Algebra I.
Lesson 2
Rational Numbers (Grades 9-12)
Instruction 2-1
Integers and the Number Line | Adding and Subtracting Integers | Inequalities and the Number Line | Comparing and Ordering Rational Numbers | Dividing Rational Numbers | Summary
| Integers and the Number Line |
| CCSTD HS Grades Algebra 13.0 |
We know the set of natural numbers or counting numbers. It
is the set of numbers 1, 2, 3, 4, …, . We use these numbers to count objects or
express the quantities.
Let look at the example below. In Table 2.1, some amounts of money shown. These
are the amounts John earned during six days.
| Date | Amount of Money and Its Source |
|---|---|
| 4-1-2003 | $120 from saving account |
| 4-2-2003 | $230 from selling items |
| 4-3-2003 | $760 from tax return |
| 4-4-2003 | $120 from selling items |
| 4-5-2003 | $240 refund from returned items |
| 4-6-2003 | $82 from selling items |
Table 2.1
All the amounts in Table 2.1 are gained. Nothing is recorded as losses. You
can make these records simpler. Place them in a table such as Table 2.2.
Both tables show the same information. But the second one is much simpler.
| Date | Amount of Money Received |
|---|---|
| 4-1-2003 | $120 |
| 4-2-2003 | $230 |
| 4-3-2003 | $760 |
| 4-4-2003 | $120 |
| 4-5-2003 | $240 |
| 4-6-2003 | $82 |
Table 2.2
As you see, we used simply whole numbers in this case. There are some cases
in which one does not gain money only. He both gains and loses. For example,
look at Table 2.3. This table shows the operations on a checking account
over six days.
| Date | Operation |
|---|---|
| 5-1-2003 | $120 Withdrawal |
| 5-2-2003 | $230 Direct Deposit |
| 5-3-2003 | $760 Direct Deposit |
| 5-4-2003 | $120 Electronic Debit |
| 5-5-2003 | $240 Electronic Deposit |
| 5-6-2003 | $82 Overdraft Charge |
Table 2.3
We can not use only the whole numbers to represent the data in Table 2.3. We
must use another set of numbers. They are called integers.
What Are Integers?
Let recall natural numbers. They are 0, 1, 2, 3, … In this set, 0 is less
than 1, 1 is less than 2, 2 is less than 3, and so forth. So, 0 is the
smallest number in this set. Is there any number less than zero? Yes.
There are many numbers less than 0. These numbers called negative numbers.
Such numbers are used to represent quantities such as losses, temperatures
below zero, debts, withdrawals, or altitudes below the sea level.
If we show $120 deposit by +$120, we can show $240 withdrawal by
–$240. If the depth of a valley is 32 feet below the sea level, then we can
show its altitude by –32 ft. Negative numbers along with natural numbers
form the set of integers.
Practice 1. Use labels counting number, whole number, integer
for each number.
(a) –32
(b) 45
(c) –76
(d) 92
Answer.
Practice 2. Which numbers are integers?
–3, 4.5, 32, –23, 65, 15, –43.6
Answer.
Real Life Application 1. Show each measure by an integer.
(a) altitude of a building 120 feet above the sea level
(b) altitude of a submarine 860 feet below the sea level
(c) 12˚ F below zero
(d) altitude of the bottom of a valley whose depth is 110 feet below the sea
level
(e) 32˚ F above zero
(f) $328 withdrawal from a saving account
Answer
Number Line
To understand the integers, it is a good idea to use a diagram. This is called a
number line. To graph a number line, first draw a ray (Figure 2.4).

Figure 2.4
Find the midpoint of this ray. Label it as 0. Choose a length on the ray as one
unit. Start from 0 and mark the ray on both directions. The distance between
each two next by marks is one unit (Figure 2.5).

Figure 2.5
Label the marks from 0 to the right as 1, 2, 3, …. Label the marks from 0 to the
left as –1, –2, –3, … (Figure 2.6). This is a display of integers on a line. It
is called a number line.
We would use this line often in problems and other topics. Each number is called
the coordinate of the point. Zero is neither positive nor negative. It is used
as a reference for splitting numbers into two signed parts.

Figure 2.6
A number written using one of the signs – or + called a signed number.
Sometimes, we drop the + signs of positive numbers. So, any whole number is also
a positive numbers.
Practice 3. Graph the members of each set on a number line.
A = {–4, –3, 0, 2, 5}
B = {–6, –3, 1, 2, 4}
C = {–5, –1, 1, 3, 5}
Solution.

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Real Life Application 2. Write integers that can be used to the real life
situations below.
(a) a temperature of 15˚ F below zero
(b) a deposit of $340
(c) a withdrawal of $120
(d) a gain of $320
(e) a debt of $86
(f) 120 feet above the sea level
(g) 82 feet below the sea level
Answer
Links for Students, Parents and Teachers
Now let's do Practice Exercise 2-1 (top).
Adding and Subtracting Integers (top)Next Page: