What are Integers?
Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. They do not include fractions or decimals.
- Positive Integers: Numbers greater than zero (e.g., 1, 2, 3, …).
- Negative Integers: Numbers less than zero (e.g., -1, -2, -3, …).
- Zero: Zero is a special integer because it is neither positive nor negative.
Examples of integers are: -5, 1, 5, -8, 97.
Examples of numbers that are not integers are: -1.43, 3/4, 3.14, 0.09
These examples show that integers are only whole numbers with positive and negative signs, and numbers with fractions or decimal points do not count as integers.
Applications in Real Life
Temperature:
Think of a thermometer. When the temperature is above freezing, it’s a positive number (e.g., 10°C). When it’s below freezing, it’s a negative number (e.g., -5°C).Bank Balance:
A positive bank balance means you have money in your account (e.g., ₹500). A negative balance means you owe money, or are in debt (e.g., -₹200).Directions:
Moving forward is like a positive number, and moving backward is like a negative number. For example, walking 3 steps forward is +3, while 2 steps backward is -2.Sports Scores:
In some games, you can gain or lose points. For example, if your team scores 5 points, that’s +5. But if they lose 3 points because of a penalty, that’s -3. Keeping track of both positive and negative points helps you understand the overall score.
Integers on the Number Line
On a number line, integers are arranged with zero in the center. Numbers to the right of zero are positive integers, and they get bigger as you move further right. Numbers to the left of zero are negative integers, and they get smaller as you move further left.
Positive Integers are whole numbers greater than zero. As you move right on the number line, the values increase (e.g., 1, 2, 3, 4, …). The farther you go to the right, the larger the number becomes. For example, 5 is larger than 2, and 100 is larger than 50.
Negative Integers are whole numbers less than zero. As you move left on the number line, the values decrease (e.g., -1, -2, -3, -4, …). The farther you go to the left, the smaller the number becomes. For example, -5 is smaller than -2, and -100 is smaller than -50.
Key Points to Remember
Zero is neither positive nor negative. It is the starting point or middle of the number line.
The Smallest Positive Integer is 1. As you move right from zero, this is the first number you encounter.
The Largest Negative Integer is -1. As you move left from zero, -1 is the first negative number you encounter.
- Even though -10 may seem “bigger” than -1 because 10 is larger than 1, on the number line, -10 is actually smaller than -1. This is because numbers decrease as you move left from zero. So, in the world of negative numbers, -1 is larger than -10, just as 1 is larger than 10 in the positive range.
- Numbers increase in size as you go right.
- Numbers decrease in size as you go left.
Ordering and Comparison
(a) All positive integers are greater than negative integers and zero. i.e., 4 > -4, 5 > 0
(b) 0 is always greater than all of the negative integers. i.e., 0 > -4
(c) In positive integers, a number with greater numerical value is greater. i.e., 24 > 19.
(d) In negative integers, a number with greater numerical value is smaller. i.e., –122 < –31
Absolute Value
The absolute value of an integer is the distance between that number and zero on the number line, without considering whether the number is positive or negative. In other words, the absolute value of a number is its numerical value, ignoring the sign. It tells us how far a number is from zero, no matter which direction (left or right) it is on the number line.
Absolute value is represented by vertical bars on either side of the number, like this: |a|.
The absolute value of 5 = |5| = 5
The absolute value of -5 = |-5| = 5
Both 5 and -5 have the same absolute value because they are the same distance from zero.
Examples:
Positive Integers:
The absolute value of a positive number is the number itself since it’s already positive.- |7| = 7
- |12| = 12
Negative Integers:
The absolute value of a negative number is the positive version of that number. You ignore the negative sign and just take the numerical value.- |-7| = 7
- |-12| = 12
Zero:
The absolute value of zero is zero, because zero is neither positive nor negative.- |0| = 0
Addition of Integers
Adding integers is like moving along a number line. When we add a positive integer to any other integer, we move to the right on the number line. This is because adding a positive number increases the value of the original number.
Adding Two Positive Integers
3 + 4 = 7
Move 4 steps to the right starting from 3, and you land on 7.
Adding a Negative and a Positive Integer
(-4) + (6) = 2
Move 6 steps to the right starting from -4, and you land on 2.
When we add a negative integer to any other integer, we move to the left on the number line. This is because adding a negative number decreases the value of the original number.
Adding Two Negative Integers
(-2) + (-3) = -5
Move 3 steps to the left starting from -3, and you end up at -5.
Adding a Positive and a Negative Integer
1 + (-7) = -6
Move 7 steps to the left starting from 1, and you end up at -6.
Tips for Adding Integers
- If the signs are the same (both positive or both negative), add the numbers and keep the sign.
- If the signs are different, subtract the smaller number from the larger number and keep the sign of the larger number.
Practice Problems
6 + (-2) = ?
-4 + (-5) = ?
7 + 3 = ?
Subtraction of Integers
Subtraction of integers means finding the difference between two numbers. When we subtract a positive integer, we move to the left of the number line.
(+5) – (+7) = -2
Move 7 steps to the left starting from 5, and you land on -2.
(-6) – (+2) = -8
Move 2 steps to the left starting from -6, and you land on -8.
