Number Systems

The Real Number Systems

Objectives:• To determine what qualifies something as a

mathematical object.• To develop some of the basic properties of

numbers and number systems

Are phone numbers considered mathematical objects?

Number System

• A set of objects used together with operations that satisfy some predetermined properties.

Objects: 0, 1, 2, -4, 0.6, ⅔, IVOperations: +, - , , ÷Properties: = , > , < , ≠

Before we invented calculators and electricity, how did the earliest cultures use math?

Number Symbols

Natural Numbers

Are also called the counting numbers because they are the numbers we use to count with.

N: 1, 2, 3, 4, 5,…

These numbers were originally used the keep track of the number of animals, as rankings (1st place, 2nd place, 3rd place, etc.), or creating calendars.

What kinds of numbers are missing from this set?

Whole NumbersThis set of numbers is the same as the set of natural (counting) numbers except for the symbol to represent nothing (0).

W: 0, 1, 2, 3, 4, 5…

Besides representing nothing, what other important role does the symbol 0 have in our current number system?

Limitations of Natural and Whole #s1) If we add together any two whole numbers, is the result always a whole number?

2) If we subtract any two whole numbers, is the result always a whole number?

3) If we multiply any two whole numbers, is the result always a whole number?

4) If we divide any two whole numbers, is the result always a whole number?

IntegersThis set of numbers includes the set of whole numbers and the negative integers, which are values below zero.

Z: … -3, -2, -1, 0, 1, 2, 3,…

Why do you think we need negative numbers? Where do we see them in our daily lives?

Limitations of the Integers1) If we add together any two integers, is the result always a integer?

2) If we subtract any two integers, is the result always a integer?

3) If we multiply any two integers, is the result always a integer?

4) If we divide any two integers, is the result always a integers?

… - 3, - 2, - 1 0 1, 2, 3…

Integers

Whole Numbers

Natural Numbers

Use a calculator to evaluate each of the fractions below. Write down the first 6 decimal places. If it appears to be a repeating decimal write it with the bar symbol (i.e. )

Undefined

Fractions are written in the form where a is called the numerator and b is called the denominator. The denominator tells us how many equal parts the whole is broken into and the numerator tells us how many parts we have.

a

b

Collins Writing (Type I -1 point):

Most algebra textbooks define rational numbers as the quotient of two integers. Using your knowledge of rational numbers and these terms, to write a definition for rational numbers in your own words.

All rational numbers can be written as a fraction or a decimal. However, only repeating or terminating (ending) decimals can be written as a fraction (i.e. as a rational number). We will learn about decimals that do not have these characteristics in the next section.

Irrational numbers cannot be written as a fraction. This set of numbers include non-repeating and non-terminating decimals.