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.

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 basic operations of mathematics include:

• Addition: +
• Subtraction: −
• Multiplication: ×
• Division: ÷

These four operations are commonly referred to as arithmetic operations.

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.

Addition is the mathematical operation which explains the total amount of objects which we put together in a collection.

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 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.

Practice Problems

Problem 1.30 (Arithmetic Drill) 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=

Problem 1.31 (Make 1000 out of 8) Eight digits “8” are written together, like below, and plus signs “+” are inserted in between to get the sum of 1000. Where were the plus signs added?

$8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8$

Problem 1.32 (Unknown Sum) In the addition problem below, A, B, and C each represent three different digits. What are the digits?

${\begin{array}{r}AAA\\+\ BBB\\\hline AAAC\end{array}}$

Problem 1.33 (Unknown Product) A six-digit number with 1 as its left-most digit is three times bigger when we put the one at the end of the number instead. What number is this?

Problem 1.34 (Fractions and Decimals) Use long division to find the decimal expansion of each fraction.

Problem 1.35 (Terminating and Repeating Decimals) You may notice from Problem 1.7 that when you convert a fraction to a decimal, you will sometimes get what is called a repeating decimal. Take for example the fraction ${\frac {3}{11}}$.

${\frac {3}{11}}=0.272727272727...$

The decimal form of ${\frac {3}{11}}$ consists of the two digits 2 and 7 in an infinitely repeating sequence. To simplify things, instead of writing the above, we denote it as 0.27.

a. Use this bar notation to write each of the repeating decimals from Problem 1.9.
b. We see that in the fraction above, the fractional part repeats after two digits. We say that this number has a period of 2. Likewise, we say that the number ${\frac {1}{7}}$ has a period of 6, because the number repeats after 6 digits. From the numbers below, which of them has the largest period?

Problem 1.36 (Mixed Fractions) Write the following improper fractions as mixed fractions.

Problem 1.37 (Sharing Pizza) Billy's family ordered a large pizza. His father had ${\frac {1}{6}}$ of it, and his mother had ${\frac {1}{5}}$ of what remained. Later on, Billy's sister ate some pizza, and then Billy had the remaining pizza when there was exactly a half of what they started with (Billy is a large kid). What fraction of what their parents had left for her did the sister have?

A set of stamps (For Problem 1.38)

Problem 1.38 (Stamp Collection) The picture to the right shows stamps, arranged in four groups of four. How many stamps are in that image? While you can count them individually, there is a much faster way of getting the total.

Problem 1.39 (Decimal Operations) Explain how addition with decimals is comparable to addition with whole numbers, how are they different? Do the same thing with multiplication with decimals.

Problem 1.40 (A Sum and a Difference) The sum of two numbers is 104 and their difference is 32. What is the value of the larger number?

Problem 1.41 (On Allison's Street) Allison's house is on the same street as the library, post office, and supermarket, as shown in the diagram below. The distance from Allison's house to each of the three buildings is different. Based on this information, at which point is Allison's house located?

Problem 1.42 (Operations on the Number Line) Determine the performed operation that is being represented in each diagram.

Problem 1.43 (Page Numbers) How many numerals are required to number all of the pages of a book containing 450 pages?

Problem 1.44 (Inverse Operations) What is the inverse operation of “I put my shoes on today, and I walk out of my house”?