# 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 (${\displaystyle +}$ )
• Subtraction (${\displaystyle -}$ )
• Multiplication (${\displaystyle \times }$ )
• Division (${\displaystyle \div }$ )

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 ${\displaystyle =}$  is not an operator.

When you see an ${\displaystyle =}$  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:

${\displaystyle 2+2=4}$

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 ${\displaystyle x}$  and ${\displaystyle y}$ , ${\displaystyle x+y=y+x}$  For example, the two equations end up with the same result.

${\displaystyle 4+5=9}$
${\displaystyle 5+4=9}$

### 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 ${\displaystyle x,y}$  and ${\displaystyle z}$ , ${\displaystyle (x+y)+z=x+(y+z)}$  For example, the two equations will end up with the same result.

${\displaystyle (1+2)+3=3+3=6}$
${\displaystyle 1+(2+3)=1+5=6}$

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:

${\displaystyle 4+0=4}$
${\displaystyle 6.24+0=6.24}$

# 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:

${\displaystyle 5-1=4}$

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.

${\displaystyle 5-2=3}$
${\displaystyle 2-5=-3}$

Here is another example using numbers with decimals.

${\displaystyle 4.3-1.2=3.1}$
${\displaystyle 1.2-4.3=-3.1}$

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 ${\displaystyle x}$  and ${\displaystyle y}$ ,

${\displaystyle x-y=-(y-x)}$

### 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.

${\displaystyle (1-2)-3=-1-3=-4}$
${\displaystyle 1-(2-3)=1-(-1)=1+1=2}$

# 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:

${\displaystyle 5*4=20}$

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 ${\displaystyle x}$  and ${\displaystyle y}$ , ${\displaystyle x*y=y*x}$  For example, the two equations end up with the same result.

${\displaystyle 3*7=21}$
${\displaystyle 7*3=21}$

### 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 ${\displaystyle x,y}$  and ${\displaystyle z}$ , ${\displaystyle (x*y)*z=x*(y*z)}$  For example, the two equations will end up with the same result.

${\displaystyle (2*3)*4=6*4=24}$
${\displaystyle 2*(3*4)=2*12=24}$

### The Multiplicative Identity

The number 1 is known as the multiplicative identity. This means if you multiply any number with one, you end up with that original number. For example:

${\displaystyle 4*1=4}$
${\displaystyle 6.24*1=6.24}$

# Number Operations/Division

Division is one of the four basic operations of arithmetic. The operation is usually shown by the division symbol (÷). This is the act of taking a number, and fragmenting it by using another number. The objective is usually to find how many times a number can fit in another number.

How many times can 2 fit into 4? This operation would be written as:

4 ÷ 2

In this case, 4 is the number we want to divide, and 2 is the number we want to divide 4 with. Thus, 4 is the dividend and 2 is the divisor.

It's important to note that this is a type of division titled: partition division.

# 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: ${\displaystyle \{1,2,3,\dots \}}$ . If we also include the number zero ${\displaystyle 0}$  in the set, it becomes the whole numbers: ${\displaystyle \{0,1,2,3,\dots \}}$ .

## 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 ${\displaystyle 0}$ . The way we can derive the next is to simply add one to the previous number. This is easily demonstrated: ${\displaystyle 0+1=1}$ , so zero's successor is one; ${\displaystyle 1+1=2}$ , so one's successor is two; ${\displaystyle 2+1=3}$ , 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:

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

Addition is the total of two numbers. For example:

${\displaystyle 1+2=3}$

## Subtraction

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

${\displaystyle 2-1=1}$

## Multiplication

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

${\displaystyle 2*1=2}$

${\displaystyle 2*2=2+2=4}$

${\displaystyle 2*3=2+2+2=6}$  etc.

## Division

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

${\displaystyle 6/2=3}$

# 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:

${\displaystyle 3-3=0}$

## 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:

${\displaystyle 9+1=10}$

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:

${\displaystyle (1*10)+(0*1)}$

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:

${\displaystyle (5*10)+(8*1)}$

that is to say:

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

Therefore, we can safely say that the position of the digits affects its value, that is, our number system (also known as decimal or base-10), is positional.

# 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

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:

${\displaystyle |-5|=5}$
${\displaystyle |0|=0}$
${\displaystyle |5+3|=|8|=8}$
${\displaystyle 36-|-55|=36-55=-19}$

# 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.[1][2] 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 ${\displaystyle 2,697,787.84}$ 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 ${\displaystyle 2,009}$ Two thousand, nine ${\displaystyle 1,987}$ One thousand, nine hundred eighty-seven ${\displaystyle 0.684}$ Six hundred eighty-four thousandths Zero point six eight four ${\displaystyle 17.04}$ Seventeen and four hundredths Seventeen point zero four ${\displaystyle 0.1}$ One tenth Zero point one ${\displaystyle 4.3}$ Four and three tenths Four point three ${\displaystyle 0.0001}$ one ten-thousandth Zero point zero zero zero one ${\displaystyle 5.000008}$ five and eight millionths Five point zero zero zero zero zero eight ${\displaystyle 0.00073}$ 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.
${\displaystyle q=\mathrm {floor} (y)=\left\lfloor y\right\rfloor =-\left\lceil -y\right\rceil \,}$
• round up (or take the ceiling, or round towards plus infinity): q is the smallest integer that is not less than y.
${\displaystyle q=\mathrm {ceil} (y)=\left\lceil y\right\rceil =-\left\lfloor -y\right\rfloor \,}$ .
• round towards zero (or truncate, or round away from infinity): q is the integer part of y, without its fraction digits.
${\displaystyle q=\mathrm {truncate} (y)=\operatorname {sgn}(y)\left\lfloor \left|y\right|\right\rfloor =-\operatorname {sgn}(y)\left\lceil -\left|y\right|\right\rceil \,}$
• 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
${\displaystyle q=\operatorname {sgn}(y)\left\lceil \left|y\right|\right\rceil =-\operatorname {sgn}(y)\left\lfloor -\left|y\right|\right\rfloor \,}$
• 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.

# Part I Review and Test

## Notes

Remember that:

Any time a number smaller than (LESS than) 0 is added to a number larger than (GREATER than) 0 you would use subtraction as shown below:

   5+(-3)=5-3, therefore 5+(-3)=2


Any number subtracted from itself always equals 0.

Any number above 9 will always have more than one digit.

The number line is very important in calculating numbers smaller than 0.

## Part I Test

Part 1 of 3.

1

 ${\displaystyle 2+2=}$

2

 ${\displaystyle 2+4=}$

3

 ${\displaystyle 5+3=}$
Part 2 of 3. Numbers greater than 9

4

 ${\displaystyle 10+11=}$

5

 ${\displaystyle 12+12=}$

6

 ${\displaystyle 14+10=}$
Part 3 of 3. Numbers smaller than 0

7

 ${\displaystyle -4-4=}$

8

 ${\displaystyle -2+5=}$

9

 ${\displaystyle 5-(-2)=}$
BONUS

10

 ${\displaystyle 4\times 5=}$

# Working with Integers

There are a number of ways to compare two arithmetic expressions. The first one can either be:

• equal to the second (=)
• greater than the second (>)
• less than the second (<)
• greater than or equal to the second (≥)
• less than or equal to the second (≤)

Examples:

5>2 (reads: "Five is greater than Two")
2<5 (reads: "Two is less than Five")
7=7 (reads: "Seven is equal to Seven")
a ≥ b (reads: "A is greater than or equal to B". This means A could be equal to B or greater than B.)
a ≤ b (reads: "A is less than or equal to B". This means A could be equal to B or less than B.)
• These are very important to understand before diving into solving equations with these symbols.

