As we all know, addiction is one of the four basic functions that make up the arithmetic universe. The most fundamental definition of addition is combining one or more entities to find one single final entity.
With the definition aside, we have all studied in our early days of school; there are four basic properties of addition that govern the process of summation in any given situation. These properties help us understand the fundamentals of addition and help us process addition functions with ease.
Let’s list what is properties of addition and discuss them in detail.
So, the following are the four key properties that we’ll be discussing in this topic:
- Commutative Property
- Associative Property
- Distributive Property
- Additive Identity
Commutative property of addition
Let’s pick the Commutative property first.
Well, as we all know, if we have two or more entities, considering integers in this case, what commutative property states that no matter the order in which we pick them and add, the result remains unchanged.
For example, this essentially means that if we take a, b and c as three integer values and add them, let’s say as a + b + c; we get value x.
Now, if we reverse the order of a, b and c, say b + a + c or c+ a +b or any other combination that we can think of, the result will remain x and not change.
Let’s give a, b, and c definitive values and see the result.
Ex 1:
a = 2 ; b = 5 ; c = 3
Now, putting the value in the first combination of a + b + c
We get: 2+ 5 +3 = 10
Now, changing the order of the numbers, say b + c + a
We get: 5 +3 + 2 = 10
Now, changing the order of the numbers again, say c + a + b
We get: 3 + 2 + 5 = 10
So we see, all three versions of addition yield the same result. Let us take another example where one integer is negative.
Ex 2:
a = 2 ; b = – 5 ; c = 3
Now, putting the value in the first combination of a + b + c
We get: 2 + (- 5) +3 => 0
Now, changing the order of the numbers, say b c a
We get: (- 5) +3 + 2 => – 5 + 5 => 0
Now, changing the order of the numbers again, say c + a + b
We get: 3 +2 + (- 5) => 5 + (- 5) => 0
As we can see, when it comes to addiction, no matter the sequence we put the integers or say entities in, the result will always be the same. And that is what the commutative law states.
Associative property of addition
Let’s move on to the next property, called associative property.
This property states that when three or more integers or entities are added together, no matter the pattern followed, the result will always be the same. It sounds quite like the commutative property, right? There is a difference. This property comes into play when a set of integers are given preference over others while adding.
For example, if integers a, b and c are considered, as per the property:
a + (b + c) = (a + b) + c
Let’s take an example to understand this better.
Like before, let’s take three integers, a, b and c, with values.
Ex 3:
a = 2 ; b = 5 ; c = 3
Now, putting the value in the first combination of a + (b + c)
We get: 2 + (5 + 3) => 2 + 8 => 10
Now, changing the order of the numbers, say (a + b) + c
We get: (2 + 5) + 3 => 7 + 3 => 10
Now, changing the order of the numbers again, say (a +c) + b
We get: (2 + 3) + 5 => 5 + 5 => 10
As we can see, the answer in all three cases remains 10 and does not change with the change in sequence the numbers are added.
Let’s take a negative integer and see the result.
Ex 4:
a = 2 ; b = – 5 ; c = 3
Now, putting the value in the first combination of (a b) c
We get: 2 + (- 5)+ 3 => (2 – 5) +3 => – 3 + 3 => 0
Now, changing the order of the numbers, say a + (b + c)
We get: 2 + (- 5) + 3 => 2 + (- 2) => 0
Now, changing the order of the numbers again, say (c +a) +b
We get: (3+ 2) +(-5) => 5 + (- 5) => 0
Hence from both the examples, we can deduce that it does not matter if the integer is positive or negative in value; change in preference of addition of integers does not affect the final result.
Distributive property of addition
Let’s check out the third property called the distributive property of addition.
As per this property, if we multiply any integer with the addition of any two integers, it is equal to the addition of the multiplied values with the third integer of the two integers that were added. Sounds confusing?
Let’s put it in terms of a, b and c.
So as per the property:
a * (b + c) = (a * b) + (a * c)
Take a numeric example to understand this better.
Ex 5:
a = 2 ; b = 5 ; c = 3
Now, putting the value in the combination:
=> a * (b + c) = (a * b) + (a * c)
We get:
=> 2 * (5 + 3) = (2 * 5) + (2 * 3)
=> 2 * 8 = 10 + 6
=> 16 = 16
So we see, both the sides of the equation fall on the same value, i.e. 16.
Let’s go with one negative integer to see if the property holds true there or not.
Ex 2:
a = 2 ; b = – 5 ; c = 3
Now, putting the value in the combination:
=> a * (b + c) = (a * b) + (a * c)
We get:
=> 2 * [(- 5) + 3] = [2 * (- 5)] + (2 * 3)
=> 2 * (- 2) = (- 10)+ 6
=> (- 4) = (- 4)
As we can see, both sides of the equation remain the same, i.e. ( – 4). Hence proving that no matter the sign of any integer, the distributive property of addition remains true in all cases.
Additive identity property of addition
Moving on to the last property called the additive identity of addition.
One of the easiest to understand properties of addition states that any given integer remains unchanged when added to zero. Quite a simple right!
So basically, let’s say there is an integer a. If we perform the operation a 0, the answer will always be a.
Let’s take an example of it. Take any integer, say 5. Now when we add 5 to 0, we know the answer will remain 5.
Due to this fact, 0 is also called the identity integer in addition, as it retains the integer’s value even after its addition to them.
Conclusion
Hope the four properties of addition are clear now for all of us. Practice similar examples to master the properties of addition so that you can ace any questions in this topic.
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