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.

**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 in…

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