# Arithmetic/Printable version

Arithmetic

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# How This Textbook Is Organized

This textbook is organized by parts and then into sections and subsections. The parts get more advanced as you work through the textbook, and the sections build off of one another until the reader has a substantial working knowledge of the material for that part of the textbook. Skipping any part of this textbook is not advised.

# Introduction to Arithmetic

Whether you're a scientist or a shopper you need arithmetic. There are all sorts of situations where it is useful to be able to count, measure, and communicate quantities. Arithmetic is the study of numerical quantities. At least a basic understanding of arithmetic is fundamental in the study of algebra and all other mathematical studies. This book assumes that the reader already understands some mathematics but wishes to relearn it in a more formal manner.

## Basic Operations

There are four basic operations one can perform on different numeric quantities (as well as other mathematical objects which you will encounter later). These are:

• Addition ($+$ )
• Subtraction ($-$ )
• Multiplication ($\times$ )
• Division ($\div$ )

There are further operations past these but they are more advanced than the scope of this book and are less likely to be used in day-to-day life.

## Equals

It is also important to understand that the equals sign means that the things on both sides are of equal value, and that $=$ is not an operator.

When you see an $=$ sign, say to yourself Same value, different appearance.

 Arithmetic ← How This Textbook Is Organized Printable version Number Operations →

# Number Operations

## Arithmetic Operations

Addition is one of the four basic operations of arithmetic. The operation is usually shown by the plus symbol (+). This is the act of combining two numbers into one. This operation may be used on any number, from natural numbers to complex numbers. For example, look at the following equation:

$2+2=4$ This means "Adding two with two gives us four" or put more simply, "Two plus two equals four". In this example, both twos are terms (addends), the numbers being added together, while the 4 is the sum, the result of the addition.

### Commutativity

When adding two numbers together, it doesn't matter what order the numbers are placed in, since the outcome will be the same. That is to say for any two numbers $x$ and $y$ , $x+y=y+x$ For example, the two equations end up with the same result.

$4+5=9$ $5+4=9$ ### Associativity

When adding multiple numbers together, the order in which you add the numbers doesn't matter, since the outcome will be the same. For any three numbers $x,y$ and $z$ , $(x+y)+z=x+(y+z)$ For example, the two equations will end up with the same result.

$(1+2)+3=3+3=6$ $1+(2+3)=1+5=6$ The number 0 is known as the additive identity. This means if you add any number with zero, you end up with that original number. For example:

$4+0=4$ $6.24+0=6.24$ $(5+4i)+0=5+4i$ # Number Operations/Subtraction

Subtraction is one of the four basic operations of arithmetic. The operation is usually shown by the minus symbol (-). This is the act of taking a number and taking away a certain amount of it. Think of it as the opposite of addition, an operation where a number is combined with another to form a resulting sum. For example, the following equation:

$5-1=4$ This means, "Taking away one from five gives us four", or put more simply, "Five minus one equals four." In this example, the 5 is the subtrahend, the number being subtracted; the 1 is the minuend, the number it is subtracted from; and the 4 is the difference, the result of the subtraction.

## Properties of Subtraction

### Non-Commutativity

Unlike addition, the order in which two numbers are subtracted does matter. In other words, unless the subtrahend and minuend are equal to each other, they are distinct elements when subtracting and cannot be switched order wise. For example, the two equations end up with different results.

$5-2=3$ $2-5=-3$ Here is another example using numbers with decimals.

$4.3-1.2=3.1$ $1.2-4.3=-3.1$ Notice how the difference of the original expression is negative of the difference of the flipped expression. This is a specific type of non-commutativity known as anticommutativity. For any numbers $x$ and $y$ ,

$x-y=-(y-x)$ ### Non-Associativity

Unlike addition, when subtracting multiple numbers, the order in which you subtract the numbers matters. Subtracting numbers in different orders may, and likely will, result in different differences. For example, the two equations will not end up with the same result if the order of which you subtract is different, even if the expression itself is the same.

$(1-2)-3=-1-3=-4$ $1-(2-3)=1-(-1)=1+1=2$ # Number Operations/Multiplication

Multiplication is one of the four basic operations of arithmetic. The operation is usually shown by the times symbol (*). This is the act of taking a number and adding it to itself a certain number of times. Think of multiplication as a shorthand way of performing repeated addition. For example, the following equation:

$5*4=20$ This means, "Multiplying five by four gives twenty," or put more simply, "five times four equals twenty." In this example, 5 and 4 are the factors, and 20 is the product, the result of the multiplication.

