Here are some examples of irrational numbers: √2, √3, π What properties do these numbers share that make them irrational?

1. Number Systems
The Real Number Systems

2. Objectives:
• To determine what qualifies something as a
mathematical object.
• To develop some of the basic properties of
numbers and number systems

3. Are phone numbers considered
mathematical objects?

4. Number System
• A set of objects used together with operations
that satisfy some predetermined properties.
Objects: 0, 1, 2, -4, 0.6, ⅔, IV
Operations: +, - ,  , ÷
Properties: = , > , < , ≠

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

6. Number Symbols

7. 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?

8. Whole Numbers
This 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?

9. Limitations of Natural and Whole #s
1) 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?

10. Integers
This 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?

11. Limitations of the Integers
1) 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?

12. … - 3, - 2, - 1 0 1, 2, 3…
Integers
Natural Numbers
Whole Numbers

13. 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. )

14. 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.

15. 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.

16. 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.

17. Irrational numberscannot be written as a
fraction. This set of numbers include non-
repeating and non-terminating decimals.