Significant Figures and Scientific Notation

Significant Figures ► When using our calculators we must determine the correct answer; our calculators are mindless drones and don’t know the correct answer. ► There are 2 different types of numbers –Exact –Measured ► Measured number = they are measured with a measuring device so these numbers have ERROR. ► When you use your calculator your answer can only be as accurate as your worst measurement Chapter Two 2

3 Exact Numbers An exact number is obtained when you count objects or use a defined relationship. Counting objects are always exact 2 soccer balls 4 pizzas Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm For instance is 1 foot = inches? No 1 ft is EXACTLY 12 inches.

4 Learning Check Classify each of the following as an exact or a measured number. 1 yard = 3 feet The diameter of a red blood cell is 6 x cm. There are 6 hats on the shelf. Gold melts at 1064°C.

5 Classify each of the following as an exact (1) or a measured(2) number. This is a defined relationship. A measuring tool is used to determine length. The number of hats is obtained by counting. A measuring tool is required. Solution

Measurement and Significant Figures ► Every experimental measurement has a degree of uncertainty. ► The volume at right is certain in the 10’s place, 10mL<V<20mL ► The 1’s digit is also certain, 17mL<V<18mL ► A best guess is needed for the tenths place. Chapter Two 6

7 What is the Length? ► We can see the markings between cm ► We must guess between.6 &.7 ► We record 1.67 cm as our measurement

Learning Check What is the length of the wooden stick? A. 4.5 cm B cm C cm

Chapter Two 9 Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

Note the 4 rules When reading a measured value, all nonzero digits should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not. ► RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, g has five significant figures. ► RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, cm has three significant figures, and mL has four. 10 Chapter Two

► RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant m has six significant figures. If the value were known to only four significant figures, we would write m. ► RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point. Chapter Two 11

Practice  All digits count Leading 0’s don’t Trailing 0’s do 0’s count in decimal form 0’s don’t count w/o decimal All digits count 0’s between digits count as well as trailing in decimal form

Examples of Rounding For example you want a 4 Sig Fig number , is dropped, it is <5 8 is dropped, it is >5; Note you must include the 0’s 5 is dropped it is = 5; note you need a 4 Sig Fig ,

Practice Rule #2 Rounding Make the following into a 3 Sig Fig number ,  ,  10 6 Your Final number must be of the same value as the number you started with, 129,000 and not 129

RULE 1. RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers. Chapter Two 15

Chapter Two Multiplication and division  1.54 =  =  =  10 6  =     = 

► RULE 2. In carrying out an addition or subtraction, the answer cannot have more significant digits BEFORE or AFTER the DECIMAL point than either of the original numbers. Chapter Two 17

Addition/Subtraction ‑

__ ___ __ Addition and Subtraction = = = – = Look for the last important digit

Chapter Two Mixed Order of Operation  = ( )  ( ) = = = = =  = =

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

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

Express in scientific notation. Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative x 10 -8

Write in scientific notation. A x B x C x 10 4 D x 10 5

Express 1.8 x in decimal notation Express 4.58 x 10 6 in decimal notation. 4,580,000 On the calculator, scientific notation is done with the button x 10 6 is typed

Use a calculator to evaluate: 4.5 x x Type You must include parentheses if you don’t use those buttons!! (4.5 x 10 -5) (1.6 x 10 -2) Write in scientific notation. 2.8 x 10 -3

Use a calculator to evaluate: 7.2 x x 10 2 On the calculator, the answer is: 6.E -11 The answer in scientific notation is 6.0 x The answer in decimal notation is

Write (2.8 x 10 3 )(5.1 x ) in scientific notation. A x B. 1.4 x C x D x 10 -3

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

Write x 10 5 in scientific notation. A x 10 2 B x 10 3 C x 10 4 D x 10 5 E x 10 6 F x 10 7 G x 10 8