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.
 

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.
 

 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.

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.


 

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