Chapter 8 Integers

8.1 Addition and Subtraction Definition: The set of integers is the set The numbers 1, 2, 3, … are called positive integers and the numbers -1, -2, -3, … are called negative integers. Zero is neither a positive or a negative integer.

Set Model In a set model, two different colored chips can be used, one color for positive numbers and another color for negative numbers. --3 +5

Using Chips One black chip represents a credit of one and one red chip represents a debit of 1. One chip of each color cancel each other out making 0.

All three examples represent +3. Using Chips Each integer has infinitely many representations using chips. +3 +3 +3 All three examples represent +3.

Number Line Representation The integers are equally spaced and arranged symmetrically about 0. Due to this symmetry, we have the concept of “opposite.” -5 -4 -3 -2 -1 1 2 3 4 5

Opposite Set Model: Measurement Model: -3 Opposites Opposites +3 -2 -1 1 2 Opposites Opposites +3

Addition of Integers Definition: Let a and b be any integers. If a and b are positive, they are added as whole numbers. If a and b are positive (thus –a and –b are negative), then where a+b is the whole number sum of a and b.

Addition of Integers Continued Adding a positive and a negative a. If a and b are positive and then where a – b is the whole number difference of a and b. b. If a and b are positive and then where b – a is the whole number difference of a and b.

Addition using the Set Model Example: Example: -3 +4 -3 -4 +1 -7

Properties Closure Property for Integer Addition. Commutative Property for Integer Addition Associative Property for Integer Addition Identity Property for Integer Addition Additive Inverse Property for Integer Addition

Additive Cancellation for Integers Theorem: Let a, b, and c be any integers. If then Proof: Let Then Addition Associativity Additive Inverse Additive Identity

Theorem: Let a be any integer. Then

Subtraction Pattern: The first column remains 4. The second column decreases by 1 each time. The column after the = increases by 1 each time.

Subtraction Take-Away: Take Away 3 Take Away 4 M Leaves 2 Leaves –1

Subtraction Adding the Opposite Let a and b be any integers. Then Missing-Addend Approach Let a, b, and c be any integers. Then if and only if

8.2 Multiplication, Division, and Order Positive Times a Negative The first column remains 3. The second column decreases by 1 each time. The column after the = decreases by by 3 each time.

Negative Times a Positive The first column remains –3 . The second column decreases by 1 each time. The column after the = increases by by 3 each time.

Chip Model Positive Times a Negative Combine 4 groups of 3 red chips

Negative Times a Positive Chip Model Negative Times a Positive Take away 4 groups of 3 black chips. Take away 4 groups of 3 blacks. Insert 12 chips of each color. Leaves 12 reds. --12

Multiplication of Integers Definition: Let a and b be any integers. If a and b are positive, they are multiplied as whole numbers. If a and b are positive (thus–b is negative), then where is the whole number product of a and b.

Multiplication of Integers Continued Multiplying two negatives a. If a and b are positive then where is the whole number product of a and b.

Properties Closure Property for Integer Multiplication Commutative Property for Integer Multiplication Associative Property for Integer Multiplication Identity Property for Integer Multiplication Distributive Property of Multiplication over Addition

Some Theorems Theorem: Let a be any integer. Then Theorem: Let a and b be any integers. Then

Two More Properties Multiplicative Cancellation Property Let a, b, c be any integers with If then Zero Divisors Property Let a and b be integers. Then if and only if or a and b both equal zero.

Division Definition: Let a and b be any integers, where Then if and only if for a unique integer c.

Negative Exponents Definition: Negative Integer Exponent Let a be any nonzero number and n be a positive integer. Then

Scientific Notation A number is said to be in scientific notation when expressed in the form where and n is any integer. The number a is called the mantissa and the exponent n is the characteristic.

Ordering Integers Less Than: Number Line Approach The integer a is less than the integer b, written if a is to the left of b on the integer number line. -5 -4 -3 -2 -1 1 2 3 4 5 Since –2 is to the left of 4 on the number line, --2 is less than 4.

Ordering Integers Less Than: Addition Approach Since –2 +6=4, The integer a is less than the integer b, written if and only if there is a positive integer p such that Since –2 +6=4,

Properties of Ordering Integers Let a, b and c be any integers, p a positive integer and n a negative integer. Transitive Property for Less Than Property of Less than and Addition Property of Less Than and Multiplication by a Positive. Property of Less Than and Multiplication