1. Introduction to Real Numbers and Algebraic Expressions
Chapter 1
Introduction to Real Numbers and Algebraic Expressions

2. 1.2
THE REAL NUMBERS
a. State the integer that corresponds to a real-world situation.
b. Graph rational numbers on the number line.
c. Convert from fraction notation to decimal notation for a rational number.
Determine which of two real numbers is greater or smaller and know how to write it.
Determine whether an inequality like 3  5 is true or false.
e. Find the absolute value of a number.

3. Natural Numbers Whole Numbers Integers
The set of natural numbers N = {1, 2, 3, …}. These are the numbers used for counting.
Whole Numbers
The set of whole numbers W = {0, 1, 2, 3, …} This is the set of natural numbers with 0 included.
Integers
The set of integers
I = {…, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, …}.

4. Integers Integers consist of the whole numbers and their opposites.
0, neither positive nor negative
Positive integers
Negative integers
Opposites
Integers to the left of zero on the number line are called negative integers and those to the right of zero are called positive integers.
Zero is neither positive nor negative and serves as its own opposite.

5. Tell which integer corresponds to each situation.
Death Valley is 282 feet below sea level.
282 below sea level corresponds to …
Margaret owes $312 on her credit card. She has $520 in her checking account.
The integers … and … correspond to the situation.

6. Tell which integer corresponds to each situation. 1
Tell which integer corresponds to each situation. 1. Death Valley is 282 feet below sea level. 282 below sea level corresponds to  Margaret owes $312 on her credit card. She has $520 in her checking account. The integers 312 and 520 correspond to the situation.

7. Fractions such as ½ are not integers.
A larger system called rational numbers contains integers and fractions. The rational numbers consist of quotients of integers with nonzero divisors.
The following are rational numbers:

8. Rational Numbers The set of rational numbers is the set of numbers ,
where a and b are integers and b is not equal to 0 (b  0).

9. To graph a number means to find and mark its point on the number line.
Graph: Graph: 2.8

10. Solution: The number can be named , or 3. 5
Solution: The number can be named , or 3.5. Its graph is halfway between 3 and 4. The number -2.8 will be closer to -3.

11. Find decimal notation for

12. Find decimal notation for Solution Because means 7  40, we divide:
We are finished when the remainder is 0.

13. Find decimal notation for Solution Divide 1  12
Since 4 keeps reappearing as a remainder, the digits repeat and will continue to do so; therefore,

14. Decimal notation for rational numbers either terminates or repeats.
Decimal notation for irrational numbers neither terminates nor repeats.
Examples of Irrational numbers:

15. Negative integers: -1, -2, -3, …
Real Numbers
Positive Integers: 1, 2, 3, …
Integers
Zero: 0
Rational numbers
Negative integers: -1, -2, -3, …
Real numbers
Rational numbers that are not integers: 2/3, -4/5, 19/-5, -7/8, 8.2,
Irrational numbers: pi, square roots, …
The set of real numbers = all numbers corresponding to points on the number line.

16. The symbol < means “is less than,”
Numbers are written in order on the number line, increasing as we move to the right For any two numbers on the line, the one to the left is less than the one to the right.
For any two numbers on the line, the one to the right is greater than the one to the right.
The symbol < means “is less than,”
4 < 8 is read “ 4 is less than 8.”
The symbol > means “is greater than,”
6 > 9 is read “6 is greater than 9.” 10 -9 -7 -5 -3 -1 1 3 5 7 9
-10 -8 -4 4 8 -2 6 -6 2

17. Use either < or > for to form a true sentence. 1. 7 3 2. 8 3 3
1.  3
3.  9

18. Use either < or > for to form a true sentence. 1. 7 3 2. 8 3 3
1.  3
3.  9
Since 7 is to the left of 3,
we have 7 < 3.
Since 21 is to the left of 9, we have  21 < 9.
8 is to the right of 3, so > 3.
7 < 3.
8 > 3.
 21 < 9

19. Use either < or > for to form a true sentence. 1. 7.2 2. 10 -9 -7 -5 -3 -1 1 3 5 7 9
-10 -8 -4 4 8 -2 6 -6 2

20. Use either < or > for to form a true sentence. 1. 7. 2 2
Use either < or > for to form a true sentence. 1.  Solution 1.  For graphing convert to decimal notation: 10 -9 -7 -5 -3 -1 1 3 5 7 9
-10 -8 -4 4 8 -2 6 -6 2
<
>

21. Order; >, < a < b also has the meaning b > a.
Write another inequality with the same meaning.
a. 4 > 10
b. c < 7

22. Write another inequality with the same meaning. a. 4 > 10 b
Write another inequality with the same meaning. a. 4 > 10 b. c < 7 Solution a. The inequality 10 < 4 has the same meaning. b. The inequality 7 > c has the same meaning.

23. If b is a positive real number, then b > 0.
Positive numbers b
Negative numbers a
a < 0
b > 0
If b is a positive real number, then b > 0.
If a is a negative real number, then a < 0.

24. 5 units from 0
5 units from 0
|5| =
|5| =
|0| =
Absolute Value
The absolute value of a number is its distance from zero on a number line. We use the symbol |x| to represent the absolute value of a number x.

25. Find the absolute value of each number. a. |8| b. |3. 6| c. |0| d

26. Find the absolute value of each number. a. |8| b. |3. 6| c. |0| d
Find the absolute value of each number. a. |8| b. |3.6| c. |0| d. |52| Solution a. |8| The distance of 8 from 0 is 8, so | 8| = 8. b. |3.6| The distance of 3.6 from 0 is 3.6, so |3.6| = 3.6. c. |0| The distance of 0 from 0 is 0, so |0| = 0. d. |52| The distance of 52 from 0 is 52, so |52| = 52.

27. Finding Absolute Value
If a number is negative, its absolute value is positive.
If a number is positive or zero, its absolute value
is the same as the number.