Lesson 5.11 Scientific Notation

SCIENTIFIC NOTATION A QUICK WAY TO WRITE REALLY, REALLY BIG OR REALLY, REALLY SMALL NUMBERS.

How wide is our universe? 210,000,000,000,000,000,000,000 miles (22 zeros) This number is written in decimal notation. When numbers get this large, it is easier to write them in scientific notation.

Scientific Notation A number is expressed in scientific notation when it is in the form a x 10 n where a is between 1 and 10 and n is an integer (negative and positive numbers)

Rules for Scientific Notation 23 X 10 5 is not in proper scientific notation. Why?

= x 10 5 Standard Form Scientific Notation

Changing Standard Notation to Scientific Notation: See if the original number is greater than or less than one. – If the number is greater than one, the exponent will be positive = x 10 5 – If the number is less than one, the exponent will be negative = 6.72 x 10 -8

Write the width of the universe in scientific notation. 210,000,000,000,000,000,000,000 miles Where is the decimal point now? After the last zero. Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1

2. 10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 10 23

Write in scientific notation x x x x 10 5

In the United States, 15,000,000 households use private wells for their water supply. Write this number in scientific notation. 1.5 X 10 7

A ribosome, another part of a cell, is about of a meter in diameter. Write the length in scientific notation. 3 X 10 -9

Express 1.8 x in decimal notation. Express 4.58 x 10 6 in decimal notation. Changing Scientific Notation to Standard Notation: ,580,000

Try changing these numbers from Scientific Notation to Standard Notation: 1)9.678 x ) x ) x ) x

The U.S. has a total of X 10 7 acres of land reserved for state parks. Write this in standard form. 12,916,000 acres

The nucleus of a human cell is about 7 X meters in diameter. What is the length in standard notation?

Write in PROPER scientific notation. (Notice the number is not between 1 and 10) x x x x 10 2

YOU TRY: Write in PROPER scientific notation. (Notice the number is not between 1 and 10) Problem 1: x Problem 2: x 10 7

Adding/Subtracting when Exponents are The SAME General Formulas (N X 10 x ) + (M X 10 x ) = (N + M) X 10 x (N X 10 y ) - (M X 10 y ) = (N-M) X 10 y

Example 1 Given: 9.49 X 10 5 – X 10 5 Subtract: 9.49 – = Answer: X 10 5

Example 2 Given: 2.56 X X 10 3 Add: = Answer: X 10 3

YOU TRY (3.45 x 10 3 ) + (6.11 x 10 3 ) (8.96 x 10 7 ) – (3.41 x 10 7 )

Adding/Subtracting when the Exponents are Different

When adding or subtracting numbers in scientific notation, the exponents must be the same. If they are different, you must move the decimal either right or left so that they will have the same exponent.

Moving the Decimal For each move of the decimal to the right you have to add -1 to the exponent. For each move of the decimal to the left you have to add +1 to the exponent. It does not matter which number you decide to move the decimal on, but remember that in the end both numbers have to have the same exponent on the 10.

Example 1 Given: 2.46 X X 10 3 Shift decimal 3 places to the left for Move: X Add: 2.46 X X 10 6 Answer: X 10 6

Example 2 Given: X 10 3 – 2.65 X Shift decimal 4 places to the right for Move: X 10 (-1+4) Subtract: X X 10 3 Answer: X 10 3

Example 3 (4.12 x 10 6 ) + (3.94 x 10 4 ) (412 x 10 4 ) + (3.94 x 10 4 ) = x 10 4 Express in proper form: 4.15 x 10 6

Example 4 (4.23 x 10 3 ) – (9.56 x 10 2 ) (42.3 x 10 2 ) – (9.56 x 10 2 ) 42.3 – 9.56 = x 10 2 Express in proper form: 3.27 x 10 3

Multiplying with Scientific Notations (N x 10 x )(M x 10 y ) = (N)(M) x 10 x+y First multiply the N and M numbers together and express an answer. Secondly multiply the exponential parts together by adding the exponents together.

Example 1 (2.41 x 10 4 )(3.09 x 10 2 ) 2.41 x 3.09 = = x 10 6

Write (2.8 x 10 3 )(5.1 x ) in scientific notation x x x x 10 11

Dividing with Scientific Notations (N x 10 x )/(M x 10 y ) = (N/M) x 10 x-y First divide the N number by the M number and express as an answer. Secondly divide the exponential parts by subtracting the exponent from the exponent in the upper number.

Example 1

Example 2: Evaluate 7.2 x x 10 2 : The answer in scientific notation is 6 x The answer in decimal notation is

Example 3 Evaluate 4.5 x x x 10 -3