Unformatted text preview:

Introduction to IntegersRepresenting IntegersModeling Integer OperationsInteger AdditionInteger SubtractionInteger MultiplicationInteger DivisionConclusionIntroduction to Integers Representing Integers Modeling Integer Operations ConclusionMATH 113Section 5.1: IntegersProf. Jonathan DuncanWalla Walla UniversityWinter Quarter, 2008Introduction to Integers Representing Integers Modeling Integer Operations ConclusionOutline1Introduction to Integers2Representing Integers3Modeling Integer OperationsInteger AdditionInteger SubtractionInteger MultiplicationInteger Division4ConclusionIntroduction to Integers Representing Integers Modeling Integer Operations ConclusionKnown Number SetsLast quarter we spent much of our time with two simple sets ofnumbers.Natural NumbersThe natural numbers are the counting numbers as shown below.N = {1, 2, 3, 4, . . .}Whole NumbersThe whole numbers add zero to the natural numbers.W = {0, 1, 2, 3, 4, . . .}During the first few weeks of this quarter, we will extend these setsof numbers to include integer, rational, and real numbers.Introduction to Integers Representing Integers Modeling Integer Operations ConclusionThe IntegersThe first new set of numbers we will consider is the integers.IntegersThe integers include positive and negative natural numbers andzero.Z = {. . ., −2, −1, 0, 1, 2, . . .}We can use a Venn Diagram to show the relationship between nat-ural, whole, and integer numbers.ZNWIntroduction to Integers Representing Integers Modeling Integer Operations ConclusionWhy Use IntegersIs the concept of a negative number really that natural or useful?ExampleSuppose I owe you $2. One might say that I have −$2. Is thisnatural?ExampleLocated in California, Death Valley is 282 feet below sea level. Itselevation is -282 feet.ExampleAs a cold front moves in, the temperature drops by 2 degrees perhour. The rate of change in temperature is −2◦/hour.Introduction to Integers Representing Integers Modeling Integer Operations ConclusionThe Set ModelWhen we started working with natural and whole numbers, therewere several different models which were useful for representing theconcept of a number.The Set ModelIn the set model, a natural number is represented as a set ofobjects. For example, we might represent the number 3 as a pile of3 cookies.ExampleDescribe how one could expand the set model to represent allintegers, including negative numbers. Draw a picture to representthe integers 0 and -2.Introduction to Integers Representing Integers Modeling Integer Operations ConclusionThe Number Line ModelAnother method which we used to represent natural numbers is theconcept of a number line.Number LinesWith whole numbers, a number line starts at 0 and extendsinfinitely to the right. The natural numbers appear at set distancesalong that number line as it moves to the right.ExampleDescribe and draw a picture of how one could expand the numberline described above to represent all integers. Use a number line torepresent -2.Introduction to Integers Representing Integers Modeling Integer Operations ConclusionInteger AdditionUsing the Set ModelWhile your book divides integer addition into four cases (+/+,+/−, −/+, and −/−), we will examine them all together usingthe set model.ExampleUse the idea of “cookies” and “anti-cookies” to model thefollowing addition problems.13 + 523 + (−2)3−7 + 44−3 + (−4)What are some of the advantages and disadvantages of the setmodel for integer addition?Introduction to Integers Representing Integers Modeling Integer Operations ConclusionInteger AdditionUsing the Number Line ModelIf the concept of an “anti-cookie” seems unnatural to you, thenumber line model for integer addition may make more sense.ExampleUse a number line to model the following addition problems.13 + 523 + (−2)3−7 + 44−3 + (−4)What are some of the advantages and disadvantages of thenumber line model for integer addition?Introduction to Integers Representing Integers Modeling Integer Operations ConclusionInteger AdditionResults of Integer AdditionWhile we did not need to consider the four cases mentionedpreviously to model integer addition, it is interesting to considerthe results of each of these types of integer addition problems.Results of Integer AdditionSuppose that a and b are both integers with a < b. Then there arefour possibilities for the sum a + b depending on the sign of a and bCase a b a + bI + + larger than a or b and +II + − smaller than a but +III − + larger than a and −IV − − smaller than a or b and −ExampleWhat happens in each of the cases above if a = b?Introduction to Integers Representing Integers Modeling Integer Operations ConclusionInteger AdditionAbsolute ValueThe number line model and the cases seen above tie in nicely withthe concept of the absolute value of an integer.Absolute ValueThe absolute value of an integer a, written |a|, is the distance fromzero to the number a on a number line.ExampleUse a number line to find |5| and | − 5|.Is there a quicker way to find an absolute value than drawing out anumber line and finding distances?Introduction to Integers Representing Integers Modeling Integer Operations ConclusionInteger SubtractionUsing the Set ModelSince subtraction is the opposite of addition, it makes sense thatwe should be able to use similar methods to model integersubtraction.ExampleUse the idea of cookies and “anti-cookies” to model the followingsubtraction problems.14 - 222 - 437 - (-2)4-3 - 2Do you think that this model works as well for subtraction as it didfor addition? Explain.Introduction to Integers Representing Integers Modeling Integer Operations ConclusionInteger SubtractionUsing the Number Line ModelThe number line model can also be adapted to subtraction simplyby remembering that if we subtract a number, we move left on thenumber line.ExampleUse a number line to model the following subtraction problems.14 - 222 - 437 - (-2)4-3 - 2Do you think that the number line model or the set model is easierto understand when subtracting? Explain.Introduction to Integers Representing Integers Modeling Integer Operations ConclusionInteger SubtractionA General Rule ExplainedModels such as the two we have just seen can be useful forexplaining why a general rule works. Consider the following rule ofaddition and subtraction.Equivalence of Addition and SubtractionIf a and b are any integers, then:a − (−b) = a + band alsoa − b = a + (−b)ExampleUse a number line to justify the two


View Full Document

WWU MATH 113 - Section 5.1: Integers

Download Section 5.1: Integers
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Section 5.1: Integers and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Section 5.1: Integers 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?