In the beginning of every basic mathematics book you will find three fundamental properties of numbers. These properties are: the Distributive property, the Associative property and the Commutative property. We usually ignore them. But actually these properties are so important that there are many mathematical proofs which wouldn't exist if these properties were not used. Basically these three properties forms the base of every mathematical calculations. In this oneHOWTO article we will tell you what commutative property means.
Origin and definition
Origin: The word commutative is derived from the word “commute” which means “to move around”. In commutative property the numbers are moved around for computation.
Definition: According to the commutative property, order does not matter during computation. The Commutative property can only be applied in addition and multiplication. It cannot be applied on division and subtraction.
Commutative property in mathematics
For Addition
The addition of two or more real numbers is always commutative.
This means: x + y = y + x where x and y are both real numbers.
- Example:
1 + 2 = 2 + 1
99 + 190 = 190 + 99
Addition of complex numbers and vectors also uses the commutative property.
For Multiplication
Multiplication of two or more real numbers is always commutative.
This means: xy = yx where x and y are both real numbers.
- Example:
1 x 2 = 2 x 1
99 x 190 = 190 x 99
Multiplication of complex numbers and scalar multiplication of vectors also uses the commutative property.
Commutative property in Sets
Union
When two sets are added together, it is known as union of sets.
A union of sets is always commutative. A union of two sets A and B is denoted by A U B and it is always commutative.
- Example:
{1, 2} U {2, 3} = {2, 3} U {1, 2}
{1, 2, 3} U {3, 4, 5} = {3, 4, 5} U {1, 2, 3}
Intersection
Intersection is the way of creating a new set by determining the common members of two sets. Intersection of two sets A and B is denoted by A ∩ B.
Intersection of two sets is always commutative.
- Example:
{1, 2} ∩ {2, 3} = {2, 3} ∩ {1, 2}
{1, 2, 3} ∩ {3, 4, 5} = {3, 4, 5} ∩ {1, 2, 3}
Commutative Property used in daily life
In our daily life we use and perform a number of activities in which commutative property can be applied. Some examples of commutative property used in daily life are as follows:
- Putting on your shoes is a commutative operation. This is because it doesn’t matter which shoe we put on first. Putting on either the left shoe followed by the right shoe or the right shoe followed by the left one will result the same i.e. having both shoes on. Similarly putting on the socks or gloves is also commutative. But putting on underwear and trousers is not commutative. You can’t put on trouser and then put on underwear. According to social etiquette you need to put on your underwear first and then put on your trousers later.
- Adding sugar and cream in to coffee is a commutative action because it doesn’t matter which one goes into coffee first. If you put in sugar and then cream you will get coffee and if you put in cream and then sugar, still you will get coffee. So, it is commutative. But every culinary action is not commutative. For example you cannot put flour into the oven and then add water to get bread. Rather you have to add water into the flour first then make dough and put into oven to get the bread. It cannot be reversed.
- The Commutative property can also be observed while paying or receiving cash. While paying for an item with cash, it doesn’t matter in which order the money is handed over. The total money will be the same even when we change the order in which the bills are handed over. Same is the case with receiving an item. It doesn’t matter which item you receive first, you will always receive all the items in the end.
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