# Arithmetic Operations

Subject classification: this is a mathematics resource. |

**Arithmetic Operations**[edit | edit source]

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[edit | edit source]

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.

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[edit | edit source]

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.

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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

1 | 2 | 3 | 4 |

chocolate candy bar 2

5 | 6 | 7 | 8 |

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.