The reading material provided on this page for 'Integers' is specifically designed for students in grades 5 to 12. So, let's begin!
1. Integers are a set of whole numbers that include all positive numbers (1, 2, 3, ...), zero (0), and all negative numbers (-1, -2, -3, ...).
2. Integers are a subset of real numbers, which include all rational and irrational numbers. However, unlike real numbers, integers do not include fractions or decimals.
3. Integers can be added, subtracted, multiplied, and divided using mathematical operations. For example, when we add two integers, the result is always an integer. When we subtract two integers, the result may or may not be an integer. When we multiply two integers, the result is always an integer. When we divide two integers, the result may or may not be an integer, depending on whether the division is exact or not.
4. Integers are used in a variety of mathematical and scientific applications, including algebra, number theory, geometry, and physics. They are also used in computer programming and data analysis, where they are used to represent discrete quantities, such as counts, indices, and identifiers
The integers are denoted by the symbol “Z “.
Z = {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}
The range of integers is from negative infinity to positive infinity.
Now, let us learn about the types of integers!
There are several types of integers based on their value, sign, and properties.
1. Positive integers: These are the integers that are greater than zero, such as 1, 2, 3, 4, and so on.
2. Negative integers: These are the integers that are less than zero, such as -1, -2, -3, -4, and so on.
3. Even integers: These are the integers that are divisible by 2, such as -6, -4, -2, 0, 2, 4, 6, and so on.
4. Odd integers: These are the integers that are not divisible by 2, such as -5, -3, -1, 1, 3, 5, and so on.
5. Prime integers: These are the integers that are only divisible by 1 and themselves, such as 2, 3, 5, 7, 11, 13, and so on.
6. Composite integers: These are integers that are not prime, i.e., they have more than two factors, such as 4, 6, 8, 9, 10, 12, and so on.
Zero is a whole number and it is neither a negative number nor a positive number.
Four basic operations on integers are addition, subtraction, multiplication and division.
1. Addition: The operation of finding the sum of two or more integers.
Example: 3 + (-5) = -2
2. Subtraction: The operation of finding the difference between two integers.
Example: 7 - (-5) = 12
3. Multiplication: The operation of finding the product of two or more integers.
Example: 3 x (-5) = -15
4. Division: The operation of finding the quotient of two integers.
Example: 15 / (-3) = -5
Note: The division of integers may not result in a whole number, so in this case, the result is rounded down to the nearest whole number.
Here are some important properties of integers:
1. Closure property: The sum or product of any two integers is also an integer.
Example: 5 - 4 = 1
-5 x (-3) = 15
The numbers thus obtained are integers.
2. Associative property: The sum or product of integers is associative, meaning that changing the grouping of the integers being summed or multiplied does not change the result.
Example: (a + b) + c = a + (b + c)
(a x b) x c = a x (b x c)
Where a, b, and c are integers.
Example: (2 + 3) + 4 = 2 + (3 + 4) = 9
(2 x 3) x 4 = 2 x (3 x 4) = 24
The grouping of numbers will not change the result.
3. Commutative property: The sum or product of integers is commutative, meaning that changing the order of the integers being summed or multiplied does not change the result.
Example: a + b = b + a
a x b = b x a
Where a and b are integers.
Example: 2 + 7 = 7 + 2 = 9
2 x 7 = 7 x 2 = 14
4. Identity property: The sum of 0 and any integer is that integer, and the product of 1 and any integer is that integer.
Example: 5 + 0 = 5
8 x 1 = 8
5. Inverse property: Every integer has an additive inverse (i.e., a negative integer that, when added to the original integer, results in 0).
Example: 3 + (-3) = 0
6. Distributive property: The product of an integer and a sum or difference of integers is equal to the sum or difference of the products of the integer and each individual term.
a x (b + c) = a x b + a x c
a x (b - c) = a x b - a x c
Where a, b and c are integers.
Example: 4 x (5 + 2) = (4 x 5) + (4 x 2) = 28
This can be summarised in the form of table given below:
1. The product of two integers is always an integer.
2. The product of a positive integer and a negative integer is always a negative integer.
3. The product of two negative integers is always a positive integer.
4. The product of zero and any integer is always zero.
5. The product of any integer and one is always the original integer.
6. The order of multiplication does not matter, meaning that a x b = b x a.
7. The distributive property applies, meaning that a(b + c) = ab + ac.
8. The product of a fraction and an integer is always a fraction.
9. The product of two fractions is always a fraction.
10. The product of a decimal and an integer is always a decimal.
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