The Basic Concepts of Integers Help

The Basic Concepts of Integers

The set of whole numbers consists of the numbers 0, 1, 2, 3, 4, 5, …. In algebra we extend the set of whole numbers by adding the negative numbers 1, 2, 3, 4, 5, …. The numbers … 5, 4, 3, 2, 1, 0 1, 2, 3, 4, 5, … are called integers . These numbers can be represented on the number line, as shown in Fig. 2-1 . The number zero is called the origin .

Basic Concepts

Fig. 2-1.

Math Note: Any number without a sign (except 0) is considered to be positive; i.e., 6 = +6. The number zero is neither positive nor negative.

Each integer has an opposite . The opposite of a given integer is the corresponding integer, which is exactly the same distance from the origin as the given integer. For example, the opposite of -4 is +4 or 4. The opposite of 0 is 0.

The positive distance any number is from 0 is called the absolute value of the number. The symbol for absolute value is | |. Hence, | 6| = 6 and |+10| = 10. In other words, the absolute value of any number except 0 is positive. The absolute value of 0 is 0, i.e., |0| = 0.

Math Note: Do not confuse the concepts of opposite and absolute value. With the exception of zero, to find the opposite of an integer, change its sign and to find the absolute value of an integer, make it positive.

Examples

Example 1

Find the opposite of 12.

Solution 1

The opposite of 12 is –12 since we change the sign.

Example 2

Find |12|.

Solution 2

|12| = 12 since the absolute value of this number is 12.

Example 3

Find the opposite of -3.

Solution 3

The opposite of –3 is +3 or 3 since we change the sign.

Example 4

Find |–3|.

Solution 4

|–3| = 3 since the absolute value is positive.

Sometimes a negative sign is placed outside a number in parentheses. In this case, it means the opposite of the number inside the parentheses. For example, –(–6) means the opposite of –6, which is 6. Hence, –(–6) = 6. Also, –(+8) means the opposite of 8, which is –8. Hence, –(+8) = –8.

Example 5

Find the value of –(+41).

Solution 5

The opposite of 41 is –41. Hence, –(41) = –41.

Example 6

Find the value of –(–17).

Solution 6

The opposite of –17 is +17 or 17. Hence, –(–17)= 17.

Practice

1. Find the opposite of –16.

2. Find the opposite of 32.

3. Find |–23|.

4. Find |11|.

5. Find the opposite of 0.

6. Find |0|.

7. Find the value of –(–10).

8. Find the value of –(+25).

9. Find –(0)

10. Find the value of –| –6| (Be careful.)

Answers

1. 16

2. –32

3. 23

4. 11

5. 0

6. 0

7. 10

8. –25

9. 0

10. –|–6| = –(+6)= –6

Practice problems for this concept can be found at: Integers Practice Test.

The Basic Concepts of Integers Help