Addition operations are denoted by the + sign. The addition operator (plus sign) will take any two numbers, called addends, as operands to work on. The result is called the sum of the two numbers.

The operation of addition is commutative. This means that the addition of two numbers will give the same sum regardless of the order in which the numbers are added.

Examples:

• ${\displaystyle 7+5=12}$  and ${\displaystyle 5+7=12}$
• ${\displaystyle 3+2+4=9}$  and ${\displaystyle 4+3+2=9}$

To add vertically, stack the numbers on top.

 52
19
----------------


Now add the numbers in the first or right column. If you get a number with tens in it, put what's in the tens place on the tens place in the problem. You should have:

 1
52
19
----------------
1


 1
52
19
----------------
71


Repeat in a similar fashion for each additional column (hundreds, thousands, etc.)until you finish the problem.

## Subtraction

Subtraction, as you probably also have seen, uses the minus (-) sign. The generic subtraction operator will take any two numbers as operands. The result is called the difference of the two numbers.

Subtraction is not a commutative operation. Changing the order of the operands will likely give a different (not the same) result.

Example:

• ${\displaystyle 7-5=2}$  and ${\displaystyle 5-7=-2}$
• ${\displaystyle 3-2-4=-3}$  and ${\displaystyle 4-3-2=-1}$
• ${\displaystyle 36-5-11=36-11-5=20}$

## Exercises

In Exercises 1–25, find the sum.

1

 ${\displaystyle 5+3=}$

2

 ${\displaystyle 8+7=}$

3

 ${\displaystyle 9+2=}$

4

 ${\displaystyle 6+3=}$

5

 ${\displaystyle 1+4=}$

6

 ${\displaystyle 2+17=}$

7

 ${\displaystyle 12+11=}$

8

 ${\displaystyle 53+8=}$

9

 ${\displaystyle 41+9=}$

10

 ${\displaystyle 84+12=}$

11

 ${\displaystyle 16+17=}$

12

 ${\displaystyle 7,576+5,345=}$

13

 ${\displaystyle 2,345+3,245=}$

14

 ${\displaystyle 8,952+9,423=}$

15

 ${\displaystyle 2,783+2,389=}$

16

 ${\displaystyle 189,583+1,574,822=}$

17

 ${\displaystyle 1.5+2.7=}$

18

 ${\displaystyle 5.4+3.9=}$

19

 ${\displaystyle 8.3+9.2=}$

20

 ${\displaystyle 2.23+4.89=}$

21

 ${\displaystyle 534.4+34.675=}$

22

 ${\displaystyle 348.904+23,498.2=}$

23

 ${\displaystyle 1.673+48,210.38=}$

24

 ${\displaystyle 10.4823+94.29478=}$

25

 ${\displaystyle 128.52+2,070.24=}$
In Exercises 26–50, find the difference.

26

 ${\displaystyle 5-3=}$

27

 ${\displaystyle 8-7=}$

28

 ${\displaystyle 9-2=}$

29

 ${\displaystyle 6-3=}$

30

 ${\displaystyle 1-4=}$

31

 ${\displaystyle 2-17=}$

32

 ${\displaystyle 12-11=}$

33

 ${\displaystyle 53-8=}$

34

 ${\displaystyle 16-17=}$

35

 ${\displaystyle 84-12=}$

36

 ${\displaystyle 3,636-511=}$

37

 ${\displaystyle 7,576-5,345=}$

38

 ${\displaystyle 2,345-3,245=}$

39

 ${\displaystyle 8,952-9,423=}$

40

 ${\displaystyle 2,783-2,389=}$

41

 ${\displaystyle 1,574,822-189,583=}$

42

 ${\displaystyle 2.7-1.5=}$

43

 ${\displaystyle 5.4-3.9=}$

44

 ${\displaystyle 8.3-9.2=}$

45

 ${\displaystyle 2.23-4.89=}$

46

 ${\displaystyle 10.38-1.673=}$

47

 ${\displaystyle 534.4-34.675=}$

48

 ${\displaystyle 348.904-23,498.2=}$

49

 ${\displaystyle 10.4823-94.29478=}$

50

 ${\displaystyle 2,070.24-128.52=}$

# Multiplication and Division

## Multiplication

Multiplication is denoted by an asterisk (*), ${\displaystyle \times }$ , or ${\displaystyle \cdot }$  sign. However, the “×” sign is normally not used in algebra, and is instead limited to very basic elementary math, as it can easily be confused with an “x” variable. The generic multiplication operator will take any two numbers, called factors, as operands. The result is called the product of the two numbers. If the multiplicants are not both written as numbers, the multiplication sign can be left out. Thus, the following example expressions are equivalent:

${\displaystyle 3*\!(a*b)=3\times \!(a\times b)=3\!\cdot \!(a\!\cdot \!b)=3\!\cdot \!(ab)=3(a\!\cdot \!b)=3(ab)=3ab}$

Multiplication is a form of repeated addition. For example ${\displaystyle 3\times 5}$  means

${\displaystyle 3+3+3+3+3\quad \operatorname {or} \quad 5+5+5}$

Multiplication is also commutative. This means that the multiplication of two numbers (factors) will give the same product regardless of the order in which the numbers are multiplied together. The following expressions are also equivalent:

${\displaystyle 3\!\cdot \!(ab)=a\!\cdot \!(3b)=b\!\cdot \!(3a)}$

Numbers with exponents that are whole numbers larger than 1 indicate the number of factors to be multiplied, thus that number is multiplied by itself as many times as the exponent shows. Numbers with an exponent of 1 have only one factor, and therefore are equal to the number. Any number with an exponent of 0 has no factors at all, and the result is 1. Examples:

${\displaystyle 5^{3}=5\times 5\times 5\qquad \qquad 5^{1}=5\qquad \qquad 5^{0}=1}$

### Long Multiplication

Long Multiplication is the multiplication of numbers larger than 12, but usually only the facts from 1 through 9 are used. Before you attempt long multiplication, please make sure you know the facts 1 through 9. The others are optional, but makes long division a bit easier. The steps for the vertical multiplication method are:

1. Write the numbers down.

  52
19
------


2. Multiply 9 times 2. If there is a tens place for the answer, regroup. Multiply 9 times 5, add the regroup, and write the numbers down. You should have:

  1
52
19
------
468


3. Multiply 1 times 2, and 1 times 5, regrouping if needed, but this time shift the answer one space to the left. If you want to, you can put a zero under the 8 instead. You should now have:

  52
19
------
468
520


 52
19
-----
468
520
-----
988


If you are multiplying decimals, then multiply without the decimal point. Count the number of decimal places in both numbers, and add the number of decimal places. In the answer, count that number of spaces to the left. Put the decimal point there.

5. In a summary:

Write the problem down, vertically. Multiply the last digit of the second number to the last digit of the first number. Regroup if there is a tens place in the answer. Multiply it by the second to last number, and ADD the regroup. Repeat the process for the second to last digit of the second number, but put a 0 at the end of the line under the number you got before, if a third line put 2 0s, and repeat until the problem is done.

