Algebra/Chapter 1/Arithmetic

The Number Line

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, as well as how these operations are related to one another.

Groups of Numbers

For the purposes of understanding operations 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:

Addition: +

The combining of values.

Subtraction: -

The taking away of one value from another.

Multiplication: ×

Repeated addition.

Division: ÷

Repeated subtraction.

These four operations are commonly referred to as arithmetic operations. These are considered the cornerstones of all of mathematics.

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 1.2.1: 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. The operation of addition therefore means that we combine the numbers together to get a final result.

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 1.2.2: 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.

Subtraction

Defining 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.

Example 1.2.3: We can define subtracting as the process of counting how many ones we are removing. $\ 5-3=(1+1+1+1+1)-(1+1+1)=2$ means 5 ones remove 3 ones, leaving a result of 2 ones.

The term we are subtracting from is called the minuend. The number that is being subtracted is called the subtracthend. The resulting number is called the difference.

Performing Subtraction

Subtraction is the mathematical operation which explains the total amount of objects which we are left with after taking a certain amount from a collection. In subtraction, the number being subtracted from is called the minuend , the amount we are subtracting is called the subtrahend , and the end result is referred to as the difference.

Multiplication

Defining Multiplication

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

Example 1.2.4: We can define multiplication as the process of repeating addition. $\ 5\cdot 3=15$ What this means is to add up '3' 5 times; or add up '5' 3 times. $\ 5\cdot 3=3+3+3+3+3=15$

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".

Performing Multiplication

The end result is referred to as the product.

Multiplication Table

The Multiplication Table shows the products of a set of two numbers. Below is the products involving the numbers from 1 to 12.

×	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

Division

Division is the opposite of multiplication.

Example 1.2.5: We can define division as the process of finding the number of equal groups. $\ {\frac {6}{3}}=2$ 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.

Operations Involving Zero

1. 1 + 0 = 1
2. 1 * 0 = 0
3. 0 / 1 = 0
4. 1 / 0 = Undefined

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.