Introduction to Integers
- There are so many situations where we have to use negative numbers.
- Negative Numbers are the numbers with the negative sign(-).
- These numbers are less than zero.
Example: We use negative numbers to represent temperature.
Here,
(i) +10 shows 10° hotter than 0,
(ii) -10 shows 10° colder than 0.
Successor and Predecessor
- If we move 1 to the right then it gives the successor of that number.
- If we move 1 to the left then it gives the predecessor of that number.
Tag me with a sign
- Accounting: In this case, we use
(i) Negative sign to represent the loss, and
(ii) Positive sign to represent the profit. - Sea Level: In this case, we use
(i) Negative sign to represent the height of the place below the sea level, and
(ii) Positive sign to represent the height of the place above the sea level.
Integers
- The collection of whole numbers and negative numbers together is called the Integers.
- All the positive numbers are positive integers.
- All the negative numbers are negative integers.
- Zero is neither a positive nor a negative integer.
Representation of Integers on Number Line
To represent the integers on a number line, follow the below steps:
- Begin by drawing a horizontal line.
- Mark a point as zero on the line.
- On the right side of zero, mark all the positive integers with equal distances, such as 1, 2, 3, etc.
- On the left side of zero, mark all the negative integers with equal distances, such as -1, -2, -3, etc.
For example, to mark (-7) we have to move 7 points to the left of zero.
Ordering of Integers
From the above number line, it is quite clear that:
- Numbers increase as we move to the right on the number line.
- Numbers decrease as we move to the left on the number line.
- So, any number on the right side of another number on the number line is greater than that number.
- Conversely, a number to the left of another number on the number line is smaller than that number.
For example,
(i) 5 is to the right of 2 so 5>2.
(ii) -4 is to the left of -1, so -4 < -1.
Some facts about Integers
- Any positive integer is always greater than any negative integer.
- Zero is less than every positive integer.
- Zero is greater than every negative integer.
- Zero doesn’t come in any of the negative and positive integers.
Addition of Integers
Addition of Two Positive Integers
If you have to add two positive integers then simply add them as
natural numbers.For eg: (+6) + (+7) = 6 + 7 = 13
Addition of Two Negative Integers
- If we have to add two negative integers, then simply add them as natural numbers.
- Then put a negative sign on the answer.
For eg: (-6) + (-7) = - (6+7) = -13
Addition of One Negative and One Positive Integer
- If we have to add one negative and one positive integer, then simply subtract the numbers.
- Put the sign of the bigger integer.
- We will decide the bigger integer ignoring the sign of the integers.
For eg: (i) (-6) + (7) = 1 (bigger integer 7 is positive integer)
(ii) (6) + (-7) = -1(bigger integer 7 is negative integer)
Addition of Integers on a Number Line
Addition of Two Positive Integers
Example: Add 3 and 4 on the number line.
Sol: To add 3 and 4, first, we move 3 steps to the right of zero then again move 4 steps to the right from point 3.
As we reached to the point 7, hence (+3) + (+4) = +7.
This shows that the sum of two positive integers is always positive.
Addition of Two Negative Integers
Example: Add (-2) and (-5) using a number line.
Sol: To add (-2) and (-5), first we move 2 steps to the left of zero then again move 5 steps to the left of (-2).
As we reached to the point (-7), hence (-2) + (-5) = -7.
This shows that the sum of two negative integers is always negative.
Addition of One Negative and One Positive Integer
(i) If a positive integer is greater than the negative integer- To add (+6) and (-2), start by moving 6 steps to the right from zero on a number line.
- Then, move 2 steps to the left from the point 6.
- As we reach the point 4, it implies that (+6) + (-2) equals +4.
(ii) If a negative integer is greater than the positive integer
- To add (-5) and (+4), start by moving 5 steps to the left from zero.
- Then, from the point (-5), move 4 steps to the right.
- When you reach the point -1, stop.
- The sum of (-5) and (+4) is (-1).
Additive Inverse
- If we add numbers like (-7) and 7 then we get the result as zero.
- So these are called the Additive inverse of each other.
- When adding (-2) and (2), first move 2 steps to the left of zero.
- Then, move two steps to the right of (-2).
- Finally, you'll reach zero.
Therefore, adding the positive and negative of the same number results in zero.
Example: What is the additive inverse of 4 and (-8)?
Sol: The additive inverse of 4 is (-4).
The additive inverse of (-8) is 8.
Subtraction of Integers
- When we subtract a larger positive integer from a smaller positive integer, the difference is a negative integer.
Example: (+5)-(+8) = -3 - To subtract a negative integer from any given integer, we just add the additive inverse of the negative integer to the given integer.
Example: (-5)-(-8) = +3
Subtraction of Integers on Number Line
If we subtract an integer from another integer then we simply add the additive inverse of that integer.
(-3) – (-2) = (-3) + 2 = -1
(-3) – (+2) = (-3) + (-2) = -5
Subtraction of Two Positive Integers
Example: Subtract 2 from 5.Sol: To subtract 2 from 5, first, we move 5 steps to the right from zero then move 2 steps back to the left.
As we reached to 3 hence, 5 – (+2) = 5 – 2 = 3
Subtraction of Two Negative Integers
Example: Subtract (-12) from (-8).
Sol: To subtract (-12) from (-8), first, we have to move 8 steps to the left of zero then move 12 steps to the right of (-8).
As we reached to 4, hence (-8) – (-12) = (-8) + (12) = 4
Subtraction of One Negative and One Positive Integer
(i) To subtract a positive integer from any other integer.Example: Subtract 3 from (-4).
Sol: To subtract (-4) from (3), first, we have to move 4 steps to the left of zero then move 3 steps more to the left.
As we reached to (-7), hence (-4) – (+3) = (-4) + (-3) = -7.
(ii) To subtract a negative integer from any other integer
Example: Subtract (-3) from (4).
Sol: To subtract (-3) from (4), first, we have to move 4 steps to the right of zero then move 3 steps more to the right.
As we reached to (7), hence (4) – (-3) = (4) + (+3) = +7