### Fast Multiplication

Fast Multiplication is the method in which you can reasonably multiply numbers greater than 10 and reasonably less than 1000 by simply multiplying by the method of "10's." This is done by recognizing how many digits there are in the numbers. Here are some steps which are useful for multiplying numbers really fast

1. See if you can recognize any zeros on the end and simply "add them" to your answer.

  45,300 x 5
The easy way to do this is "taking away" the 2 zeros for now and reserving them for later.

the number is now 453 x 5, which is much too mind boggling to do.
Now here comes the interesting part of the method
of "10's"


2. Break down the number into its "10's" parts

  What this means is basically breakup the number by its place value.

  453 = 4 (hundreds place) + 5 (tens place) + 3 (ones place)

  Knowing that, this become 400, 50 and 3.


3. Multiply and apply the "10's part"

  okay now simply multiply:

  So, here is a step where we essentially take out the "0's" out for a bit and put it back
in when were done.

  so, its now 5 x 400. in order to make it easier, "take out" the zeroes for now and
multiply 4 x 5 = 20. Now
heres the magic. Since you magically took away the 2 zeroes, you will now suddenly make
the 2 zeroes reappear!
20 + "00" = 2000! AMAZING! (the quotes means they're magic zeroes, and simply not the value
of zero!)

  50 is done the same away. Take away the "0" and multiply 5 x 5 = 25. Now add it back, 25 +
"0" = 250

  Simply 3 x 5 = 15


  2000
250
15
-----
2265!


5. Now take the two zeroes you reserved in the beginning (from the original 45,300), and tack them onto the end to get your answer: 226,500.

Step 2: Multiplying numbers that are not zero friendly

  2102 x 52


Using the step from before recognize that:

2102 = 2 (thousand) + 1(hundred) + 0 (tens) + 2 (ones)
52 = 5 (tens) + 2 (ones)

Now, to make it easier on yourself, circle the number 2 of "52" and put it in your magic
hat. (2)

Now the problem becomes 2102 x 50. Look familiar? First of all, take out the magic "0" and
put it in the hat, too.
Since we recognized that 50 is basically 5 with an added magical "0" to it, we now see the
problem as

2102 x 5!

Now break down the bigger, uglier number and start multiplying:
2000 x 5 (take away the magic zeroes) = 2 x 5 = 10 + "000"(now put them back!) = 10,000 (notice it has 4 zeroes)
100 x 5 (take away the magic zeroes) = 1 x 5 = 5 + "00"  (now put them back!) =   500 (2 zeroes)
2 x 5  (sadly, no magical zeroes)  = 2 x 5 = 10                             =    10 (1 zero)

 Remember, after every step, be sure to put your friendly magical "0" back in:
10,000 + "0" =100,000
500 + "0" =  5,000
10 + "0" =    100
(notice how the number of zeroes on the left side equal the number of zeroes on the right
side)

Now add them all together:

 100000
5000
100
-----
105100....... That's not all yet folks! Do you remember the 2 in your magic hat? Lets get it to work:

  2 x 2102 =
2000 x 2 = 2 x 2 = 4 = 4000
100 x 2 = 1 x 2 = 2 =  200
2 x 2             =    4
total:  4204

  So the answer should be
105,100
4,204
-------
109,304! Wow!


### Outside Resources

To practice this concept, I recommend Developmental Mathematics, volume 8, along with cards to memorize multiplication facts. This is my favorite for developing multiplication skills, but if you prefer something else, try Miquon Math. This is a complete program for grades 1-3 and is an excellent program for developing advanced math skills. Another option is Progress in Mathematics, a standards-based math program for Sadlier-Oxford. Of course, advance multiplication skills until the student/s is doing 5-digit multiplication with ease.

## Division

Division uses the ÷ sign. It may also be signified by the slash(/), :, or the fraction bar. The generic division operator will take any two numbers as operands. The number before the ÷ sign is called the dividend and the number after the ÷ sign is called the divisor. The result is called the quotient of the two numbers.

Division is not a commutative operation. Switching the dividend and the divisor will likely give a different quotient (but sometimes not). The division with a divisor of 0 is not defined (to be more precise, it's indeterminate, which means that there is more than 1 answer, therefore we don't give it a value). There is no answer for it.

Example:

• ${\displaystyle 4/2=2}$  and ${\displaystyle 2/4=0.5}$

### Long Division

In arithmetic, long division is an algorithm for division of two real numbers. It requires only the means to write the numbers down, and is simple to perform even for large dividends because the algorithm separates a complex division problem into smaller problems. However, the procedure requires various numbers to be divided by the divisor: this is simple with single-digit divisors, but becomes harder with larger ones.

A more generalized version of this method is used for dividing polynomials (sometimes using a shorthand version called synthetic division).

In long division notation, 500 ÷ 4 = 125 is denoted as follows:

${\displaystyle {\begin{matrix}\quad 125\\4{\overline {)500}}\\\end{matrix}}}$

The method involves several steps:

1. Write the dividend and divisor in this form:

${\displaystyle 4{\overline {)500}}}$

In this example, 500 is the dividend and 4 is the divisor.

2. Consider the leftmost digit of the dividend (5). Find the largest multiple of the divisor that is less than the leftmost digit: in other words, mentally perform "5 divided by 4". If this digit is too small, consider the first two digits.

In this case, the largest multiple of 4 that is less than 5 is 4. Write this number under the leftmost digit of the dividend. Divide the multiple by the divisor (in this case, 4 divided by 4) and write the result (in this case, 1) above the line over the leftmost digit of the dividend.

${\displaystyle {\begin{matrix}1\\4{\overline {)500}}\\4\end{matrix}}}$

3. Subtract the digit under the dividend from the digit used in the dividend (in this case, subtract 4 from 5). Write the result (remainder) (in this case, 1) underneath and in the same column, then drop the second digit of the dividend (in this case, the first zero) to the right of it. This gives you a new number to divide by the divisor.

${\displaystyle {\begin{matrix}1\\4{\overline {)500}}\\{\underline {4}}\\\;\,10\end{matrix}}}$

4. Now apply the same steps 2 and 3 to this new number, and write the results in the corresponding columns (in this case, the unit column is aligned with the second digit of the original dividend): multiple and remainder underneath the new number, and the answer above the line.

${\displaystyle {\begin{matrix}\,\,12\\4{\overline {)500}}\\{\underline {4}}\\\;\,10\\\quad {\underline {8}}\\\quad \;\,20\end{matrix}}}$

5. Repeat step 4 until there are no digits remaining in the dividend. The number written above the bar is the quotient, and the last remainder calculated is the remainder for the entire problem.

${\displaystyle {\begin{matrix}\quad 125\\4{\overline {)500}}\\{\underline {4}}\\\;\,10\\\quad {\underline {8}}\\\quad \;\,20\\\quad \;\,{\underline {20}}\\\qquad 0\end{matrix}}}$

## Is One a Factor of Everything?

To answer the question above, yes. 1 is a factor of any number n. For proof, consider this statement: "any number n is a factor of itself when multiplied by one".

Proof. Let n = 27, without loss of generality (sometimes written "WLOG", for short. Here, it means that you can choose whatever integer you want). Then, 27 * 1 = 27 since any number multiplied by 1 is itself.

