Scientific Notation

Scientists often work with very large or very small numbers. 6,020,000,000,000,000,000,000,000,000,000,000 is a very large number. 0.000000000000000000624 is a very small number. (The more zeros there are after the decimal the smaller the number.) Using zeros to keep track of place value can be tedious with such large numbers. Scientific notation uses powers of ten to make it easier to work with such numbers. Look at each number written in scientific notation. 6,020,000,000,000,000,000,000,000,000,000,000 = 6.02 x 1033 0.000000000000000000624 = 6.24 x 10-19

6,020,000,000,000,000,000,000,000,000,000,000 = 6.02 x 1033 Standard or decimal notation. Scientific notation. Notice how the large number has a positive exponent and the *small number has a negative exponent. *smaller than one 0.000000000000000000624 = 6.24 x 10-19 Standard or decimal notation. Scientific notation.

A number written in scientific notation is written in the following form: n x 10p This exponent is positive if the number is large and negative if the number is smaller than one. This must be a number greater than 1 but less than 10. To get this number you move the decimal to where there is only 1 digit in front of it. To get the exponent you count how many places you had to move the decimal.

Write the number 625,000,000 in scientific notation. Put the decimal here to get a number greater than 1 but less than 10. 6 25000000 . Now count how many spaces you had to move the decimal. 6 . 2 5 0 0 0 0 0 0 The decimal started out here. And I moved it to here. How many places did I move it?

6 . 2 5 0 0 0 0 0 0 How many places did I move it? 8 places Since the decimal was moved 8 places and the number was a large number the exponent will be 8. 625,000,000 = 6.25 x 108

. Write 245,000,000,000 in scientific notation. 2 4 5 0 0 0 0 0 0 0 0 0 . Move the decimal from the end to behind the 2. Now, drop all the zeros. 2.45 So far we have 2.45 x 10p To figure out what the exponent is count how many places the decimal was moved. The decimal was moved 11 places so the exponent is 11. The exponent is positive because we are dealing with a large number. 2.45 x 1011

You try it. Write 714,000 in scientific notation. Did you get 7.14 x 105 ?

Let’s do a small number. Write 0.00000000008834 in scientific notation. 0.00000000008834 Move the decimal to here to get 8.834 Now count how many places the decimal was moved. It was moved 11 places. Since we are dealing with a small number the exponent will be negative. So the exponent is -11. 0.00000000008834 = 8.834 x 10-11

Your turn. Write 0.0000025 in scientific notation. You should have gotten 2.5 x 10-6

Write 2.67 x 106 in standard notation. Now lets work backwards. Instead of writing a number in scientific notation you can take a number that is already in scientific notation and write in standard or decimal notation. Write 2.67 x 106 in standard notation. To convert from scientific notation to standard notation you move the decimal from its current position. You will move the decimal the number of places according to the exponent. Since the exponent here is 6 you will move the decimal 6 places. Since the exponent is positive you will move the decimal to the right. (Remember numbers to the right are positive.) 2. 6 7 _ _ _ _ The decimal will have to go here.

Write 2.67 x 106 in standard notation. Now lets work backwards. Instead of writing a number in scientific notation you can take a number that is already in scientific notation and write in standard or decimal notation. Write 2.67 x 106 in standard notation. To convert from scientific notation to standard notation you move the decimal from its current position. You will move the decimal the number of places according to the exponent. Since the exponent here is 6 you will move the decimal 6 places. Since the exponent is positive you will move the decimal to the right. (Remember numbers to the right are positive.) . 2 6 7 _ _ _ _ Now, fill the gaps in with zeros. The decimal will have to go here.

