Arithmetic Operations

Arithmetic Operations
There are four (4) arithmetic operations: Addition (+), Subtraction (-) , Multiplication (*) , and Division (/). These operations can be more concretely (you can see it) with algebraic manipulatives.

Addition
The addition operation is the sum of two or more numbers. The numbers that you add together are called addends. Numbers that are added can be added in any order. This means that you can reverse the order of addition and get the same result.


 * $$A + B = C$$

Where
 * A
 * B
 * C
 * + . Addition Operator


 * Example
 * 2 + 3 = 5


 * 3 + 2 = 5


 * 1 + 2 + 3 = 6


 * 3 + 2 + 1 = 6


 * (-10) + 3 = -7


 * 3 + (-10) = -7

a + b = b + a

b + a = a + b

x + y + z = z + y + x

z + y + x = x + y + z

Subtraction
The subtraction operation is the difference of two or more numbers. It can be thought of as the "distance" between numbers. The order of subtraction matters. Numbers cannot be reversed and get the same result.
 * $$A - B = C$$

Where
 * A
 * B
 * C
 * - . Subtraction Operator


 * Example
 * 5 - 3 =  2


 * 3 - 5 =  -2


 * 6 - 3 - 2 =  1


 * 2 - 3 - 6 =  -7

All subtraction can also be written as addition:


 * a - b =  a + (-b)


 * b - a =  b + (-a)


 * x - y - z =  x + (-y) + (-z)


 * z - y - x =  z + (-y) + (-x)

Multiplication
The multiplication operation is the product of two or more numbers. The numbers that you multiply together are called factors. Remember the word factor; factor is a very important word in mathematics. Numbers that are multiplied can be multiplied in any order and still get the same result, just like addition! It represents repeated addition:

Example
5 * 3 = 5 + 5 + 5

5 * 3 = 3 + 3 + 3 + 3 + 3

3 * 5 = 5 + 5 + 5

3 * 5 = 3 + 3 + 3 + 3 + 3

(-5) * 3 = (-5) + (-5) + (-5) = -15

3 * (-5) = (-5) + (-5) + (-5) = -15

There is no written way to represent negative 5 sets of 3, but it can be seen using manipulatives.

It could be thought of as (-1)(3) + (-1)(3) + (-1)(3) + (-1)(3) + (-1)(3) = -15

Division
The division operation is the quotient of two or more numbers. Division creates a different kind of number called a fraction. The top (or first) number of the fraction is called the numerator and the bottom (or second) number of the fraction is called the denominator. It represents repeated subtraction until we get to 0, or until we cannot subtract another denominator from the numerator. Just like with subtraction, the order of division matters. We cannot reverse the order of subtraction and get the same result. The number of times that we subtract the denominator is the quotient:

Example
8 / 2 = 8 - (2) - (2) - (2) - (2) ---> subtract four times by 2 ---> = 4

We say that: 2 divides into 8, 4 times

You can also think of it as 4 groups of 2 will "fit" into 8.

The reverse or reciprocal does not give the same result:

2 / 8 <--- 8 does not divide into 2 evenly

Think of this as 2 objects, like chocolate candy bars, that are split up into a total of 8 equal pieces (not 8 pieces for each) and let's say that the two candy bars are divided up between 8 people. Each person would get one-fourth (1/4) of a candy bar.

chocolate candy bar 1

chocolate candy bar 2

Sometimes the denominator does not divide evenly into the numerator:

8 / 3 = 8 - (3) - (3) ---> We can only subtract 2 groups of 3 evenly, it is not possible to subtract another group of 3 evenly.

So 3 will only "fit" evenly into 8, 2 times, and there is a "left over" or remainder from the division. The quotient can be written as

2 Remainder 2

or 2 ⅔ <--- since we are dividing by 3.