## Properties of Multiplication

### Commutativity

When multiplying two numbers together, it doesn't matter what order the numbers are placed in, since the outcome will be the same. That is to say for any two numbers $x$ and $y$ , $x*y=y*x$ For example, the two equations end up with the same result.

$3*7=21$ $7*3=21$ ### Associativity

When multiplying multiple numbers together, the order in which you multiply the numbers doesn't matter, since the outcome will be the same. For any three numbers $x,y$ and $z$ , $(x*y)*z=x*(y*z)$ For example, the two equations will end up with the same result.

$(2*3)*4=6*4=24$ $2*(3*4)=2*12=24$ The number 1 is known as the multiplicative identity. This means if you add any number with one, you end up with that original number. For example:

$4*1=4$ $6.24*1=6.24$ $(5+4i)*1=5+4i$ # Types of Numbers

We can classify numbers into several different kinds.

# Introduction to Natural Numbers

The ability to count things has been essential throughout the ages. Over time, several systems for counting things were developed; the first of which was the natural numbers. As a set, the natural numbers can be written like so: $\{1,2,3,\dots \}$ . If we also include the number zero $0$ in the set, it becomes the whole numbers: $\{0,1,2,3,\dots \}$ .

## Formulation

The whole numbers can be formed in many ways. The easiest way is to use what is called an inductive definition. This is when we define the first of a series of numbers, and then make it possible to derive any given number's successor so that given any number we can always find the next. The first of the whole numbers is $0$ . The way we can derive the next is to simply add one to the previous number. This is easily demonstrated: $0+1=1$ , so zero's successor is one; $1+1=2$ , so one's successor is two; $2+1=3$ , so two's successor is three; and this can be continued "ad infinitum," which is just a Latin phrase meaning "to infinity".

## Uses

The natural numbers are used for three main purposes: for counting, for ordering, and for defining other concepts. Counting is the natural way to measure the quantity of a set of several discrete, individually identifiable objects. To count a specific set of objects using the natural numbers, you must simply assign one and only one natural number to one element of the group of objects, starting with one. To the next object, selected arbitrarily, that has not yet been assigned a number, you would assign the next number in the group of natural numbers and then proceed to move on to the next, until all of the items have been assigned a number. (If we can never reach the end, we cannot describe the count in terms of any natural number. There are ways of dealing with "infinite" sets, but for now we stick to "finite" sets.) The attentive will notice that this is an inductive definition: we define the first term and come up with a way of deriving any given term's successor. Counting sometimes goes by the fancy name "enumeration."

Ordering (also called "ranking") is the assignment of natural numbers to members of a group not arbitrarily, but with some property in mind. To do this, you select the object that has the most extreme value of that property (i.e. the smallest, the smartest, the fattest, etc. . .) and assign it the natural number one, then you set it aside and move on to the remaining object with the greatest (remaining) value of that property and assign it the next natural number, in this case, two. You then set it aside too, and proceed to the remaining object with the greatest (remaining) value of that property and assign it, once again, the next natural number, repeating this step until all objects have been ranked. (If we are only interested in the first few rankings, we can stop before we have ranked all of a large number of objects.) Once again, we use an inductive definition. In most natural languages, different words are used for numbers as quantities ("three") and for numbers as ranks ("third"). We call the former "cardinal numbers" and the latter "ordinal numbers," although they are both just natural numbers being applied in slightly different ways.

It should be noted that in all of the above cases zero does not come into play. Zero is a unique case where, in the case of counting, you have not yet assigned any number to an object. For example, if you are attempting to count the amount of apples you own, and you own no apples, then the amount you count is zero. With ordering, the number zero is never used because if you have nothing to order, you are done before you start, and no object will ever be ranked in 0th place.

The natural numbers also play an integral part in the definition of many other mathematical concepts, including the very concept of mathematical induction we have used to define counting and ordering. Because the procedure used on this page uses mathematical induction, in a more formal situation we must use another method to define the natural numbers, in order to avoid a "circular definition" (where a concept is defined in terms that depend on the concept being defined). A formal definition of natural numbers can be based on the "successor" idea.

## Properties

One notices that the natural numbers go on forever, with any singular one of them having an infinite number of successors, as any successor has a successor, and that successor has a successor onwards to infinity. Yet in spite of the infinite size, we can still count the numbers. This makes the set countably infinite. Mathematicians have created a whole set of special numbers called the cardinal numbers to describe the different sized infinities; in this case, the set of natural numbers is aleph-null sized. This is important to remember for further studies in mathematics.

