Properties Of Operations

I. Introduction

Let's look at some simple, but important, properties of mathematical operations. These properties are applicable for binary operators. As a recap, binary operators are the ones that take two values, called

operands

and gives a result based on the operation. Some examples of common binary operations are

addition

subtraction

multiplication

and

division

We are going to understand 3 properties of binary operators, viz, $\underline{Associative\ Property}$ , $\underline{Commutative\ Property}$ and the $\underline{Distributive\ Property}$.

II. Commutative Property

When changing the order of operands does not change the result, the operation is called commutative.

For example, $4 + 5 = 9$. If we change the order of operands and write the expression as $5 + 4$, we still get the same result $9$.

Hence, addition is commutative. In general, we can write $A + B = B + A$, which is true for all values of $A$ and $B$.

Multiplication is also commutative, since $A \times B = B \times A$ for all values of $A$ and $B$.

Let's look at division and subtraction.

$20 \div 4 = 5$, but if we change the order of operands, we get $4 \div 20 = \dfrac{1}{5} = 0.20$. Hence, division is $NOT$ commutative.

Similarly,

$10 - 7 = 3$, whereas changing the order of operands, we get $7 - 10 = -3$. They are not same, hence subtraction is also $NOT$ commutative.

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Now let us try the following problem:

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III. Associative Property

Associative operation means that when there are more that one operator of the same type, the sequence in which you perform the operation does not impact the result. You get the same result for any sequence of you choice.

That is if you select addition:

$A + (B + C) = (A + B) + C$

For example:

$5 + (4 + 3) = 5 + 7 = 12$

$(5 + 4) + 3 = 9 + 3 = 12$. Hence addition is associative.

Even multiplication is associative, since $(A \times B) \times C = A \times (B \times C)$ for all values of $A$, $B$ and $C$.

Let's take a look at subtraction.

$A - (B - C) \neq (A - B) - C$

(Note: the symbol $\neq$ means $\unicode{0x2018}not\ equal\ to\unicode{0x2019}$)

Let's take an example to see this

$10 - (4 - 3) = 10 - 1 = 9$

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whereas,

$(10 - 4) - 3 = 6 - 3 = 3$

which are not equal.

Therefore, subtraction is not associative. Division is also not associative. Can you think of some more examples to illustrate this?

IV. Distributive Property

Unlike Associative or Commutative properties, Distributive property is applicable for two operations.

We will take an example to understand this better. Let's say you have an expression within brackets that has multiple terms separated by the same operation, like $(5 + 10 + 2)$. You are required to multiply the result by $3$, so we can write the expression as:

$3 \times (5 + 10 + 2)$ which is same as $3 \times 17 = 51$. Let's see different way of calculating the result:

$3 \times (5 + 10 + 2)$

$= 3 \times 5 + 3 \times 10 + 3 \times 2$

$= 15 + 30 + 6$

$= 51$

So, we get the same result as before.

We can see that the operation multiplication by $3$ gets distributed over all the terms inside the bracket that are separated by the $+$ operation, so we can say that $\unicode{0x201C}Multiplication\ is\ distributive\ over\ addition\unicode{0x201D}$.

Now let's look at an example of non-distributive operation.

Let's evaluate the expression $4 + (5 \times 3)$.

$4 + (5 \times 3)$

$= 4 + 15$

$= 19$.

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If we try to distribute the operation $+$ with the values inside the bracket like before, we will get:

$4 + (5 \times 3)$

$= (4 + 5) \times (4 + 3)$

$= 9 \times 7$

$= 63$ ✗.

This is not same as our first evaluation, and is incorrect.

Therefore we can say that $\unicode{0x201C}Addition\ is\ NOT\ distributive\ over\ multiplication\unicode{0x201D}$

At this stage we can make a list of the distributive operations that we will need, with one example for each case:

$\underline{Multiplication\ is\ distributive\ over\ addition}$.

$5 \times (7 + 2) = 5 \times 9 = 45$

$5 \times (7 + 2) = 5 \times 7 + 5 \times 2 = 35 + 10 = 45$

In general we can write:

$A \times (B + C) = A \times B + A \times C$

$\underline{Multiplication\ is\ distributive\ over\ subtraction}$.

$7 \times (10 - 6) = 7 \times 4 = 28$

$7 \times (10 - 6) = 7 \times 10 - 7 \times 6 = 70 - 42 = 28$

In general we can write:

$A \times (B - C) = A \times B - A \times C$

$\underline{Division\ is\ distributive\ over\ addition}$.

$(12 + 8) \div 4 = 20 \div 4 = 5$

$(12 + 8) \div 4 = 12 \div 4 + 8 \div 4 = 3 + 2 = 5$

In general we can write:

$(A + B) \div C = A \div C + B \div C$

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$\underline{Division\ is\ distributive\ over\ subtraction}$.

$(42 - 18) \div 6 = 24 \div 6 = 4$

$(42 - 18) \div 6 = 42 \div 6 - 18 \div 6 = 7 - 3 = 4$

In general we can write:

$(A - B) \div C = A \div C - B \div C$

There are many more operations in maths that have one or more of these properties, but for now, we need to understand these properties with respect to addition, subtraction, multiplication and division. You can try out some examples of each of these properties by taking some numbers by yourself.