Scientific Notation & Dimensional Analysis Chapter 2.2

What is scientific notation? A short-hand way to write a very small or large number without all the zeros =6.78x =7.9589x10 12 Numbers expressed as a multiple of two factors. A number between 1 to 10 and ten raised to a power. The exponent or power tells you how many times the first factor must be multiplied by 10. Numbers greater than 1→ exponent=positive (+) Numbers less than 1 → exponent=negative (-)

Converting data into scientific notation STEP 1→Move the decimal place so only one number is between 1 and 9. STEP 2→Remove the extra zeros at the end or beginning or the number. STEP 3→Multiply the result by 10 n where n is the number of decimal places moved. If the number was less than 1, then n will be negative. (The decimal place moved to the right). If the number was greater than 1, then n will be positive. (The decimal place moved to the left).

EXAMPLE: The Distance From the Sun to the Earth 93,000,000

Step 1 Move decimal Leave only one number in front of decimal

Step 2 Write number without zeros

Step 3 Count how many places you moved decimal Make that your power of ten

The power of ten is 7 because the decimal moved 7 places. 93,000,000 → Standard Form 9.3 x 10 7 → Scientific Notation

More Examples 9.85 x 10 ? 6.4 x 10 ? 2.7 x 10 ? 4.29 x 10 ? 198,500, ,000,000, Write in decimal form w/o zeros. Determine the power of ten x x x x 10 -9

Practice Problems Write the number in scientific notation. 734,000, ,000,000,000 90,000,000,

Answers 1)734,000,000 = 7.34 x )870,000,000,000 = 8.7 x )90,000,000,000 = 9.0 x ) = 4.65 x ) = 3.4 x

Converting data out of scientific notation Move the decimal place the number of decimal places indicated by the exponent. If exponent is negative, the decimal place will move to the left and the result will be less than 1. If exponent is positive, the decimal place will move to the right and the result will be greater than 1.

Practice Problems Write the number in standard form. 9.5 x x x 10 10

Answers 9.5 x 10 6 = 9,500, x = x = 67,750,000,000

Using a calculator for scientific notation The EE button on your calculator will allow you to enter a number in scientific notation. EXAMPLE → 3.95 X Enter 3.95 Press EE Enter (-) Enter 7 Proceed with manipulation as usual.

Practice Problems Add 2.7 x 10 7, 5.35 x 10 6, & 7.49 x 10 8 FINAL ANSWER: 7.8 x 10 8 Divide 5.65 x 10 7 by 3.9 x FINAL ANSWER: 1.4 x 10 10

Dimensional Analysis How to convert from one set of units to another AKA Factor Label Method

What is a conversion factor? Two numbers that are equivalent but written in different units. Written as a fraction and the same no matter which number is written on top. Example: 365 days = 1 year 365 days OROR 1 year 365 days

Steps of Dimensional Analysis Read the problem. Determine the given. Determine the unknown. Think about the conversions factors that you know to get from the given to the unknown.

Example: How many yards are in 56 inches? Write the given with it’s unit in the first numerator of your “railroad” track. 56 inches

What conversion factors do you know that will get you closer to the unit you want in the final answer? I have 56 inches I want yards HaveWant I Know (conversion factors) I know 12 in = 1 footand 3 feet = 1 yard

Pick a conversion factor that will get you closer to the unit you want in the final answer. This becomes the unit at the top of the new fraction. I know 12 inches = 1 foot 56 inches inches 12 1 foot

Is the unit on TOP the one you WANT to end up with? YES NO then do the math find another conversion factor that gets you closer

My “given” unit cancelled and I’m left with feet. I WANT yards I HAVE feet 56 inches inches12 1 foot I KNOW 3 feet = 1 yard so... 3 feet 1 yard

The last step is to check your cancelled units and do the math! 56 x 1 x 1 ÷ 12 ÷ 3 = 1.6 yards Congratulations! Now you can solve word problems by dimensional analysis! 56 inches inches12 1 foot 1 yard 3 feet

Rules of Dimensional Analysis 1.Write ALL numbers as fractions. 2.Include units with all numbers (No Naked Numbers!). 3.Figure out what conversion factors are needed to get to the wanted units. 4.Arrange conversion factors so the units cancel. 5.Set up the entire problem first, then do the math. 6.Multiply the numbers on top (avoid calculation error-don’t stop in middle of calculation). 7.Divide result of top by result of bottom. Show WORK and DON’T skip steps.

If you are 16 years old, what is your age in seconds? Given: 16 years Wanted: Age in seconds Conversion factors: days/yr, hours/day, minutes/hour, & seconds/minutes 16 years365 days24 hours60 minutes60 seconds 1 year1 day1 hour1 minute = seconds What is that in scientific notation and sig figs? = 5.0 x 10 8 seconds

You want download your music collection to iTunes. You have 225 CDs. If each CD has 12 songs and it takes 85 seconds to download 1 song, how many hours will it take to download your entire collection? Given: 225 CDs Wanted: Hours to Download Conversion factors: songs/CD, sec/song, sec/min, min/hour =63.75 hours 225 CDs12 songs85 seconds1 minute1 hour 1 CD1 song60 seconds60 minutes

The density of copper is 8.96 g/mL. What is its density in kg/m 3 ? 8.96 g1 kg1 mL1,000,000 cm 3 1 mL1000 g1 cm 3 1 m 3 =8960 kg/m 3 or 8.96 x 10 3 kg/m 3