MTH 231 Section 2.3 Addition and Subtraction of Whole Numbers

Overview In this section we introduce the operations of addition and subtraction on the set of whole numbers. Operations that are performed using two objects are called binary operations. We will define addition and subtraction using a variety of conceptual models and manipulative activities.

The Set Model of Addition Let A and B be disjoint sets, with n(A) = a and n(B) = b. Then the sum of a and b is Key question: Why is it necessary for A and B to be disjoint sets?

A Picture

Using Manipulatives Any two sets of objects that are somewhat similar but distinguishable in some way (coins, counters, blocks, tiles, etc.) can be used to represent the two sets under discussion.

The Number-Line Model of Addition The whole numbers can be represented as distances on a number line as shown below: Addition can be visualized as a combination of two distances to get a total distance.

An Example Here is a visualization of 4 + 3:

Properties of Whole-Number Addition 1.The set of whole numbers has the closure property under addition. This means that anytime you add two whole numbers, the resulting sum will be a whole number. 2.The commutative property, a + b = b + a, tells us that changing the order of the whole numbers you are adding does not change the resulting sum (the number-line model shows this property very well).

Addition Is Commutative

Properties (Continued) 3.The associative property, (a + b) + c = a + (b + c) tells us that changing the grouping of the whole numbers you are adding does not change the sum.

Properties (Continued) 4. The additive identity property, a + 0 = a, tells us that there exists a “special” whole number such that adding any other whole number to that “special” number gives you back the number you started with. (Key Question: since Addition is commutative, does it follow that 0 + a = a?)

Subtraction of Whole Numbers One way to introduce subtraction is to define subtraction in terms of a related addition problem. For example, 9 – 4 = 5 because = 9. More formally, we define the difference of a and b, written a – b, as the unique whole number c such that a = b + c. Some terminology: in the expression a – b, a is called the minuend and b is called the subtrahend.

Subtraction: Four Models 1.Take away: David has $7 and spends $5 on a slice of pizza and a drink. How much money does David have left? 2.Missing addend: Maria has fed 5 of her new baby kittens. If there are 7 baby kittens in all, how many more kittens does Maria have to feed? 3.Comparison: Norma has 7 pencils and Andre has 5 pencils. How many more pencils does Norma have than Andre? 4.Number line (measurement): Tony did 7 homework problems. He did five of the problems before dinner. How many problems did he do after dinner? Each of these problems can be modeled using either sets or a number line.

Example 1

Example 2

Example 3

What About The Properties? 1.Is the set of whole numbers closed under subtraction? 2.Is subtraction commutative? 3.Is subtraction associative? 4.Is there a subtractive identity element?