Scientific Notation

An ordinary quarter contains about 97,700,000,000,000,000,000,000 atoms. The average size of an atom is about centimeters across. The length of these numbers in standard notation makes them awkward to work with. Scientific notation is a shorthand way of writing such numbers. Why do we need scientific notation?

Vocabulary WordDefinition scientific notation A form of writing very large & very small numbers. Written as: Decimal number greater than or equal to 1 and less than 10. Multiplied by power of 10. Ex: 1800 = 1.8 x 10 3

Translating Standard Notation to Scientific Notation  10 To Scientific Notation: 1)Move decimal point so only one non-zero digit is to left of decimal. 2)Set up scientific notation. 3)Exponent = # places you moved decimal point. *If very small #, use – exponent. *If very large #, use + exponent. Write in scientific notation. So written in scientific notation is 7.09  10 –3.

Translating Standard Notation to Scientific Notation Think: The number is very large so make exponent positive. 14,500,000 Think: The decimal needs to move 7 places to get a number between 1 and  10 Set up scientific notation. Check 1.45  10 7 = 1.45  10,000,000 = 14,500,000 Write 14,500,000 in scientific notation. So 14,500,000 written in scientific notation is 1.45  10 7.

Think: The number is very small so make exponent negative Think: The decimal needs to move 4 places to get a number between 1 and  10 Set up scientific notation. Check 8.11  10 = 8.11  = –4 Write in scientific notation. So written in scientific notation is 8.11  10 –4. Translating Standard Notation to Scientific Notation

Think: The number is very large so make exponent positive. 225,000 Think: The decimal needs to move 5 places to get a number between 1 and  10 Set up scientific notation. Check 2.25  10 5 = 2.25  100,000 = 225,000 Write 225,000 in scientific notation. So 225,000 written in scientific notation is 2.25  10 5.

135,  100,000 Think: Move the decimal right 5 places because I am multiplying by a power of 10 and there are 5 zeroes represented by the exponent   = 100,000 Note: There are 5 zeroes, which are represented by the exponent. 5 Translating Scientific Notation to Standard Notation Write the number in standard notation.

To Standard Notation: 1.) Write the decimal. 2.) Move the decimal point the number of places given by exponent. - Exp. = very small #s (< 1) + Exp.= very large #s 2.7  10 –3 10 = – Write the number in standard notation. Translating Scientific Notation to Standard Notation

Positive exponents make the decimal larger because you are multiplying by powers of 10 (move to right). Negative exponents make the decimal smaller because you are dividing by powers of 10 (move to left). REMEMBER: A negative exponent is the same as dividing by that number with a positive exponent. Ex: = A number less than 1 will have a negative exponent when written in scientific notation. Helpful Hint

2.01  ,100 Think: Move the decimal right 4 places as positive exponent = very large #  10 4 Translating Scientific Notation to Standard Notation Write the number in standard notation.

Think: Move the decimal left 5 places as negative exponent = very small #. 1.9  10 –5 1.9  10 –5 Write the number in standard notation. Translating Scientific Notation to Standard Notation

Lesson Quiz Write each number in standard notation   10 – ,000,000 17,  10 –3 5.7  10 7 Write each number in scientific notation. 5. Order the numbers from least to greatest. T 2  10 –4, 9  10 –5, 7  10 –5 7  10 –5, 9  10 –5, 2  10 –4

A pencil is 18.7 cm long. If you were to lay 10,000 pencils end-to-end, how many millimeters long would they be? Write the answer in scientific notation. 187 mm  10,000 1 centimeter = 10 millimeters 18.7 centimeters = 187 millimeters Multiply by 10 to change from cm to mm. 1,870,000 mm Additional Example: Application Find the total length. Multiply.

Think: The decimal needs to move 6 places. In scientific notation the 10,000 pencils would be 1.87  10 6 mm long, laid end-to-end. Additional Example Continued 1.87  10 Set up scientific notation. Think: The decimal needs to move right to change 1.87 to 1,870,000, so the exponent will be positive.

A certain cell has a diameter of approximately 4.11 x meters. A second cell has a diameter of 1.5 x meters. Which cell has a greater diameter? = x x Compare powers of 10. Additional Example: Life Science Application Compare the values between 1 and 10. The first cell has a greater diameter > x > 1.5 x 10 -5