Two Kinds of Numbers (Notes are in red)

There are two types of numbers: 1 There are two types of numbers: 1. Exact numbers Obtained by counting or by definition. There is no uncertainty in them. 2. Inexact numbers Obtained by comparing to a standard unit. They always contain uncertainty.

1. Exact Numbers * Math is based on exact numbers* 1 dozen = 12 1 yard = 3 feet * Math is based on exact numbers*

How many people are in your math class?

Your answer is easy to figure, and there is no uncertainty in it Your answer is easy to figure, and there is no uncertainty in it. You can count the number of people. The answer is a small whole number. The answer is exact.

Counting numbers (examples): The number of home runs hit by a player The number of pages in a newspaper The number of states in the U.S. The number of books on a shelf.

How many weeks are in a year?

This answer is also easy to come up with. It is 52 This answer is also easy to come up with. It is 52. By definition, there are 52 weeks in a year. The number doesn’t change, there is no uncertainty. It is exact. (1 kilogram = 1,000 grams) (1 gallon = 4 quarts) (1 mile = 5,280 feet)

Defined numbers (examples): 12 inches per foot 52 cards in a deck 2 Defined numbers (examples): 12 inches per foot 52 cards in a deck 2.54 centimeters per inch 360 degrees in a circle

Calculating with exact numbers is easy; use the answer from your calculator. How many people would be in 8 families just like yours? How many weeks are there in five years?

2. Inexact Numbers (We always measure to the first degree of uncertainty) 24.5 oC 195 mL The object's length is measured to be 41.64 cm. *Science is based on measurements*

How far is it from San Francisco to New York City?

This answer is not as easy to arrive at.

You can use a rule of thumb that the U. S * You can use a rule of thumb that the U.S. is about 3000 miles wide, east to west. * You can drive the distance and check your odometer. * You can scale off the distance from a map and come up with an estimate.

But these answers will all be different But these answers will all be different. There will be uncertainty in each, and answers are not exact.

Measured quantities (examples): The mass of copper in a coin The size of an atom The pressure of the air in a tire The speed of light per second

What’s the reason for the uncertainty in measurements What’s the reason for the uncertainty in measurements? Brainstorm your ideas, the teacher will list them on the board.

Some reasons for uncertainty…

In general, measuring to the first degree of uncertainty means we always have an uncertain digit in the value of the quantity. (By not using First Degree of Uncertainty we reduce precision by a factor of 10) 14.67 cm Certain digits Uncertain digit

Things that we measure are continuous - like time, temperature, length, mass, and pressure. Their values are in a continuous spectrum that can be expressed to many decimal places. But in practice we can’t measure that precisely.

Measurements are always uncertain, never exact Measurements are always uncertain, never exact. There is no such thing as a perfect measurement because there is no such thing as a perfect measuring tool, nor is there a perfect measurement technique. The act of estimating to the first degree of uncertainty limits the exactness of a measurement.

You will see for yourself, in this class, that different students can measure the same object and come up with slightly different values for the measurement. We can’t know the exact correct value.

Yet the history of science tells us that small differences in measurements can mean the difference between understanding nature and developing a valid theory or not understanding it and creating a poor or no theory. We must make inexact measurements useful by understanding uncertainty and knowing how to use it.

Calculating with measurements presents problems of consistency and precision. Any measurement contains uncertainty. And when you calculate with measurements the uncertainty carries through to the answer.

For example: If you measure one side of a table to be 73 For example: If you measure one side of a table to be 73.88 cm, and the width to be 32.69 cm, your calculator tells you the area is 2415.1372 cm2 width side area

2415. 1372 cm2 That’s a very satisfying number 2415.1372 cm2 That’s a very satisfying number. It seems like we really understand the area. But how much of the number has physical meaning? If we’re estimating the sides to a hundredth of a cm, how can we know the area to a ten thousandth of a cm2?

Significant Figures …provide rules for using measurements so that we can understand uncertainty and use inexact amounts in logical, consistent ways to get meaningful answers. …these rules are never used with exact numbers.