Arithmetic/Properties of Operations

Why Do Operations Have Properties?

Before continuing to learn about the properties of operations you must understand two basic questions:

What is the point of operations having properties?
→ Without operations having properties, we would not know their usage. Understanding their usage helps lay the foundation for solving word problems later on.

Why do operations have properties?
→ Operations have properties which define their usage. The usage of operations is the very essence of them, without usage properties are useless (vice versa).

By understanding properties we are able to enter the realm of higher-level thinking.

This is because properties illustrate general cases which allow us to lead to more mathematical generalizations.

Definitions of Mathematical Properties:

Addition:

1. Commutative Property: $a+b=b+a$
2. Associative Property: $(a+b)+c=a+(b+c)$
3. Identity Property of zero: $0+a=a(=a+0)$
4. Inverse Property: For every $a$, there exists a $-a$ such that $a+(-a)=0$

Multiplication:

1. Commutative Property: $ab=ba$
2. Associative: $(ab)c=a(bc)$
3. Identity: $a\cdot 1=1\cdot a=a$
4. Inverse Property, for every $a\neq 0$, there exists a $({\dfrac {1}{a}})$ (or $a^{-1}$), such that ${a}({\dfrac {1}{a}})=1.$

In general, the inverse property of multiplication can be stated more rigorously as saying that two numbers are multiplicative inverses of each other, if and only if their product is 1.

Remember that combining addition and multiplication yields the...
→ Distributive Rule: $a(b+c)=ab+ac.$

Most rules can be derived by paying attention to expressions and clever use of the properties.

Example:

Show that $a\cdot 0=0$.

Proof.
$a=a\cdot 1=a\cdot (1+0)=a\cdot 1+a\cdot 0.$

is equivalent to:

$a=a+a\cdot 0$
subtract a from both sides, yielding

$a\cdot 0=0$.

QED

Properties of Addition and Subtraction

Sure, you've added 4 + 3 = 7, but have you tried 3 + 4? You get the same answer right? Yes, because of the commutative law of addition.

For any numbers, a and b, a plus b is equal to b plus a. From this we know that addition is commutative, meaning that the operation of addition can be performed in any order with the same result.

What about 5 + 6 = 11? That's right, 6 + 5 = 11 as well.

There is another property of addition, the associative property. Here is an algebraic example:

For any numbers $a,b,c$: $a+(b+c)=(a+b)+c$

Can the same be said about subtraction? Well, let's try it... 7 - 5 = 2, does 5 - 7 = 2? Well... no. If you look at the number line, you will notice that when you do 5 - 7, you go below 0 into the realm of the negative numbers. Specifically, your solution is -2. Even though -2 is same in absolute value as 2, it isn't the same number. Therefore, subtraction is not commutative. Is it associative? No, because the associative law depends on the commutative law in order to work (since it really is just an extension of the commutative law.)

Exercises

1

8 + 9 =

2

50 + 30 =

3

45 + 9 =

4

36 + 11 =

5

2 + 3 =

6

(5 + 7) + (4 + 3) + (8 + 2) + 5 =

7

6 - 3 =

8

77-11 =

9

66 - 5 =

10

(80 - 13) - (36 - 5) - 11 + (36-11) =

Associative property

The associative property of real numbers is: $(a+b)+c=a+(b+c)$ for all real numbers $a,b,c$. This implies that the order in which addition is done does not matter.

Note however that this only applies to addition and multiplication. In fact, multiplication shares the same property. $(ab)c=a(bc)$ for all real numbers $a,b,c$.

The position of the parenthesis does change but the order by which you operate does not.