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Negative Numbers and 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 ori…

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