Let's say you had a bag of 12 marbles. If you now have 2 bags of marbles, you have 24 marbles because each bag has 12 marbles (this is 2 * 12 = 24). However, if you only have 1 bag, you only have 12 marbles due to the same reasoning as before, that is, 1 * 12 = 12. Therefore, we conclude that any number n can be factored to n * 1.

58 million times 1? Answer: 58 million. The size of the number doesn't matter (as in this example: 1*(-245) = -245.) Though, negative numbers are beyond the scope of this page, but as you continue to increase your understanding of mathematics, you will learn of negative numbers and absolute value.

## Is Zero a Factor of Anything?

To answer the question above, no.

A number is a factor of another number if it can be multiplied by a whole number to give the number is it a factor of. Anything multiplied by zero is also zero, which means that the only number that could possibly have zero as a factor is zero itself.

Phrased differently, the multiples of 0 go 0, 0, 0, 0... and so on, never becoming any larger than 0. Since a factor is the reverse of a multiple, there are no numbers other than 0 with a factor of 0 as well.

## The Greatest Common Factor

The Greatest Common Factor (GCF) of two whole numbers, is the highest whole number that divides both original numbers.

## Finding the GCF of a number

1. Write down the prime factorisation of both numbers;
2. Find all equal prime factors;
3. Multiply the prime factors to get the GCF;

### Example

1. ${\displaystyle 28=2\times 2\times 7}$
${\displaystyle 98=2\times 7\times 7}$
2. Notice that two and seven are used in both the equations?
3. ${\displaystyle 2\times 7=14}$
Therefore, the GCF of 28 and 98 is 14.

If both numbers don't have equal prime factors, then the GCF is 1 (since every whole number is a multiple of 1, that is, n = n * 1, as we saw previously).

## The Least Common Multiple

As the name implies, the Least Common Multiple (LCM) of two whole numbers, is the smallest (least) shared (common) factor (multiple) by both numbers.

1. Write down the prime factorisation of both numbers;
2. Find the smallest prime factor for each one of them;
3. If they are equal, then the LCM is equal to that factor;

### Example

1. ${\displaystyle 28=2\times 2\times 7}$
${\displaystyle 98=2\times 7\times 7}$
2. Notice that both numbers have 2 as their smallest prime factor.
3. Therefore, the LCM of 28 and 98 is 2.

If all prime factors are different, then the LCM is 1 (again, because every whole number is a multiple of 1).

# More About Multiplication/Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic is stated as follows:

Every natural number, except 1, is the product of one or more prime numbers.

Very simple, right?

# Properties of Operations

## Why Do Operations Have Properties?

Before continuing to learn about the properties of operations you must understand two basic questions:

What is the point of operations having properties?
Without operations having properties, we would not know their usage. Understanding their usage helps lay the foundation for solving word problems later on.

Why do operations have properties?
Operations have properties which define their usage. The usage of operations is the very essence of them, without usage properties are useless (vice versa).

By understanding properties we are able to enter the realm of higher-level thinking.

This is because properties illustrate general cases which allow us to lead to more mathematical generalizations.

Definitions of Mathematical Properties:

1. Commutative Property: ${\displaystyle a+b=b+a}$
2. Associative Property: ${\displaystyle (a+b)+c=a+(b+c)}$
3. Identity Property of zero: ${\displaystyle 0+a=a(=a+0)}$
4. Inverse Property: For every ${\displaystyle a}$ , there exists a ${\displaystyle -a}$  such that ${\displaystyle a+(-a)=0}$

Multiplication:

1. Commutative Property: ${\displaystyle ab=ba}$
2. Associative: ${\displaystyle (ab)c=a(bc)}$
3. Identity: ${\displaystyle a\cdot 1=1\cdot a=a}$
4. Inverse Property, for every ${\displaystyle a\neq 0}$ , there exists a ${\displaystyle ({\dfrac {1}{a}})}$  (or ${\displaystyle a^{-1}}$ ), such that ${\displaystyle {a}({\dfrac {1}{a}})=1.}$

In general, the inverse property of multiplication can be stated more rigorously as saying that two numbers are multiplicative inverses of each other, if and only if their product is 1.

Remember that combining addition and multiplication yields the...
Distributive Rule: ${\displaystyle a(b+c)=ab+ac.}$

Most rules can be derived by paying attention to expressions and clever use of the properties.

Example:

Show that ${\displaystyle a\cdot 0=0}$ .

Proof.
${\displaystyle a=a\cdot 1=a\cdot (1+0)=a\cdot 1+a\cdot 0.}$

is equivalent to:

${\displaystyle a=a+a\cdot 0}$
subtract a from both sides, yielding

${\displaystyle a\cdot 0=0}$ .

QED

## Properties of Addition and Subtraction

Sure, you've added 4 + 3 = 7, but have you tried 3 + 4? You get the same answer right? Yes, because of the commutative law of addition.

What about 5 + 6 = 11? That's right, 6 + 5 = 11 as well.

There is another property of addition, the associative property. Here is an algebraic example:

For any numbers ${\displaystyle a,b,c}$ : ${\displaystyle a+(b+c)=(a+b)+c}$

Can the same be said about subtraction? Well, let's try it... 7 - 5 = 2, does 5 - 7 = 2? Well... no. If you look at the number line, you will notice that when you do 5 - 7, you go below 0 into the realm of the negative numbers. Specifically, your solution is -2. Even though -2 is same in absolute value as 2, it isn't the same number. Therefore, subtraction is not commutative. Is it associative? No, because the associative law depends on the commutative law in order to work (since it really is just an extension of the commutative law.)

### Exercises

1

 8 + 9 =

2

 50 + 30 =

3

 45 + 9 =

4

 36 + 11 =

5

 2 + 3 =

6

 (5 + 7) + (4 + 3) + (8 + 2) + 5 =

7

 6 - 3 =

8

 77-11 =

9

 66 - 5 =

10

 (80 - 13) - (36 - 5) - 11 + (36-11) =

## Associative property

The associative property of real numbers is: ${\displaystyle (a+b)+c=a+(b+c)}$  for all real numbers ${\displaystyle a,b,c}$ . This implies that the order in which addition is done does not matter.

Note however that this only applies to addition and multiplication. In fact, multiplication shares the same property. ${\displaystyle (ab)c=a(bc)}$  for all real numbers ${\displaystyle a,b,c}$ .

The position of the parenthesis does change but the order by which you operate does not.

# Order of Operations

## Order of operations

The order of operations is the order in which all algebraic expressions should be simplified. Oftentimes, the meaning of a complex expression changes depending upon the order in which it is calculated. The order of operations is:

Parentheses means brackets()
Exponents (and Roots) means power
Multiplication & Division

EXAMPLE: 2 + 2 × 5 is equal to 12

This means that expressions within parentheses are evaluated first, then exponents (including roots, i.e. radicals), then multiplication and division (at the same level), and finally addition and subtraction (at the same level). If there are multiple operations at the same level on the order of operations, move from left to right.

There is a number of different abbreviations for memorizing the order. PEMDAS, BEDMAS and BODMAS (B is Brackets) are common. Another common way to remember the order is the mnemonic

• "Please Excuse My Dear Aunt Sally,"

with the beginning letters standing for each operation. Whichever mnemonic you use, be aware that multiplication does not always come before division, and addition does not always come before subtraction. For example:

If you have an expression like

• 3 × 3 - 5 + 2

you work like this: First notice that, there are no Parentheses or Exponents, so we move to Multiplication and Division. There's only the one multiplication, so we do that first and end up with 9 - 5 + 2. Now we move to Addition and Subtraction, working left to right. So firstly we do the subtraction to get 4 + 2, and finally the addition to give 6. If we had blindly done the addition first, we would have got the answer 2, which is wrong!

