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Integers

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Integers:

Corresponding to natural numbers 1, 2, 3, 4, 5,.. etc  we create new numbers —1, —2,  —3,  —4,  —5, … etc  called minus one, minus two, minus three, minus four, minus live, etc  respectively such that
1 + (—1) = 0
3 + (—3) = 0
4 +(—4) = 0 and soon.
—1 is also called negative of one and —1 and 1 are called the opposites of each other.
Similarly, —2 is also called negative of 2 and —2 and 2 are called the opposites of each other.
Also, —3 is called the negative of 3 and —3 and 3 are called the opposites of each other and soon.
Combining these new numbers with whole numbers, we obtain a new collection of numbers which is written as —3, —2, —1, 0, 1, 2, 3
These numbers are called integers.

Integers are “whole” numbers that have no fractional part. They can be positive or negative.

The set of all integers is denoted by Z
Positive Integers:
The numbers 1, 2, 3, 4, 5, 6…. i.e., natural numbers are called positive integers.
Negative Integers:
The numbers -4, -2, —3, —4, —5, —6 are called negative integers.

integers representation
The number 0 is simply an integer. It is neither positive nor negative
Positive integers are also written as +1, +2, +3… However, the plus sign (+) is usually omitted and understood.

Remark: We use the symbol ‘—‘ to denote the negative integers and the same symbol is used to indicate subtraction. But, the context will always make it clear whether we mean negative integer or subtraction.

Ordering Of Integers:

We know that 7 > 4 and from the number line shown above, we observe that 7 is to the right of 4.
Similarly, 4 > 0 and 4 is to the right of 0. Now, since 0 is to the right of –3 so, 0 > – 3. Again, – 3 is to the right of – 8 so, – 3 > – 8.
Thus, we see that on a number line the number increases as we move to the right and decreases as we move to the left.
Therefore, – 3 < – 2, – 2 < – 1, – 1 < 0, 0 < 1, 1 < 2, 2 < 3 so on.

integers
Hence, the collection of integers can be written as…, –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5…
Thus we arrive at the following fact:
Negative of a negative integer is a positive integer i.e., -(-a) = a for any integer a.
Clearly, 0 is the image of itself. Therefore, negative of zero is zero itself.

Absolute value of an Integer:

The absolute value of an integer is the numerical value of the integer regardless of its sign.
The absolute value of -7, written as |-7|, is 7.
The absolute value of -9, written as |-9|, is 9.
The absolute value of 8, written as |8|, is 8.
The absolute value of 0, written as |0|, is 0.

Note: The absolute value of an integer is always positive.

For any integer a, we have
|a| = {a, if a is positive or zero
= {-a, if a is negative

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