BIG NUMBERS and SMALL NUMBERS (Scientific Notation)

In science, you sometimes have to deal with very LARGE numbers How far is the sun from the earth?

150,000,000,000 meters

Or there may be times that you’ll use very small numbers… How wide is one atom of gold?

meters

Scientific notation is a system used to avoid dealing with all the ZEROS in very big and very small numbers

In scientific notation, numbers have TWO parts n { a number between 1-10 that may be followed by decimals} X 10 x a power of 10

How do you write numbers in scientific notation?

Problem Write 150,000,000,000 in scientific notation: Step 1 Move the decimal point from its original position until it is behind first nonzero digit (1). This gives you a number between 1 and From here To here The number becomes 1.5

Step 2 Count the number of places that the decimal point moves to the left; this becomes the positive exponent (The decimal pt. moves 11 places; the exponent of 10 must be 11) Answer: 1.5 x 10 11

As you can see, there are two ways to write the SAME number (standard notation) or 1.5 x (scientific notation)

Problem: Write in scientific notation Step 1 Move the decimal point from its original position until it is behind first nonzero digit (2). This gives you a number between 1 and From here To here The number becomes 2.74

Step 2 Count the number of places that the decimal point moves to the right ; this becomes the negative exponent (The decimal pt. moves 10 places; the exponent of 10 must be 10) Answer: 2.74 x

Again, there are obviously two ways to write this small number (standard notation) or 2.74 x (scientific notation)

Remember Big numbers have positive exponents Small numbers have negative exponents

Now let’s practice ! Write in scientific notation. Correct answer is: x 10 7

Let’s try another one… Write in scientific notation. Correct answer is: x 10 -6

You’re on your own… Write these numbers in scientific notation: (1) 6,700 (2) 123,000 (3) (4) 9,362,000 (5)

One more thing Be sure you also know how to change a number in scientific notation back to standard form. For example: Scientific notation: 8.32 x 10 4 How do you write this number in standard form? Answer: Standard form:

How do you change numbers in scientific notation to standard form?

Big numbers For numbers with positive exponents For numbers with positive exponents Move the decimal point from its current position to the right. The number of decimal places moved must be the same as the exponent. Fill the spaces with zeros x 10 5 From here …. move decimal point 5 places to the right (you need to write 3 zeros) Answer:

Shall we give it a try? Change 5.02 x 10 6 to standard form 5.02 x 10 6 Move the decimal point 6 places to the right You will need to write in 4 more zeros Answer: or

Small numbers For numbers with negative exponents Move the decimal point from its current position to the left. The number of decimal places moved must be the same as the exponent. Fill the spaces with zeros x From here …. move decimal point 4 places to the left (you need to write 3 zeros) Answer:

Let’s try this problem… Change 9.12 x to standard form 9.12 x Move the decimal point 3 places to the left You will need to write 2 more zeros Answer: or

Arrange these numbers from the largest to the smallest. 6.5 x x x x x x x 10 6

Multiplying Numbers in Scientific Notation ( 4.2 x 10 3 ) ( 6.01 x 10 4 ) 1) Multiply the first factors : (4.2 x 6.01) 2) Add the powers of 10: ANSWER: 25.2 x 10 7

Dividing Numbers in Scientific Notation (3.0 x 10 5 ) / (6.0 x 10 2 ) 1)Divide the first factors: 3.0 / ) Subtract the powers of – 2 ANSWER: 0.5 x 10 3 = 5.0 x 10 2

Adding and Subtracting Numbers in Scientific Notation If you are adding and subtracting numbers in scientific notation without a calculator: first,adjust the numbers so that the exponents are the same ( 5.4 x 10 3 ) + ( 8.0 x 10 2 ) 1) Adjust second number 8.0 x 10 2 = 0.8 x ) Add the two numbers (5.4 x 10 3 ) + (0.8 x 10 3 ) ANSWER: 6.2 x 10 3