Algebra/Chapter 1/Arithmetic

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Algebra Chapter 1: Elementary Arithmetic Section 2: Operations in Arithmetic

Exponents and Roots

1.2: Operations in Arithmetic

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Arithmetic is the process of performing certain operations on quantities. In this section, we will cover four of the arithmetic operations: addition, subtraction, multiplication, and division.

Groups of Numbers

For the purposes of understanding opertions involving numbers, we will first discuss what we mean by an "operation" as well as the groups of numbers that they occur in.

In an operation, we take one or more numbers of interest, and perform a procedure on them to receive a new number, or the result of the operation. A group of two numbers may be combined together with a given operation to produce a third number.

The Four Basic Operations

The four operations of mathematics take two different values, and convert them to a new value. These include:

Division: ÷

These four operations are commonly referred to as arithmetic operations.

Addition

Defining Addition

$\ 1+1=2$

To define the number one is a rather difficult task, but we all have a good intuitive sense of what "oneness" is. Oneness is the property of having or thinking of a singular quantity. For example, think of when you have one dollar, one bushel of potatoes, or one light year. From here we can recursively (that is, relating to the last one) define the natural numbers by assigning a new name for each new number of ones that we have:

1	1	one
2	1 + 1	two
3	1 + 1 + 1	three
⋮	⋮	⋮
n	1 + 1 + … + 1	n ones

Example 1.1: Now that we have named the numbers using the number one we can define addition as the process of counting how many ones we have. For example, $\ 5+3=(1+1+1+1+1)+(1+1+1)=8$

In the above, notice that we represent the numbers 5 and 3 as a repeated addition of 1 in parentheses, which we subsequently added together.

3 + 2 = 5 with apples, a popular choice in textbooks.

Performing Addition

Addition is the mathematical operation which explains the total amount of objects which we put together in a collection. In addition, the numbers being added together are called addends or terms, and the end result is referred to as the sum.

Example 1.2: Tony and Aaron's aunt came to visit. She gave each of the boys 25 marbles. Tony won 12 marbles from Aaron during the visit. How many marbles did Tony have after they played? $\ 25+12=37$

Addition has several important properties. One of these properties is that the order you add the numbers will not effect the final result. Refer to the diagram of the apples above. Much like how we can add 3 apples to a group of 2 will result in 5 apples in total, we can add 2 apples to a group of 3 to get 5 apples in total.

Addition Problems

Subtraction

Subtraction is the term we use to describe how we "take away" one or more numbers from another. The term is used in two types of situations. The first is to answer the question of "how much is less?", and the second is "how much is needed?"

It can likewise be defined as counting initial quantity of ones and removing some amount of them. For example:

$\ 5-3=(1+1+1+1+1)-(1+1+1)=2$ means 5 ones remove 3 ones, leaving a result of 2 ones.

Subtraction Problems

Multiplication

Multiplication is a shorthand (that is, a faster way of writing something) for repeated addition. For example:

$\ 5\cdot 3=15$

What this means is to add up '3' 5 times; or add up '5' 3 times.

$\ 5\cdot 3=3+3+3+3+3=15$

Multiplication Problems

Note that in some regions and cases, it is better to use the cross symbol ($\times$) or the letter "x" instead of the dot. Most regions use the dot instead of the cross symbol because the cross symbol looks like the letter "x".

Division

Division is the opposite of multiplication.

$\ {\frac {6}{3}}=2$

This example asks if six is 1+1+1+1+1+1, and three is 1+1+1, then how many sets of three can we break six into? The answer is of course 2, because

$\ 6=1+1+1+1+1+1=(1+1+1)+(1+1+1)=3+3$; two sets of three.

Division Problems

Division is the first operation where a problem arises. In all the previously defined operations (addition, subtraction, and multiplication) we could perform the operation on any pair of numbers we chose. However, with division we cannot divide by zero. Much will be said about this fact throughout the course of this book, and even through your studies in all of mathematics.

Basic Applications of the Four Operations

Converting between Fractions and Decimals

While fractions and decimals both essentially represent the same thing, one may be better suited for a problem than the other.

Quiz

Perform the following operations specified below.

Example: 3+2=5

1

3+7=

2

100-1=

3

33+10=

4

19-3=

5

2+67=

6

2*7=

7

10*5=

8

30+50=

9

41-10=

10

15/5=

11

31+46=

12

1/2 of 80 =

13

67-40=

14

8+ +4=17

15

100/10=

16

58+32=

17

45+16=

18

1/4 of 16 =

19

60+ =89

20

+25=37

21

54-15=

22

60/5=

23

92-85=

24

-50=50

25

66-38=