Q. 2 - Calculus

Question

Properties of addition: State the associative law for addition, the commutative law for addition, and the distributive law for multiplication over addition of real numbers. (You may have to think back to a previous algebra course.)

Step-by-Step Solution

Answer

• The commutative law state that the order in which you add or multiply two real number does not affect the result. 
  The Commutative Law of Addition: $a+b=b+a$
  
• The associative law state the when you add or multiply any three real numbers, the grouping (or association) of the numbers does not affect the result.
  The Associative Law of Addition: $(a+b)+c=a+(b+c)$
  
• The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together. 
  The Distributive Law: $a(b+c)=ab+ac and (b+c)a=ba+ca$
  

Step 1. Given Information

We have to state 

• the associative law for addition, 
• the commutative law for addition, and 
• the distributive law for multiplication over addition of real numbers. 

Step 2. Solution:

The Commutative Laws (or the Commutative Properties)

The commutative law state that the order in which you add or multiply two real number does not affect the result.

The Commutative Law of Addition:

$a+b=b+a$

Example:

$$\begin{gathered} 3+5=5+3=8 \\ 20+(-3)=(-3)+20=17 \end{gathered}$$

The Associative Laws (or the Associative Properties)

The associative law state the when you add or multiply any three real numbers, the grouping (or association) of the numbers does not affect the result.

The Associative Law of Addition:

$(a+b)+c=a+(b+c)$

Example:

$$\begin{gathered} (2+3)+5=5+5=10 \\ 2+(3+5)=2+8=10 \end{gathered}$$

The Distributive Law or Distributive Property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together. 

Let's see what happens if we do $4(7+3)$ :

$4(7+3)=4\left(10\right)=40$

This follows PEMDAS (the order of operations).

But  $4\left(7\right)+4\left(3\right)=28+12=404\left(7\right)+4\left(3\right)=28+12=40$

Let's write this as

$4(7+3)=4\left(7\right)+4\left(3\right)4(7+3)=4\left(7\right)+4\left(3\right)$

We say we "distribute" the 44 to the terms inside.

This translates to the general case:

$a(b+c)=ab+ac and (b+c)a=ba+ca$

This is known as the Distributive law or the Distributive Property