The rationale for the grouping (apart from parentheses, which are obviously first) is that multiplication is repeated addition and exponentiation is repeated multiplication. Also, division is multiplication by the reciprocal and subtraction is addition of the negative, so these operations are equivalent. In fact PEMA would be a better phrase ("Please Excuse My Aunt"), but in lower arithmetic courses MDAS is often taught without explaining reciprocals.

Parentheses are curved symbols, ( and ), that are put around part of an expression in order to convey that the expressions inside them should be evaluated first. Within a set of parentheses, the order of operations should be followed. Square brackets, [ and ], are sometimes used around parentheses to avoid confusion: [(3+5)×2]2 means the same as ((3+5)×2)2. The fraction bar and radical bar (often called a vinculum) groups expressions like parentheses.

For example, the expression 2×(6+7)-82 should be solved in the following order:

 2×(6+7)-82 {first compute the expression inside the parentheses (6+7)} = 2×(13)-82 {second, calculate the exponent 82} = 2×(13)-64 {third, calculate the multiplication 2×(13) = 26-64 {finally, calculate the subtraction} = -38 {our final answer}

If the desired order for solving the expression were different (based on the initial problem), parentheses would be positioned differently, or even omitted.

The meaning of the fraction and radical bars must be deciphered carefully. The part of the expression directly below or above the bar is to be treated as parenthesised. (Care must be taken in writing expressions with a bar.)

The expression ${\displaystyle {\sqrt {a+b}}\times c}$  means c times the root of a + b, not the root of a + b × c or even the root of c times the sum a + b, since the bar is above the a and the b, but not the c.

The expression ${\displaystyle {\frac {4+5}{1+2}}}$  could be written in one line as
(4+5)/(1+2) = 9/3 = 3, not as 4+5/1+2 = 4+5+2 = 11. As you see, the expression above the bar is evaluated, as is the expression below the bar. Finally we can divide.

Because of order of operations -22 = -(22) = -4, not (-2)2 = +4: the negative sign can be considered to have an implicit 0 in front, making the expression 0 - 22.

When it comes to distributing a power, use the raise a power to a power rule. Example: (xy^2)to the fourth power (^4) =(x)^4 (y^2)^4 =x^4×y^8 (originally it was y^6; this would only be true for y^2 * y^4, where you add the exponents)

The order of operations is very important, and you must remember the order when using simple calculators. Expressions such as 2+3 × 5 vary on the order used. Entering [2] [+] [3] [×] [5] on most calculators would result in adding three to two and then multiplying by five, resulting in 25; the proper evaluation sequence would be 2+(3×5), multiplying three and five and then adding that to two to get 17. Some scientific calculators and most graphing calculators use the proper order of operations, but four-function calculators typically use "left-to-right" evaluation, which can return incorrect results.

# Estimation

## Estimation

Estimation involves working out a rough answer to a calculation. The most common way to estimate a solution to a calculation is to round the numbers up or down to numbers which are easier to calculate with.

An example: Estimate the answer to 99 × 9. In this case we can easily see that 9 is almost 10, and multiplying by 10 is really easy, so we can replace our calculation with 99 × 10 which is far more easy to calculate. As we rounded a number up we know that our estimate is going to be larger than the real answer.

Why don't we just call this a guess? The difference is that a guess is just that, a completely wild guess. An estimate is based on some extra information. So whilst you might guess that
99 × 9 is something around 1000 by just pulling a number out of the air, we estimate that 99 × 9 is close to 99 × 10, then we work out 99 × 10 exactly, which gives 990 as an estimate of
99 × 9.

When we estimate, we want our estimate to be close enough to the actual value so as to be useful. (This also applies to approximation.) In our example above, the answer deviates from the actual value by about 10%, which might not be acceptable. Estimation often involves some guesswork, so we might not actually know how accurate our estimate is. (And yes, that figure 10% is itself an estimate.)

A better way to estimate 99 × 9 is to say that 99 is close to 100, then we work out 100 × 9 exactly, which gives 900 as an estimate of 99 × 9. This estimate is 1.0101...% off. The reason this is a better estimate is that 99 deviates from 100 much less than 9 deviates from 10 in percentage terms (although the absolute difference is one in each case).

1

 36+11=

2

 44+42=

3

 45+38=

4

 46+79=
Minus.

5

 36-11=

6

 66-5=

# A Look Ahead at the Decimal System

So far, every number we have discussed has been an integer. What if something costs between two and three dollars? This is when decimals are used. A decimal point, which looks like a full stop (or period), separates the integral part of the number from the decimal part. Instead of each number being ten times as large, they become ten times as small.

• 1.5 : 1 + (5 ÷ 10)
• 3.85 : 3 + (8 ÷ 10 + 5 ÷ 100)

# Introduction to Exponents

## What Are Exponents

Exponents are a short hand that tells us how many times we should multiply a number with itself. If we have to write out every multiplication expressions could get quite long. For example, the number 5×5×5×5×5×5 is often shortened to 56. The number 6 is called an exponent. It tells us how many times we should multiply 5 by itself.

## A Focus on Squaring Numbers

In multiplication, there are some common names when we use small numbers. For example, multiplying by two is called "doubling", and by three is "tripling". In the same way, exponentiating with two is called "squaring" and by three is called "cubing".

So, just how ${\displaystyle 5*2=5+5}$ , ${\displaystyle 5^{2}=5*5}$ . Overall, ${\displaystyle x^{2}=x*x}$ . So, 6^2 is 6*6 (36), and 10^2 is 10*10 (100).

You might also notice that the square of a number, as we have learned it, is always positive. For example, 2 squared is 2*2 which is 4; at the same time, -2 squared is -2*-2, which is also 4. Actually, (-x)^2 and (x)^2 equal the same thing.

## A Focus on Cubing Numbers

Unlike squaring numbers, cubing them (raising them by three) preserves the original numbers' sign. For example, ${\displaystyle 2^{3}=2*2*2=8}$ , while ${\displaystyle (-2)^{3}=-2*-2*-2=-8}$ .

In fact, you might notice that when you raise a number to any even power (2, 4, 6, ...), the sign must be positive, whereas when you raise it to an odd degree (1, 3, 5, 7, ...) the sign will be the same as the original number's.

## Properties of Exponents

There are, however, several useful tricks for manipulating exponents. Say we have some powers of ${\displaystyle x}$ , for example, ${\displaystyle x^{a}}$  and ${\displaystyle x^{b}}$ . If we want to figure out the product of these, ${\displaystyle x^{a}*x^{b}}$ , we can see that it is actually ${\displaystyle x}$  multiplied by itself ${\displaystyle a}$  times, multiplied by itself ${\displaystyle b}$  more times.

In other words, there are a x's in the first term, and b x's in the second, and multiplying the two means that there will be a+b x's in the answer. Therefore, ${\displaystyle x^{a}*x^{b}=x^{a+b}}$ .

