NUMBER SYSTEMS Ch 1-1 The Integers Ch 1-2 The Rational Numbers C. N. Colón Algebra St. Barnabas H.S. Bronx, New York

OBJECTIVE To recognize integers and the subsets of integers found in the Set of Real Numbers To define and evaluate absolute value expressions To define and identify a domain To define and evaluate inequalities

What is a Set? A set is a collection of objects called elements. The elements have something in common.

Let’s look at a Set? Living Things Plants Animals Bacteria Mammals Reptiles Fish Whales Humans Horses Barnabites Tree Diagram Ariana Grande Taylor Swift

Venn Diagram for Quadrilaterals Concave Trapezoids Convex Parallelograms Rhombuses Rectangles Squares

Decimals will terminate or repeat Set of Real Numbers Rational Irrational Decimals: non-terminating and non-repeating any number which can be written as a fraction. , 7, -0.4 Integers Ex:Fractions/Decimals …-3, -2, -1, 0, 1, 2, 3... , -0.33, - 2.1 Whole Ex: Negative Integers Decimals will terminate or repeat 0, 1, 2, 3... …-3, -2, -1 Natural Zero 1, 2, 3... (counting numbers)

Real This Venn Diagram displays the Sets of Real Number (Rational, Irrational, Integers, Wholes, and Naturals) Real Rational -2.65 5 Integers -3 -19 Wholes 5 Irrational Naturals 1, 2, 3...

You will learn about imaginary numbers in Algebra II Real Rational -2.65 Integers -3 -19 Wholes Irrational Naturals 1, 2, 3...

The square root of any negative number will yield “No Real Solution” RULE The square root of any negative number will yield “No Real Solution”

Identify all of the sets to which each number belongs. (Real, Rational, Irrational, Integers, Wholes, Naturals) 1) -6 Real, Rational, Integer 2) Real, Rational 3) 14 Real, Rational, Integer, Whole, Natural 4) 6 Real, Irrational

RATIONAL NUMBERS The rational numbers are all numbers that can be expressed in the form of a fraction and the denominator does not equal 0. Remember that a ratio is the comparison of two numbers by division.

RATIONAL NUMBERS The set of rational numbers is everywhere DENSE. This means that if you had any two rational numbers, you would always be able to find an infinite number of rational numbers in between them. You can find the exact number in between any two rational numbers by finding their mean, or average.

What is the exact number that lies in between 24.2 and 20.6? To find the answer just take the average of both numbers which means add them together and then divide the sum by two!

RATIONAL NUMBERS Rational numbers can be expressed as decimals that will either REPEAT or TERMINATE.  = .33 or 2 = 2.75

RATIONAL NUMBERS An integer can become a decimal if you simply place a decimal after it. 8 = 8.0 Any fraction can be become a decimal if you drop and dot, then divide. 7 5 . 2 8 20 20

TYPES OF SETS An infinite set is a set whose elements cannot be counted since there is no end to the set. An example is the set of whole numbers or the set of counting numbers. They keep going on and on… infinitely… Let W = the whole numbers W = {0, 1, 2, 3…} Let C = the counting numbers C = {1, 2, 3…}

TYPES OF SETS A finite set is a set whose elements can be counted. An example is the set of digits. Since we only have the digits 0 to 9, then there are only a specific or finite number of elements in that set. Let D = the set of digits D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

TYPES OF SETS The empty set or null set is a set that has no elements and is written using the symbols { } or ø Let S = engineers in room 204 S = { } since there are none in this room S = ø another way of writing it

Domain: the set of values which may be meaningfully substituted for a given variable. Which set of numbers would best describe the variable? 1) Let x = the number of Barnabites in this room. D: x Naturals “ is a member of ” 2) Let w = your weight D: w Positive Real Numbers 3) Let t = the low temperature in New York today D: t Integer 4) Let n = the lowest recorded temperature in Alaska. D: n Integers

Symbols of Inequality < > < > If two numbers are not equal the relationship between them can be expressed as an inequality. An inequality can be written in several different ways. INEQUALITIES < > < > less than greater than at most at least fewer than more than no more than no less than less than or equal to greater than or equal to Note that the symbols point to the smaller number and open up to the larger number

The hungry alligator always eats the largest number.

Example: How to graph an inequality Graph t < 3 over the given domain. 1) D: Reals -3 -2 -1 0 1 2 3 4 2) D: Integers -3 -2 -1 0 1 2 3 4 3) D: Wholes -3 -2 -1 0 1 2 3 4

ASSIGNMENT p. 9 #4-56 (multiples of 4) + #53 p. 15 #4-64 (mo4)