Write 2.67 x 106 in standard notation. Now lets work backwards. Instead of writing a number in scientific notation you can take a number that is already in scientific notation and write in standard or decimal notation. Write 2.67 x 106 in standard notation. To convert from scientific notation to standard notation you move the decimal from its current position. You will move the decimal the number of places according to the exponent. Since the exponent here is 6 you will move the decimal 6 places. Since the exponent is positive you will move the decimal to the right. (Remember numbers to the right are positive.) . 2 6 7 _ _ _ _ Now, fill the gaps in with zeros. So, 2.67 x 106 = 2, 670,000

Write 7.882 x 10-7 in standard notation. Since the exponent is -7 we have to move the decimal 7 places to the left. (Remember, negative numbers are to the left.) _ _ _ _ _ _ 7 . 8 8 2 Move the decimal from here...

Write 7.882 x 10-7 in standard notation. Since the exponent is -7 we have to move the decimal 7 places to the left. (Remember, negative numbers are to the left.) _ _ _ _ _ _ 7 . 8 8 2 … to here.

. Write 7.882 x 10-7 in standard notation. Since the exponent is -7 we have to move the decimal 7 places to the left. (Remember, negative numbers are to the left.) . _ _ _ _ _ _ 7 8 8 2 … to here.

. Write 7.882 x 10-7 in standard notation. Since the exponent is -7 we have to move the decimal 7 places to the left. (Remember, negative numbers are to the left.) . _ _ _ _ _ _ 7 8 8 2 Now fill in the gaps with zeros. So, 7.882 x 10-7 = 0.0000007822

Your turn. Write 3.567 x 10-4 in standard notation. You should have gotten 0.0003567. If you didn’t get that answer go back and review the process again. Try the next problems: Write 550,000,000 in scientific notation. 5.5 x 108 Write 0.00000113 in scientific notation. 1.13 x 10-6 Write 3.09 x 105 in standard notation. 309,000 Write 9.6 x 10-3 in standard notation. 0.0096

Multiplying Numbers in Scientific Notation You can multiply numbers that are in scientific notation. For example: (2.35 x 105)(1.6 x 103) Since we are allowed to rearrange multiplication we can re-write this as: (2.35 x 1.6)(105 x 103) Remember, when you are multiplying powers of 10 you just add the exponents. 3.76 x 105 + 3 3.76 x 108

Multiply (3.11 x 104)(2.42 x 103) First rearrange: (3.11 x 2.42)(104 x 103) Multiply the decimals: 7.5262 Now multiply the powers of 10. Just add the exponents. 104 + 3 = 107 Now put it all together: 7.5262 x 107

Now how about some division?!? Your turn! Multiply: (1.02 x107 )(6 x 103) The answer is: 6.12 x 1010 multiply add the exponents Now how about some division?!?

Dividing Numbers in Scientific Notation Divide: First break it apart: Now divide the powers of 10. Remember, to divide powers of 10 you subtract the exponents. Divide the decimals: 1.3398285714 105-3 = 102 Now put them back together: 1.3398285714 x 102

Your turn! Divide: Did you get 6.6 x 103 ? If not compare your work to mine. 106-3 = 103 6.6 x 103

BEWARE!!! Sometimes when you multiply or divide numbers in scientific notation you may end up with an answer that appears to be in scientific notation but really isn’t. Check your product or quotient to be sure it is in scientific notation. If it is not then make the adjustment to the decimal needed to put the number in scientific notation. Ex: You multiply two numbers together and get 34.7 x 104 . This is not in scientific notation because 34.7 is not a number greater than 1 and less than 10. Here you have to move the decimal one place to the left so you have 3.47. Since you are moving the decimal you must change the exponent. Since 34.7 is bigger than one the exponent is positive so you add one to the exponent. The new answer is 3.47 x 105

Homework tip: Keep in mind that when you are multiplying or dividing numbers in scientific notation and you add or subtract the exponents you could end up with a negative exponent. It is ok to have a negative exponent. Example: 9.45 x 103 + (-9) 9.45 x 10-6

Homework tip: Keep in mind that when you are multiplying or dividing numbers in scientific notation and you add or subtract the exponents you could end up with a negative exponent. It is ok to have a negative exponent. Example: 4.675 x 104 - 11 4.675 x 10-7