In the same way, ${\displaystyle \left({\frac {x^{a}}{x^{b}}}\right)}$  means that there are a x's on the top, divided by b x's on the bottom. As you may see, some of the x's on the bottom will cancel those on the top, particularly, b. This means that the fraction ${\displaystyle \left({\frac {x^{a}}{x^{b}}}\right)}$  will turn to ${\displaystyle \left({\frac {x^{a-b}}{1}}\right)=x^{a-b}}$

To find out what radicals are, let's look at what multiplication and division are. Multiplication is simply adding two numbers a certain number of times, i.e. ${\displaystyle 5*4=5+5+5+5}$ . Though we can't think of division in this way usually, we know that it is the "opposite" of multiplication.

This means that if we multiply some number by, say, 2, and then divide it by two, we will get that same number. It is in this way that multiplication and division "undo" each other. Mathematically, we could say ${\displaystyle x*a\div a=x}$ , or "x times a, divided by a, equals x".

So, radicals are like division to exponents; in the same way that division undoes multiplication, and vice versa, radicals undo exponents. Radicals are usually written like ${\displaystyle {\sqrt {x}}}$ .-

## A focus on square roots

So, we know that ${\displaystyle 3^{2}=9}$ ; this means, also, that ${\displaystyle {\sqrt {9}}=3}$ . In the same way, ${\displaystyle {\sqrt {16}}=4}$ , since 4 squared equals 16.

## A focus on cube roots

In the same way, we know that ${\displaystyle 2^{3}=2*2*2=8}$ , thus ${\displaystyle {\sqrt[{3}]{8}}=2}$ . Similarly, ${\displaystyle {\sqrt[{3}]{27}}=3}$ .

## Domains: What is the square root of 2?

You may be wondering how to take the square root of numbers like 2 or 3. Surely there isn't any integer that, when you multiply it by itself, you'll get two. So what could it be? Possibly a fraction?

Actually, no. The square root of two is the square root of two. There is no fraction that exactly equals it. The same is true for the square roots of 2, 5, 6, 8, etc.

## Another look ahead at irrational numbers

There are actually many numbers that can't be represented as fractions (actually, there are as many of these numbers as those that can be represented with fractions). These numbers are called irrational numbers, since they can't be made into a "rational" fraction.

## Domains: What is the square root of a negative number?

If you remember from the exponents chapter, we said that any number squared, to the fourth, or any even number, is positive. Square root finds that number, given its square. But what about numbers like -1, -2, or -543?

Well, we know that no rational, or irrational, number squared equals a negative number. That much is true. But, just like how we extended rational numbers to the irrationals as well when we tried to square root 3, we're going to have to extend our notion of rationals and irrationals (which, combined, make the "real" numbers). This group of numbers that, squared, gives a negative number is called the imaginary numbers.

Just as how ${\displaystyle {\sqrt {2}}}$  cannot be written as a rational fraction, ${\displaystyle {\sqrt {-1}}}$  cannot be written as a real number. It is simply ${\displaystyle {\sqrt {-1}}}$ . For shortness, we call this number ${\displaystyle i}$ , which stands for "imaginary" (since you can't have "i" potatoes, or "i" anything, really).

# Bases/Another Look at Decimal (Base 10)

In order to understand bases, you should look at decimal to see how they work first.

For example, look at the number 123:

The 1 in the 100s place means 1 x 100
1 x 100 = 100

The 2 in the 10s place means 2 x 10
2 x 10 = 20
100 + 20 = 120

The 3 in the 1s place means 3 x 1
3 x 1 = 3
100 + 20 + 3 = 123

Notice how 100 = 102, 10 = 101, and 1 = 100.

# Fractions

## Fractions

A fraction consists of one quantity divided by another quantity. The fraction "three divided by five" or "three over five" or "three fifths" can be written as:

${\displaystyle {\frac {3}{5}}}$
or as: 3/5

In this section, we will use the notation 3/5. Note that some fractions have special names.

1/2 - Instead of calling this a “twoth”, or a ”second”, it's called a half.
3/4 - As well as three fourths, it can also be called three quarters.

### Numerator and Denominator

The first quantity, the number on top of the fraction, is called the numerator. It tells you the number, or how many of something you've got. The other number, on the bottom, is called the denominator. The denominator tells you the denomination of the fraction, which is really just a fancy way of telling you the type of the fraction. In exactly the same way, we know a £5 note and a £10 pound note are different because they are different types.

So, look at a number like 3/5. The numerator is 3. So whatever we've got, we've got three of them. The denominator is 5, so we've got fifths, whatever they are. Put the two together, we've got three of those things called fifths. We've got three fifths, we've got 3/5. Same value, different appearance.

Several rules for calculation with fractions are useful:

## Changing the type of a fraction

If both the numerator and the denominator of a fraction are multiplied or divided by the same number, then the fraction does not change its value.

This means we can make some fractions simpler, by making the numbers involved smaller. When the numbers are as small as possible, the fraction is said to be expressed in its lowest terms, or reduced. In a reduced fraction, the numerator and denominator are not divisible do not share any common divisors greater than one. Let's look at an example:    4/6 = 2/3.

Remember what that ' = ' sign means? It means that 4/6 and 2/3, although they look different, have the same value. Same value, different appearance.

This is because

${\displaystyle {\frac {4}{6}}={\frac {4/2}{6/2}}={\frac {2}{3}}}$ .

Try it out yourself with a calculator or with long division. From here, there is nothing we can divide both numbers by, so the fraction is reduced, which is our goal. To find out what to divide them by, you need to find a common divisor, which is a number they are both divisible by.

## Mixed fractions

Sometimes you may meet 'mixed fractions' like ${\displaystyle 1{\frac {3}{5}}}$ . This really means ${\displaystyle 1+{\frac {3}{5}}}$ , or in words one and three-fifths. Mixed fractions are hard to work with... how would you go about working out the following?

${\displaystyle 1{\frac {3}{5}}\times 2{\frac {2}{7}}}$

The answer is to convert the mixed fraction into a standard fraction of the form numerator/denominator. For example,

${\displaystyle 1{\frac {3}{5}}=1+{\frac {3}{5}}={\frac {5}{5}}+{\frac {3}{5}}={\frac {8}{5}}\,\,\,\,}$

and

${\displaystyle 2{\frac {2}{7}}=2+{\frac {2}{7}}={\frac {14}{7}}+{\frac {2}{7}}={\frac {16}{7}}}$

Now, it's simple to multiply the fractions together:

${\displaystyle 1{\frac {3}{5}}\times 2{\frac {2}{7}}={\frac {8}{5}}\times {\frac {16}{7}}={\frac {128}{35}}={\frac {105}{35}}+{\frac {23}{35}}=3{\frac {23}{35}}}$

Keep in mind that with mixed fractions, the whole number represents the quotient whereas the fraction represents the remainder. If you were to divide 25 by 5, you would be left with only 5 (and no remainder). However, if you were to divide 27 by 5, you would be left with 5 2/5.

## Exercises

Which of the following is the same as ${\displaystyle 3{\frac {3}{5}}}$ ?

 5/18 9/5 18/3 18/5 11/5

Before we go into fractions, let's have a think about what addition is.

Answer these really simple questions; and don't forget units!

1 What is 1 bird + 5 birds?

2 What is 3 elephants + 9 elephants?

3 What is 6 birds + 2 elephants?

You probably did all this in your head without even thinking about it - so what has it all got to do with fractions?