# Introduction to Operations

There are four basic operators, which are used to work with numbers for uses besides counting. They are:

• $+$ addition
• $-$ subtraction
• $\times$ multiplication (The symbols * and are also used to multiply)
• $\div$ division (The symbol / is also used to divide)

Addition is the total of two numbers. For example:

$1+2=3$ ## Subtraction

Subtraction is the opposite of addition, which is the taking away of one number from another. For example:

$2-1=1$ ## Multiplication

Multiplication is the adding of a number a certain number of times. So:

$2*1=2$ $2*2=2+2=4$ $2*3=2+2+2=6$ etc.

## Division

Division is the splitting of a number into a number of equal parts. For example:

$6/2=3$ # Zero and Numbers Greater Than Nine

This page explores more numbers by getting higher and lower than the previously introduced numbers 1-9.

## Zero

What happens when you want to show with math that there is nothing? This is where we use the number zero (0). Zero can be achieved by subtracting any number by itself. For example:

$3-3=0$ ## Numbers Larger than Nine

So far we have learned about nine different numbers. But what if we want to count higher than nine? This is where we use numbers that have more than one digit. The simplest of these numbers are:

$9+1=10$ At first, this looks like nonsense. How can nine and one make "one-zero"? But the "one-zero", called ten is actually a number in itself. Since there is more than one digit, all but the first digit represent numbers larger than nine. 10 can be rewritten as:

$(1*10)+(0*1)$ that is to say:

The number 10 is equal to the total of one ten and zero ones.

Let's use another number, 58 (fifty-eight), to explain more. The number 58 can be rewritten as:

$(5*10)+(8*1)$ that is to say:

The number 58 is equal to the total of five tens and eight ones.

# Numbers Less Than Zero

Although we cannot have less than nothing in real life, we have to assume that since you can go all the way in one direction in counting up, that you can continue to count down in the other direction.

For example, when you're giving somebody apples, you can count those apples up; otherwise, if you take apples away from somebody, you carry out a negative action and, therefore, you should apply the negative counting. This way, such numbers are exactly called negatives, and they represent values less than zero.

If you have given somebody an apple, two apples, or three apples, your apple-giving may be assessed with the numbers 1, 2, or 3, respectively. But if you have taken 2, 4, or even 5 apples away from somebody, the actual "apple-giving" will deserve, indeed, negative assessment and, this way, may be represented by negative numbers, which are, in our case, -2, -4, and -5 respectively.

## Examples

-3 (pronounced "Negative Three" or "Minus Three")
-5
-10
-11

5 - 7 = -2
10 - 19 = -9
36 - 55 = -19
36 - 111 = -75
36 - 555 = -519

# The Number Line An example of a number line. Notice the arrows indicating that the line goes infinitely in both directions.

The number line is a one-dimensional graph to show the relative positions of numbers. As the line goes left, the numbers have less value; as the line goes right, the numbers have more value. The line continues infinitely (without end) in both directions.

The number line can be made in different intervals, that is, how many numbers the graph goes up and down by. The top image shows a number line going in intervals of one, while the bottom line graph goes in intervals of ten.

## Absolute Value

As you can see, the positive and negative numbers are of equal distance from the number zero. The distance of a number from zero is called the absolute value. The absolute value of a number is always positive or zero. This may also be referred to as the magnitude of the number. It is represented by two lines on the left and right of a number. When solving problems with absolute values, always solve the absolute number first. For example:

$|-5|=5$ $|0|=0$ $|5+3|=|8|=8$ $36-|-55|=36-55=-19$ # Spelling Out numbers

In this section you will learn how to spell out numbers in English. Before we can spell out we must learn how numbers are arranged. We must know what we mean by units, tens, hundreds place and so on.

• 0 – Zero
• 1 – One
• 2 – Two
• 3 – Three
• 4 – Four
• 5 – Five
• 6 – Six
• 7 – Seven
• 8 – Eight
• 9 – Nine

Places: Using the number 1,234

The place names are:

• 4 – Ones
• 3 – Tens
• 2 – Hundreds
• 1 – Thousands

It is spelled out as: One Thousand Two Hundred and Thirty Four

Note: After the Thousands, place names can vary depending on local customs. Place a comma after every third number starting at the end of the number to show place value family names change. For example: 598,482,975 or 78,109,200,330.

The word and is used only for the decimal point; And is also used to separate a whole number and a fraction. A comma is used at every third place, starting at the decimal point and moving left. Frequently a comma or space is used at every third place moving to the right of the decimal point.

