2. We are the only people who don’t use the metric system; everyone else in the world does. Just like us, the metric system has its base units. Base units for the metric system: Length – meters (m) Liquids – liters (L) Mass – grams (g) Temperature – degrees Celsius (°C) Fortunately, there are conversions from our units to the metric system: 1 m = 3.28 feet 1 L = 0.264 gallons 1 g = 0.0022 lbs °C = (5/9) (°F – 32)

3. I just decided I want to convert 4 feet into meters. How would I do that? Well, from the list on the previous slide, 1 m = 3.28 feet. That doesn’t help me at all because I want 4 feet! D: First, I want to know how many meters are in one foot. Since I know it for 3.28 feet, I just have to divide both sides by 3.28: Wonderful! Now I know I have 4 feet that I want to convert to meters, so I can just multiply both sides by 4: 4 feet = 4(0.3049 m) = 1.22 m So, 4 feet is equal to 1.22 meters. If I wanted to convert 7 gallons to liters, would I go about it the same way? 3.28 feet1 m 3.28 = 1 foot = m = 0.3049 m 1 3.28

4. Those prefixes are extremely important! You really never hear anyone say “1000 m”. Usually, that length is referred to as 1 kilometer (km). That prefix “kilo-” has a meaning associated with it. Since 1000 m = 1 km, it must be that “kilo-” means “1000”. 1 kilo- = 1000 1 mega- = 1000 kilos 1 giga- = 1000 megas 1 deci- = 0.1 1 centi- = 0.01 1 milli- = 0.001 1 micro- = 0.000001 1 nano- = 0.000000001 Conversions within the metric system work just like converting from feet to meters.

5. How would I convert 400 meters to kilometers? I know that 1 kilo- = 1000, so 1 kilometer must equal 1000 meters. I have 400 meters, though. To find how many kilometers are in one meter, we have to divide both sides by 1000: Now that we know how many km are in 1 m, it’s time to find out how many are in 400 m: 400 m = 400 (0.001 km) = 0.4 km There you have it. 400 m is 0.4 km. What about converting 5 kilograms into grams? 1000 meters1 kilometer 1000 = 1 meter = kilometers = 0.001 km 1 1000

6. Sometimes, numbers can be VERY large, or VERY small. In Astronomy, you deal with very large numbers, like distances or sizes. For example, the average distance from the Earth to the Sun is 149,597,900 km, and the distance from Earth to the next nearest star (Proxima Centauri) is 39,900,000,000,000 km. (If we were to convert these numbers to meters, would we expect these numbers to be larger or smaller? By how much?) In physics, particularly quantum mechanics, you deal with things like electrons, whose mass is 0.000000000000000000000000000000911 kg. That’s a lot of zeros! Numbers like these are difficult to work with, and it’s very likely a “0” can be forgotten (especially in the case with the electron mass). Scientific notation is the easy way to keep track of all those zeros!

7. The entire basis for scientific notation is writing everything in powers of ten. For example, 1000 = 10 × 10 × 10 = 10 3. This is a very easy way to keep track of zeros if you have a whole bunch of them! In the example with 1000, the actual number 1000 is a 1 followed by three zeros. Interestingly enough, the power in 10 3 is 3. Scientific notation is, in a sense, counting the number of zeros you have and that number becomes your power of 10. A “googol” is the name for the number that is a 1 followed by 100 zeros. Unless you’re really good at keeping track of how many zeros you write down when writing large numbers, scientific notation is the best way to represent numbers that are large. So, since a googol is a 1 followed by 100 zeros, we can just simply write it as 10 100.

8. Small numbers (like the mass of an electron) can also be written in scientific notation. It’s slightly more complicated, though. This time, instead of counting how many zeros come after a 1, we’re counting the number of decimal places we’d have to move to make the number greater than 1: For example, 0.001 has two zeros before the 1. So, we know when we write this number in scientific notation, we’re going to need a 3 as the power since we have to move over for the two zeros plus the space we have to move for the 1: 0.001 However, it makes no sense to write 10 3 for 0.001, since 10 3 already means 10 × 10 × 10 = 1000. So, when we write small numbers as a power of ten, we write that power with a negative sign: 0.001 is a decimal and not a whole number, so 0.001 = 10 -3. You’ll notice that 0.001 = 0.1 × 0.1 × 0.1 = × ×, and = 10 -1, so mathematically this makes sense. 1 10 1 1 1

9. A few slides ago, I pointed out that the distance from the Earth to Proxima Centauri is 39,900,000,000,000 km. How would this look in scientific notation? If we count all the zeros in this number, we find that there are 11 of them. So, we could write this as 399 × 10 11 km, since 39,900,000,000,000 km = 399 × 100000000000 km, and 100000000000 is a 1 followed by 11 zeros. However, when writing scientific notation, we don’t want to see things like “399 × 10 11 ”, since the 399 also has a power of ten associated with it: 399 = 3.99 × 100, or 3.99 × 10 2. So, 399 × 10 11 can be written as 3.99 × 10 2 × 10 11, or just 3.99 × 10 13. The distance from the Sun to the Earth is 149,597,900 km. How would this number be written in scientific notation? 149,597,900 km = 1.495979 × 100,000,000 km = 1.495979 × 10 8 km, which we can round to 1.5 × 10 8 km.

10. Suppose I have 8.3 milliliters (mL) of a fluid, and I want to convert this to liters. What would this number look like in scientific notation? This question has two parts! The first part is converting milliliters to liters, so let’s take care of that. We know that 1 L = 1000 mL, so 1 mL = 0.001 L. Since we have 8.3 mL of fluid, we have to multiply both sides of the equation by 8.3: 8.3 mL = 8.3(0.001 L) = 0.0083 L Now for part two: writing this number in scientific notation. Since this number is a decimal, we need to count how many spaces we have to move until we move past a nonzero number (in this case, the 8): 0.0083 L Oh! It looks like we only had to move over three decimal places. Therefore, 0.0083 L = 8.3 × 10 -3 L. What if I want to convert the mass of the electron (0.000000000000000000000000000000911 kg) into grams, and then write that in scientific notation. Would the process be the same as this example?