The major arithmetic properties in mathematics include cumulative, associative, distributive and identity. Let’s look in detail at all these properties.
A mathematical operation is said to be cumulative, even when the order of the operations alter, yet the results don’t change. You can change the placement of the addends in any way you want and still it won’t affect the result in anyway. In respect to both addition and multiplication, the result won’t be affected even when the places of factors are changed. Look at the examples below:
a + b = b + a
a * b = b * a
If an expression contains the same kind of operation which occurs more than once, it doesn’t matter how the order of operations are performed. But remember the sequence of the operations should not be changed. The parentheses can be rearranged in any manner and the result will always be the same. Let’s take a look at the following example:
a + (b + c) = (a + b) + c
a * (b * c) = (a * b) *c
For example, group and add:
So, let’s try adding this above expression using the associative property. You should notice that the addition operator occurs more than once here. Whether you add 1+6 and 3+3 or 6+3 or 1+3, the results are always 13.
When you have more than one type of operator including addition and multiplication in an expression, it is called as a distributive property. In this property, if you have a number that multiplies a sum inside a parenthesis, then according to the property, you can then remove the parentheses and multiply each value separately and still the result will always be the same. Look at the example below:
a * (b + c) = a * b + a * c
Identity element is also called a neutral element as it is one that, if combined with other elements, it remains unchanged. For addition, the identity element is 0 and for multiplication, the identity element is 1.
a + 0 = a
a * 1 = a