To add or subtract two fractions, you first need to change the two fractions so that they have the same type. The simplest way to do this is to multiply the numerator and denominator of each fraction by the denominator of the other.

For instance,

${\displaystyle {\frac {2}{3}}+{\frac {1}{4}}={\frac {2\times 4}{3\times 4}}+{\frac {1\times 3}{4\times 3}}={\frac {8}{12}}+{\frac {3}{12}}={\frac {(8+3)}{12}}={\frac {11}{12}}}$

A more advanced way is to use the LCM of the denominators, which will be explained later in this section. Then you can add or subtract the numerators and put the common denominator as the denominator of the solution.

There are two things you should know before you learn about how to add mixed numbers.

This is a mixed number, 5 3/4

We will be using 5 3/4 and 3 2/3 in our problem

Set = one mixed number

To add two mixed numbers, first you can turn the mixed number into an improper fraction. To do that you multiply the whole number by the denominator of your fraction. For this problem you would get 20 for the first set and 12 for the second set. Before you move on don’t forget to add the numerator to your other number such as 20+3 for the first problem. After that you change those numbers into fractions like this 23/4 and 12/3. Next you add them. You would get 35/9 after you add in this problem. Finally, you can turn that back into a mixed number by dividing the denominator by the numerator. After you divide you should get 3 8/9 as your mixed number. Also, no you cannot simplify. Now you know how to add mixed numbers!

# Multiplying Fractions

## Multiplying fractions

To multiply two fractions:

• multiply the numerators to get the new numerator, and
• multiply the denominators to get the new denominator.

For instance,

${\displaystyle {\frac {2}{3}}\times {\frac {1}{4}}={\frac {2\times 1}{3\times 4}}={\frac {2}{12}}={\frac {1}{6}}}$ .

## Dividing fractions

To divide one fraction by another one, flip numerator and denominator of the second one, and then multiply the two fractions. The flipped-over fraction is called the multiplicative inverse or reciprocal.

For instance,

${\displaystyle \left({\frac {2}{3}}\right)/\left({\frac {4}{5}}\right)={\frac {2}{3}}\times {\frac {5}{4}}={\frac {2\times 5}{3\times 4}}={\frac {10}{12}}={\frac {5}{6}}}$ .

To simplify a compound fraction, like ${\displaystyle {\frac {\left({\frac {3}{5}}\right)}{\left({\frac {1}{4}}\right)}}}$ , just remember that a fraction is the same as division, and divide (3/5) ÷ (1/4), which comes to 12/5.

# Percents Decimals and Fractions

## Conversions between percents, decimals and fractions

The perhaps easiest way to convert from:

Decimal to fractions is by simply moving the period over 2 and putting it over a 100

(ex) Express .50 as a fraction

   .500   = 50.0 = 50/100 = 1/2
. >. (this means the moving of the dot)


usually the best way to set it all up is the following

                             ------------------------------
| fraction | Decimal | percent |
|    ?     |   ?     |   50%   |
------------------------------
(move 2 to the right)    <--   x       -->(move dot over 2 to the right)
(and put over 100)

When converting decimal to fraction                  When converting from decimal to percent
always be sure it has 2 place to the                 this always works
right of the (.)                                     ? = 50.% if its not known, do the "reverse"
(ex)   .5 = .50   = ? =  50                         .50 <--.
.-->   ---  ---
100  100


When reading decimals, it is similar to verbalizing a fraction. For example, 0.2 is spelled out as "two-tenths", hence 2/10, which simplifies to 1/5. What if you had the real number 0.002? Same principle: "two-thousandths" implies 2/1000, which reduces to 1/500.

Previous: Grammar
Next: Simple symbolic manipulation

# Lowest Common Multiple

To find the Lowest Common Multiple (LCM) of several numbers, we first express each number as a product of its prime factors.

For example, if we wish to find the LCM of 60, 12 and 102 we write

${\displaystyle {\begin{matrix}60=2^{2}\cdot 3\cdot 5\\12=2^{2}\cdot 3\\102=2\cdot 3\cdot 17\end{matrix}}}$

The product of the highest power of each different factor appearing is the LCM.

For example in this case, ${\displaystyle 2^{2}\cdot 3\cdot 5\cdot 17=1020}$ . You can see that 1020 is a multiple of 12, 60 and 102 ... the lowest common multiple of all three numbers.

Another example: What is the LCM of 36, 45, and 27?

Solution: Factorise each of the numbers

${\displaystyle {\begin{matrix}36=2^{2}\cdot 3^{2}\\45=5\cdot 3^{2}\\27=3^{3}\end{matrix}}}$

The product of the highest power of each different factor appearing is the LCM, i.e.;

${\displaystyle 2^{2}\cdot 5\cdot 3^{3}=540}$

## Properties of the LCM

If the LCM of the numbers is found and 1 is subtracted from the LCM then the remainder when divided by each of the numbers whose LCM is found would have a remainder that is 1 less than the divisor. For example if the LCM of 2 numbers 10 and 9 is 90. Then 90-1=89 and 89 divided by 10 leaves a remainder of 9 and the same number divided by 9 leaves a remainder of 8.

# Highest Common Factor

The highest common factor (HCF) or greatest common divisor (GCD) can be found in a similar fashion. In this case, the product of the prime factors common to all the numbers gives the HCF.

For example, if we wish to find the HCF of 60, 12 and 102 we write

${\displaystyle {\begin{matrix}60=2^{2}\cdot 3\cdot 5\\12=2^{2}\cdot 3\\102=2\cdot 3\cdot 17\end{matrix}}}$

Now the HCF is ${\displaystyle 2\cdot 3=6}$ .

Two numbers which have an HCF of 1, for example 12 and 5, are said to be coprime - they have no factors in common (except 1).

# Absolute Values

## Absolute value

The absolute value of a number is found by applying a simple rule: If you see a negative sign in front of the number, change it to a plus sign. If you see a plus sign, leave it alone. So, for example, the absolute value of -17 is +17. The absolute value of +36 is +36.

Another way to understand the absolute value of a number is to think about the number line:

The absolute value of a number is the distance from zero to that number on the number line.

The absolute value of x is usually written as |x|. On calculators and computers it is sometimes written as abs (x).

Questions: (note: if you don't see an answer box, do it on paper)
Calculate the absolute value of the following numbers:

1

 |-5|=

2

 |9|=

3

 |-3.8|=

4

 |-139,462|=

5

 |5/8|=

6 What is the absolute value of 0?

 |0|=

7 Why?

Calculate the following:

8

 |27|=

9

 |-1.9|=

10

 |3 - 7|=

11

 |36| - |-11|=

12

 |3 - 0.5|=

13

 abs (-6)=

14 Draw a graph of abs(x) from -5 to +5.

15 Can abs(x) ever be less than zero?

 yes no

16 How can you see that from your graph?

# Exponents

## Exponents

Exponents or 'powers' are a process of repeated multiplication, in much the same way that multiplication is a process of repeated addition.

Exponents are normally written in the form ${\displaystyle a^{b}}$ , where ${\displaystyle a}$  is the base and ${\displaystyle b}$  is the exponent. In contexts where superscripts are not available, such as in many contexts in computers, ${\displaystyle a^{b}}$  is commonly written as "a^b" or less often as "a**b". If you're not familiar with algebra, you can just imagine the letters a and b as representing numbers. We pronounce ${\displaystyle a^{b}}$  as a to the power of b, a to the b or a exponent b.