When we subtract a negative integer, we move to the right of the number line and add its opposite positive integer.
(+4) – (-3) = +7
Move 3 steps to the right starting from 4, and you end up at +7.
(-6) – (-2)
Tips for Subtracting Integers
Subtracting an integer is the same as adding its opposite! When you subtract integers, turn the subtraction problem into an addition problem:
Subtracting a number = Adding its opposite.
- If you see “minus a positive,” change it to “plus a negative.”
- If you see “minus a negative,” change it to “plus a positive.”
Example:
-6 -(+4) = -6 + (-4) = -10
7−(−3) = 7 + 3 = 10
Practice Problems
6 – (-4) = ?
-5 – 7 = ?
-2 – (-3) = ?
Multiplication of Integers
Multiplying integers is just like regular multiplication, but we need to remember a few rules about signs (positive and negative).
The Rules for Multiplying Integers
- Positive × Positive = Positive
- Example: 3×4=12
- Negative × Negative = Positive
- Example: (−3)×(−4)=12
- Positive × Negative = Negative
- Example: 3×(−4)=−12
- Negative × Positive = Negative
- Example: (−3)×4=−12
Quick Trick
The rules for Integer Division are the same as Integer Multiplication.
- If the signs are the same (both positive or both negative), the answer is positive.
- If the signs are different (one positive and one negative), the answer is negative.
Examples
2×5=10 (Positive × Positive = Positive)
(−2)×(−5)=10 (Negative × Negative = Positive)
2×(−5)=−10 (Positive × Negative = Negative)
(−2)×5=−10 (Negative × Positive = Negative)
Practice Problems
4 × (−3) = ?
(−6) × (−2) = ?
(−7) × 5 = ?
Division of Integers
Division is splitting a number into equal parts. When dividing integers (positive and negative numbers), there are just a few rules to remember.
The Rules for Dividing Integers
- Positive ÷ Positive = Positive
- Example: 12÷3=4
- Negative ÷ Negative = Positive
- Example: (−12)÷(−3)=4
- Positive ÷ Negative = Negative
- Example: 12÷(−3)=−4
- Negative ÷ Positive = Negative
- Example: (−12)÷3=−4
Quick Trick
- If the signs are the same (both positive or both negative), the answer is positive.
- If the signs are different (one positive and one negative), the answer is negative.
8÷2=4 (Positive ÷ Positive = Positive)
(−8)÷(−2)=4 (Negative ÷ Negative = Positive)
8÷(−2)=−4 (Positive ÷ Negative = Negative)
(−8)÷2=−4 (Negative ÷ Positive = Negative)
Practice Problems
15÷(−3) = ?
(−20)÷5 = ?
(−18)÷(−6) = ?
Practice Quiz on Integers
FAQs on Integers
Natural numbers: These are the set of positive integers starting from 1 (i.e., 1, 2, 3, …).
Whole numbers: These include all natural numbers along with zero (i.e., 0,1, 2, 3, …).
Integers: These extend whole numbers to include negative numbers as well (i.e., …, −3, −2, −1, 0, 1, 2, 3, …).
Yes, zero (0) is an integer, and it is neither positive nor negative.
Closed under addition: Yes, adding any two integers results in another integer. (5 + 6 = 11)
Closed under subtraction: Yes, subtracting any two integers results in another integer. (5 – 6 = -1)
Closed under multiplication: Yes, multiplying any two integers results in another integer. (8 x 4 = 32; -5 x 4 = -20)
Not closed under division: The division of two integers may not result in an integer. (9 / 5 = 1.8 which is not an integer)
The opposite of any integer n is -n. i.e., for any integer a, a + (-a) = 0 = (-a) + a.
Here, (–a) is the additive inverse of a. For example, the opposite of 5 is −5, and the opposite of −7 is 7. The opposite of 0 is 0 itself.
For any integer a, a + 0 = a = 0 + a. Here, 0 is the additive identity.
The absolute value of an integer is its distance from zero on the number line, disregarding the sign. For example, the absolute value of −4 is 4, and the absolute value of 5 is 5. The absolute value of zero is zero.
Consecutive integers are integers that follow one another in order, with a difference of 1 between them. For example, 3, 4, 5 are consecutive integers, and −2, −1, 0 are consecutive integers.
The difference between any two consecutive even integers is always 2 (e.g., 2, 4, 6).
The difference between any two consecutive odd integers is also always 2 (e.g., 1, 3, 5).
Division by zero is undefined in mathematics because no number multiplied by zero can give a non-zero result. For example, 12 / 4 = 3 (i.e., 4 x 3 = 12). But for 12 / 0, there is no number x such that 0 × x =12
In many navigation and mapping systems, we use a coordinate plane (Cartesian plane) with two axes:
X-axis: Represents horizontal movement (east-west).
Y-axis: Represents vertical movement (north-south).
Assigning Directions to Integers:
Positive integers represent movements north and east.
Negative integers represent movements south and west.
For example:
Moving north means increasing the y-coordinate (positive integers on the y-axis).
Moving south means decreasing the y-coordinate (negative integers on the y-axis).
Moving east means increasing the x-coordinate (positive integers on the x-axis).
Moving west means decreasing the x-coordinate (negative integers on the x-axis)