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Integers

Introduction to Integers and their Absolute Value

Natural numbers

The counting numbers 1, 2, 3, ... are called natural numbers. 

The set of natural number is denoted by the letter N.

∴ N = {1, 2, 3, ...}

1 is the smallest natural number. The set of natural numbers, N is an infinite set.

Whole numbers

The numbers 0, 1, 2, 3, ... are called whole numbers.

The set of whole numbers is denoted by the letter W.

∴ W = {0, 1, 2, 3, ...}

0 is the smallest whole number. The set of whole numbers, W is an infinite set.

Integers

We had observed that adding any two whole numbers always gives a whole number. We can examine whether this case is true for the operation ‘subtraction’. Let us consider the following examples:

13 − 12 = 1

13 − 13 = 0

12 − 13 = ?

We can observe that in the last case, the operation ‘subtraction’ cannot be performed in the system of whole numbers i.e., when a bigger whole number is subtracted from a smaller whole number. In order to solve such type of problems, the system of whole numbers has to be enlarged by introducing another kind of numbers called negative integers. These numbers are obtained by putting “−” sign before the counting numbers 1, 2, 3, … That is, negative integers are −1, −2, −3 …

The most common real life example of negative integers is the temperature of our surroundings. In winters, sometimes the temperature drops down to a negative value say −1, −3. So, in such cases negative integers are highly used.

All positive and all negative numbers including zero are called integers (or directed numbers or signed numbers). That is, the numbers …−3, −2, −1, 0, 1, 2, 3… are called integers. The collection or set of all integers is an infinite set and usually it is denoted by I or Z.

Convention: If there is no sign in front of a number, then we treat it as a positive number.

However, the number ‘0’ is taken as neutral i.e., 0 is always written without any sign.

I or Z = {… −3, −2, −1, 0, 1, 2, 3, …}

Absolute value of an integer

The absolute value of an integer is its numerical value regardless of its sign. The absolute value of an integer n is denoted as |n|.

Therefore, |−10| = 10, |−2| = 2, |0| = 0, |7| = 7 etc.

Note: The absolute value of any integer is always non-negative.

Opposite of an integer

Numbers which are represented by points such that they are at equal distances from the origin but on the opposite sides of it are called opposite numbers.

Thus, the opposite of an integer is the integer with its sign reversed. The opposite of integer a is a and the opposite of integer b is +b or b as a and a; b and +b are at equal distance from the origin but on the opposite sides.

Thus, opposite of 5 is 5, opposite of 8 is 8.

Let us discuss some examples based on these concepts. 

Example 1:

Write the absolute value of  4, 19, 23 and −1.

Solution:

The absolute value of 4 = |4| = 4.

The absolute value of −19 = |−19| = 19.

The absolute value of 23 = |23| = 23.

The absolute value of −1 = |−1| = 1.

 

Example 2:

The absolute value of two integers are 11 and 0. What could be the possible value(s) of the those integers?

Solution:

If the absolute value of an integer is 11, then the possible values of that integer could be ±11 i.e., 11 or −11.

If the absolute value of an integer is 0, then the possible value of that integer could be 0.

 

Example 3:

What are the opposite of integers 51, −927 and  −7?

Solution:

The opposite of 51 is −51.

The opposite of −927 is 927.

The opposite of −7 is 7.

A line which is used to represent numbers graphically is called a number line. This line can be of any length and it has both positive and negative numbers along with zero. The numbers on number line are marked off at equal distances from each other. 

However, we do not know where to mark negative and positive numbers on this number line. 

To know where to mark and how to locate integers on a number line, let us learn go through the following video.


We can also find the predecessor and successor of a number using a number line. Let us see how.

  1. To find the predecessor of a number using a number line, we have to move 1 unit to the left of the given number. The result of this activity gives us the predecessor of the number.
  1. To find the successor of a number using a number line, we have to move 1 unit to the right of the given number. The result of this activity gives us the successor of the number.

Let us discuss some examples based on the location of integers on the number line.

Example 1:

The following figure is a horizontal number line representing integers.

Observe the number line and give answers to the following questions.

(a) If M is 4, then which points represent the integers –4, –6, and 7?

(b) Which point on the number line represents neither a negative number nor a positive number?

(c) Write the integers for the points J, D, R, and Q.

(d) Is X a positive or a negative integer?

Solution:

(a) It is given that point M represents the integer 4 i.e., M represents +4. By moving 1 unit to the left of M, we will reach at point J. This point J represents the location of the integer 3. When we keep on moving 3 units to the left, we will be at point O. This point O represents the location of the integer 0.

To locate the integer –4 on the number line, we will move 4 units to the left of O. On doing so, we will reach at point T. Thus, point T represents the location of the integer –4 on the number line. In this way, we will locate the integers –6 and 7 on the given number line. After doing so, we will obtain point G as –6 and K as 7.

Thus, the points T, G, and K represent the location of the integers –4, –5, and 7 respectively on the number line.

(b) We know that 0 can be written as +0 or –0. Therefore, 0 is such a number that isneither negative nor positive. On the given number line, the position of 0 is represented by point O.

Therefore, O is…

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