Scientific Notation 2007 Thomas M Kenyon Use with

Scientific Notation © 2007 Thomas M. Kenyon Use with permission: tkenyon@crcs. wnyric. org

Scientists and other people use scientific notation when they are talking about very large or very small numbers. Examples of large numbers: Speed of light = 30, 000 meters per second Distance to the sun = 93 million miles Number of atoms in 12 grams of carbon: 60200000000000 Examples of very small numbers: Charge on an electron = 0. 000000000160 Coulombs mass of a proton = 0. 0000000000000167 kilograms These numbers are very hard to read! Every zero matters – one too many or one too few zeros means you’re off by a factor of 10. If you had a million dollars in the bank, you want to see $1, 000 If a banker leaves off just one zero, that’s $100, 000 dollars, shorting you by $900, 000!!! The number of zeros is very important!

To make it easier to read this large (or small) numbers, scientists and mathematicians use scientific notation. In scientific notation, a number is written in this format: Always one non-zero digit in front of the decimal Decimal The rest of the point digits. For LARGE numbers, the exponent is positive. For SMALL numbers, the exponent is negative.

Practice: Find the value of each of the following using a calculator: Note: when the exponent on 10 is positive in scientific notation, that means we’re multiplying by a “big” number, i. e. 10, 1000, or even bigger. Thus a positive exponent means we are dealing with a big number.

Practice: Find the value of each of the following using a calculator: Note how your calculator displays numbers in scientific notation. That is the only way it can display this number.

Note: “three ten-thousandths” or 0. 0003 In scientific notation, when the exponent on 10 is a negative number, that means we’re dividing by some power of 10. When we divide by a big number, our result is a little number.

(Review of places) tenths thousandths hundred-thousandths ten-thousandths millionths

Converting to scientific notation Positive Exponents (big numbers): The “unwritten” decimal for this number is here. The decimal belongs there for scientific notation. . To find the exponent, count how many places the decimal was moved over. 8 places.

Negative Exponents (little numbers): The decimal belongs there for scientific notation. . The exponent is negative, because we’re dealing with a small number. To find the exponent, count how many places the decimal was moved over. 10 places.

Converting from Scientific Notation: example: This is a “big” number, because the exponent on 10 is positive. Put some zeros on the end. (don’t worry about having too many zeros) . Now, move the decimal over 5 places. So, this number converts to 392, 000 If you had any extra zeros you can ignore them (since they’re after the decimal) or you can erase them. With practice, you’ll be able to write down the exact number of zeros that you need.

Converting from Scientific Notation: example: This is a “little” number, because the exponent on 10 is negative. Put some zeros in front. (don’t worry about having too many zeros) . Now, move the decimal over 5 places to the left. So, this number converts to 0. 0000853 If you had any extra zeros you can ignore them or you can erase them. With practice, you’ll be able to write down the exact number of zeros that you need.

Doing math with numbers in scientific notation. Mr. Kenyon highly recommends learning to use the EE button on your calculator, because your calculator knows PEMDAS. example: Wrong: 4. 84 x 10^3^4 because the calculator knows PEMDAS Okay: (4. 84 x 10^3)^4 use the parentheses. Better: 4. 84 EE 3 ^ 4 calculator treats 4. 84 E 3 as one number You only put the exponent after hitting the EE button, not 10 Answer: 5. 487587354 E 14 means

Dividing (this is when you need to be most careful) Clean up the top and bottom separately!! Then divide. An example from physics: Force of attraction between the earth and the moon. On your calculator, do the numerator: 6. 67 EE-11*5. 98 EE 24*7. 35 EE 22 Ans: 2. 9316651 E 37 write it down! Always watch out for your calculator to give an answer in scientific notation! Now, on your calculator, do the denominator: 3. 84 EE 8^2 Last, divide: 2. 9316651 EE 37 ÷ 1. 47456 EE 17 Ans: 1. 47456 E 17 = 1. 988162638 E 20