The term commutative is derived from “commute” or simply “moving around”, so commutative property can be defined as the term, which refers to moving around of stuffs. The rules of addition simply states “a + b = b + a” in numbers this means, 2 + 3 = 3 + 2. However, in case of multiplication, the rule states that “ab = ba” in numbers it is simply 2 * 3 = 3 * 2. The reference of commutative property is to move things around, any computation, which requires moving around of stuffs, this computation is said to use the commutative properties.
Mathematics has various binary operations, which are commutative, that means change of operands do not change the result. The commutative property is a fundamental property for binary operations, and a lot of mathematical proofs are dependent on it. The familiarized use of the property is mostly “1 + 2 = 2 + 1” or “2 * 3 = 3 * 2”, there are varied usage of these properties in many advanced forms of mathematics. There is an importance in naming the properties because operations involving subtraction and division are not possible, which are known as non-commutative properties. The concept of using simple operations like addition and multiplication for numbers, are commutative properties, which were assumed and used implicitly for many years. The name for this property was not coined until the 19th century, until the time mathematics started becoming more formalized. There is another corresponding property that used to exist for many binary operations, where a binary relation was said to be symmetric if there were relations applied regardless of the order in which the operand was used.
The use of this term “commutative” has been used in different ways.
Binary operations * in a set A is termed commutative, if, x * y = y * x, while an operation is not satisfying this property, then this is termed as non-commutative. There are a lot of commutative properties used in binary operations, like the addition of real numbers is commutative, because y + x = x + y , and the same example can be provided in numerical form where 4 + 8 = 8 + 4, since both the equation gives the result 12. There are many examples of commutative properties used in addition, where it is given that the change in order of addends does not change the result of the sum in between two variables. Addition of 2 and 3 will give, simultaneously addition of 3 and 2 also gives 5. The addition of two numbers is only commutative if the resulting sum is equal in either of the cases.
The commutative properties of addition is related to the change in position of the addends used in a sum, whereas the result remains same irrespective of the position it is used while the calculation is done. There can be more than two numbers and still the commutative properties will hold in case of more than two numbers whose orders are changed, like addition of 3 + 2 + 7 will give 12 and addition of 2 + 3 + 7 will also give 12.
Difference between Associative and Commutative Properties
There are a lot of properties in mathematics, which use probability and statistics, where these two properties commutative and associative are basically used with arithmetical integers, real numbers, rationals, although there is a presence of these two properties in advanced level of mathematics. Both of these properties, the commutative property and the associative property are quite similar in nature and often mixed up with one another. Therefore, it is important to understand the basic difference in between these two properties.
The commutative property is concerned with the order of mathematical operation, where a binary operation is involved with addition of two elements irrespective of the order of the addends, in which the sum was done.
The associative properties state that, grouping of factors in an operation is changeable without any effect to the outcome of the equation. This can be seen through an expression, where the equation a + ( b + c ) = (a + b ) + c. Irrespective of the pair of values used in the equation to add first, the result is similar.
For example, let us consider the equation 2 + 3 + 5. Irrespective of the order in which, these values are used, the result of the equation will be 10.
( 2 + 3 ) + 5 = ( 5 ) + 5 = 10
2 + ( 3 + 5 ) = 2 + ( 8 ) = 10
Like the commutative property, examples of associate properties include addition and multiplication of integers, real numbers and rational numbers. Although, unlike the commutative property, the associative property can also be applied for function composition and matrix multiplication. Like the commutative property used in equations, equations of associative properties cannot have subtraction of real numbers. For example, in the arithmetic problem, ( 6 – 3) – 2 = 3 – 2 = 1, if there is a change in the grouping by the parentheses, this will be 6 – ( 3 – 2 ) = 6 – 1 = 5, that changes the end result of the equation.
The difference between associative property and commutative property can be found out by simply checking whether the change in order of the elements is also changing the grouping of the elements. The reordering of elements can only be applied with commutative property, whereas the regrouping of the elements is used by associative properties. The presence of parentheses does not necessarily change the application of associative property, for instance ( 2 + 3 ) + 4 = 4 + ( 2 + 3 ). This equation is a simple example of the commutative property for addition of real numbers. Careful attention given to an equation provides with necessary details on the order of the elements, based on which the position is changed, but no change in the grouping of the elements, in case of associative property on the other hand will require grouping of elements altogether.
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