Additionally, no comma is used before and. Also, all decimals end in ths except unitary decimals that end in th.

Alternatively, you may say the whole number, followed by a Point and the digits of the decimals from leftmost to right. This is a much more natural and informal way of saying numbers.

 Decimal numeral Reading thereof Alternative $2,697,787.84$ Two million, six hundred ninety-seven thousand, seven hundred eighty-seven and eighty-four hundredths Two million, six hundred ninety-seven thousand, seven hundred eighty-seven point eight four $2,009$ Two thousand, nine $1,987$ One thousand, nine hundred eighty-seven $0.684$ Six hundred eighty-four thousandths Zero point six eight four $17.04$ Seventeen and four hundredths Seventeen point zero four $0.1$ One tenth Zero point one $4.3$ Four and three tenths Four point three $0.0001$ one ten-thousandth Zero point zero zero zero one $5.000008$ five and eight millionths Five point zero zero zero zero zero eight $0.00073$ seventy-three hundred-thousandths Zero point zero zero zero seven three

## References and notes

1. Business Mathematics, 10th Edition. Authors are Charles D. Miller, Stanley A. Salzman, and Gary Clendenen. Published in 2006 by Pearson Education, Inc. ISBN 0-321-27782-1
2. Any thorough English arithmetic text discusses the reading of decimal numerals.

# Rounding

## Rounding

To round a number, you need to look at the digits right of the decimal. If the first digit right of the decimal is equal or higher than 5 in rank (see Spelling out numbers), we add 1 to the integer part of the number, lopping off digits right of the decimal; this is known as rounding up. If the first digit right of the decimal place is less than 5 in rank, we simply rewrite the number again excluding all the digits right of the decimal place; this is known as rounding down. Note that after the number is rounded up or down all of the digits right of the decimal are discarded and only the digits left of the decimal are written.

## Rounding to integer

The most basic form of rounding is to replace an arbitrary number by an integer.

There are many ways of rounding a number y to an integer q. The most common ones are

• round down (or take the floor, or round towards minus infinity): q is the largest integer that does not exceed y.
$q=\mathrm {floor} (y)=\left\lfloor y\right\rfloor =-\left\lceil -y\right\rceil \,$ • round up (or take the ceiling, or round towards plus infinity): q is the smallest integer that is not less than y.
$q=\mathrm {ceil} (y)=\left\lceil y\right\rceil =-\left\lfloor -y\right\rfloor \,$ .
• round towards zero (or truncate, or round away from infinity): q is the integer part of y, without its fraction digits.
$q=\mathrm {truncate} (y)=\operatorname {sgn}(y)\left\lfloor \left|y\right|\right\rfloor =-\operatorname {sgn}(y)\left\lceil -\left|y\right|\right\rceil \,$ • round away from zero (or round towards infinity): if y is an integer, q is y; else q is the integer that is closest to 0 and is such that y is between 0 and q. The signum function is used to determine the sign
$q=\operatorname {sgn}(y)\left\lceil \left|y\right|\right\rceil =-\operatorname {sgn}(y)\left\lfloor -\left|y\right|\right\rfloor \,$ • round to nearest: q is the integer that is closest to y.

The first four methods are called directed rounding, as the displacements from the original number y to the rounded value q are all directed towards or away from the same limiting value (0, +∞, or −∞).

If y is positive, round-down is the same as round-towards-zero, and round-up is the same as round-away-from-zero. If y is negative, round-down is the same as round-away-from-zero, and round-up is the same as round-towards-zero. In any case, if y is integer, q is just y. Here the table with the rounding methods:

y round
down
(towards −∞)
round
up
(towards +∞)
round
towards
zero
round
away from
zero
round
to
nearest
+13.67 +13 +14 +13 +14 +14
+13.50 +13 +14 +13 +14 +14
+13.35 +13 +14 +13 +14 +13
+13.00 +13 +13 +13 +13 +13
0 0 0 0 0 0
−13.00 −13 −13 −13 −13 −13
−13.35 −14 −13 −13 −14 −13
−13.50 −14 −13 −13 −14 −14
−13.67 −14 −13 −13 −14 −14

The choice of rounding method can have a very significant effect on the result.

# Truncating

Truncating is a simpler alternative to rounding. Digits to the right of the place required are just dropped — disregarding whether the first digit to the right is higher than 5 in rank or not.

## Examples

What is 98.57 truncated to the tenths place?

The 7 in the hundredths place is dropped. The answer is 98.5.

Truncate 33.504 to the units place.

All digits to the right of the units place (3) are dropped. the answer is 33.