform the "opposite" operation with it on both sides of the equation.  For example if you had g-1=w and wanted to isolate g, add 1 to both sides (g-1+1 = w+1). Simplify (because (-1+1)=0) and end up with g = w+1. 

More complex example This equation is used to calculate the density of icebergs We will isolate ρobject which is the density of the iceberg itself.

Start by isolating the equation in brackets Divide both sides by Htotal =

Next we will isolate the faction that contains ρobject Subtract 1 on each side =

Multiply both sides by ρfluid to get rid of the fraction Cancel out ρfluid

We will now remove the negative sign by multiplying by -1 =

Rearrange the equation a little And we are done