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The word integer is derived from a Latin word meaning whole. This is a form of a number that can be expressed without a fractional component. For example, 2, 8,90, -6, and many more, but numbers like 9.75, √3, etc., are not regarded as integers. Integers include 0 and are denoted by a capital alphabet Z. It also consists of positive natural numbers, their additive inverse, and negative integers. It is interesting to note that integers are a subset of all rational numbers and rational numbers are a subset of the real numbers. The alphabet Z is represented in different ways to denote different types of integers like Z+, Z+ or Z> (positive integers).
Properties of integers
Classification of integers is of 3 types; positive integer, negative integer, and 0. There are 5 properties of integers that determine the operations performed by the integers. It is these properties of integers that help us to solve equations and expressions.
Closure property- For addition and subtraction closure property states that if 2 integers are added or subtracted the outcome is always an integer. For example, 5-2= 3 where 3 is an integer. For multiplication of integers also the closure property is valid as a product of 2 integers is also an integer. This closure property is not always valid for division as the quotient is not always an integer and can be a fraction or decimal, for example, 5/10= ½
Commutative property- This property is valid for operations like addition and multiplication. According to this property, the order of the terms does not make any difference in the answer. For example, a+b = b+a and a×b = b×a, but this property is not valid for subtraction and division
Associative property- This property is about the grouping of numbers and their effect on the answer. The associative property is valid for addition and multiplication but not valid for subtraction and division. For example a + ( b+c) = (a+b) + c and similarly for multiplication
Distributive property- It tells us about the distribution of one operation over the other within a bracket. Mostly for the distributive property of integers the multiplication is used over addition and subtraction. For example, a × (b±c) = a×b ± a×c
Identity property- According to this property addition of any integer to 0 will give the same integer and here 0 is known as the additive identity. The multiplicative identity states that the multiplication of any integer with 1 gives us the same integer and here 1 is known as the multiplicative inverse
Property | Operations on integers | |||
Name | Addition | Subtraction | Multiplication | Division |
Closure | X+Y ϵ Z | X-Yϵ Z | X× Yϵ Z | X÷ Y ϵ Z |
Commutative | X+Y = Y+X | X-Y ≠ Y-X | X×Y = Y×X | X÷Y ≠ Y÷X |
Associative | (A+B)+C= A+(B+C) | (A-B)-C≠ A-(B-C) | (A×B)×C= A×(B×C) | (A÷B)÷C≠ A÷(B÷C) |
Distributive | A× (B+C)= AB + BC | A×(B-C)= AB- BC | Not applicable | Not applicable |
How are Integers Used in Everyday Life?
Through some examples, we will explain to you how are integers used in everyday life.
One good example of an integer in real life is the sea level; like for mountains, the distance is generally said as +1354m above the sea level, while the submarine can be at a distance of -55m below the sea level
Another example is the temperature, as the temperature can be expressed as positive or negative that is below or above 0 degrees
The speed limit can also be expressed in terms of the integer. If you increase speed more than a specified speed, it is regarded as positive, and if we decrease the speed below a given limit, it is regarded as negative
In games, if we win, then the points increase; on the other hand, if we lose the game, the points decrease, which is also an example of integers
The floors of the building can also be considered as an example of integers
Conclusion
The word integer is derived from a Latin word that means whole. Integers can be expressed without a fractional component, for example, 2, 8,90, -6, and many more but numbers like 9.75, √3, etc., are not regarded as integers. Integers are classified as 0, positive, and negative integers. The integers follow 5 properties named closure property, associative property, distributive property, commutative property, and identity property. All these properties are used for different calculations used while solving algebra. It is also used in our daily lives during the calculation of profit/ loss, measuring the distance with respect to sea level, determining the temperature, calculating the distance travelled, and many more such examples. In fact, we explained the concept with a few examples of how are integers used in everyday life.