### Integer exponents

When the exponent is a positive integer, then it is just a simple case of multiplying the base by itself a certain number of times. For example,

${\displaystyle 3^{4}=3\times 3\times 3\times 3=81\,}$

Here, 3 is the base, 4 is the exponent (written as a superscript), and 81 is 3 raised to the 4th power. Notice that the base 3 appears 4 times in the repeated multiplication, because the exponent is 4.

Some more examples:

${\displaystyle {\begin{matrix}12^{2}=12\times 12=144\\2^{8}=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=256\\1^{5}=1\times 1\times 1\times 1\times 1=1\end{matrix}}}$

#### Multiplying exponents

If you have two or more exponents with the same base, then multiplying them has the same effect as adding their exponents.

For instance ${\displaystyle (a^{b})*(a^{c})\,}$  is the same as ${\displaystyle a^{b+c}\,}$ . For example,

${\displaystyle (3^{4})*(3^{2})=(3*3*3*3)*(3*3)=(3*3*3*3*3*3)=(3^{6})=(3^{4+2})\,}$

#### Dividing exponents

If you have two or more exponents with the same base, then dividing them has the same effect as subtracting their exponents.

For instance ${\displaystyle (a^{b})/(a^{c})\,}$  is the same as ${\displaystyle a^{b-c}\,}$ . For example,

${\displaystyle (3^{6})/(3^{2})=(3*3*3*3*3*3)/(3*3)=(3*3*3*3)=(3^{4})=(3^{6-2})\,}$

#### Exercises

What is?

1

 ${\displaystyle 4^{3}=}$

2

 ${\displaystyle 3^{4}=}$

3

 ${\displaystyle 1^{250}=}$

4

 ${\displaystyle {250}^{1}=}$
Write these numbers as powers of 2

5

 ${\displaystyle 128=2^{\wedge }}$

6

 ${\displaystyle 8=2^{\wedge }}$

7

 ${\displaystyle 1024=2^{\wedge }}$

8 What is?

 ${\displaystyle (2^{3})*(2^{2})=}$

9 What is?

 ${\displaystyle (2^{6})/(2^{2})=}$

10 Harder:

 Why does ${\displaystyle 3^{0}=1\,}$  (clue: think about ${\displaystyle {3^{2}}/{3^{2}}\,}$ , for example) (answer on paper)

#### Negative exponents

Negative exponents work slightly differently. Let's say you want to calculate ${\displaystyle 3^{-2}}$ . To do that, you take ${\displaystyle 1/3^{2}}$  to get your answer. We do the exponent first, see Order of Operations

${\displaystyle 3^{-2}={\frac {1}{3^{2}}}={\frac {1}{9}}}$

The commutative property doesn't apply in exponents. See for yourself! Try to calculate 23, and then see if it is the same as 32 (answer here). The distributive and associative properties don't apply either.

Exponents do, however, have their own set of axioms that they consistently follow. Consistent with the preceding examples, one can state generally that:

${\displaystyle (a^{b})\times (a^{c})=a^{b+c}\,\,\,\,}$  and
${\displaystyle (a^{b})/(a^{c})=a^{b-c}\,\,\,\,}$

It's also easy to see that ${\displaystyle {\begin{matrix}(a^{b})^{c}=a^{b\times c}\end{matrix}}}$

### Fractional exponents

So far, we have only seen exponents as whole numbers, but exponents can be fractional as well. With a fractional exponent, the numerator acts as a normal whole-number exponent, while the denominator acts as a root.

In general, ${\displaystyle a^{p/q}={\sqrt[{q}]{a^{p}}}\,\,\,\,}$  for any real number ${\displaystyle q}$  ≠ 0.

Let's look at ${\displaystyle 8^{2/3}}$  as an example. First, we raise 8 to the power of the numerator, 2. Then, since the denominator is 3, we take the third root of this number. The expression is read as the cubed root of eight squared, and written as:

${\displaystyle 8^{2/3}={\sqrt[{3}]{8^{2}}}={\sqrt[{3}]{64}}=4}$

It should then be evident that when the numerator of the fractional exponent is 1, the expression is a simple root. That is, ${\displaystyle 1/2}$  is a square root, ${\displaystyle 1/3}$  is a cubed root, ${\displaystyle 1/4}$  is a fourth root, etc.

For example, ${\displaystyle 9^{1/2}}$  would be read as the square root of nine, and written as:

${\displaystyle 9^{1/2}={\sqrt[{2}]{9^{1}}}={\sqrt {9}}=3}$

A radical is a special kind of number that is the root of a polynomial equation. First, let us look at one specific kind of radical, the "square root". It is a special kind of number related to squaring.

When we have a number, say 2, what positive number will give us, when we square it, the number 2?

• 1 × 1 = 1, so the number must be higher than 1.
• 2 × 2 = 4, so the number must be lower than 2.
• 1.5 × 1.5 = 2.25, so the number must be lower than 1.5.

We could continue like this forever, and with each step get closer and closer to the answers (this kind of process for refining the answer is called iteration). The number we are looking for is approximately 1.41421...

Obviously this is very difficult for us to work with, so we have a special notation. We write for a number a, ${\displaystyle {\sqrt {a}}}$  to represent the number when squared will give us a back.

Since this is the inverse operation from squaring, it can also be denoted as a1/2 and
${\displaystyle (a^{1/2})^{2}=a^{1}=a}$ .

We can extend this idea to other kinds of radicals. ${\displaystyle {\sqrt[{3}]{a}}}$  indicates the number ${\displaystyle x}$  such that
${\displaystyle x^{3}=a}$ . For example, ${\displaystyle {\sqrt[{3}]{7}}=1.91293...}$

Note that it is not possible to find any real number whose square would be negative. Multiplying with a negative number changes the sign of the number being multiplied and two negative signs thus eliminate each other. For example, -7 × -7 = 49 and also 7 × 7 = 49 . Therefore the square root of a negative number is an undefined operation unless imaginary numbers are allowed as answers.

To simplify a radical, you look for the largest perfect square factor of the number under the radical.

${\displaystyle {\sqrt {162}}}$

Once you find a perfect square factor, you can express the number under the radical as the product of two factors.

${\displaystyle {\sqrt {81*2}}}$

Then you can take the square root of the perfect square factor out of the radical and place it outside.

${\displaystyle 9{\sqrt {2}}}$

Once you have factored out all perfect square factors, you have simplified the radical.

## Rationalizing the Denominator

"Rationalizing the denominator" is simply taking the roots out of the denominator. This is needed because it is not proper to leave roots in the denominator. To rationalize the denominator you simply multiply the numerator and the denominator by the root that is in the denominator. For example:

${\displaystyle {4 \over {\sqrt {5}}}={4{\sqrt {5}} \over ({\sqrt {5}})^{2}}={4{\sqrt {5}} \over 5}}$

If you can simplify, do so.

${\displaystyle {10 \over {\sqrt {5}}}={10{\sqrt {5}} \over ({\sqrt {5}})^{2}}={10{\sqrt {5}} \over 5}={2{\sqrt {5}}}}$
${\displaystyle {4 \over {\sqrt {8}}}={{4} \over {\sqrt {(4*2)}}}={{4} \over {2}{\sqrt {2}}}={{4{\sqrt {2}}} \over {2{{\sqrt {2}}^{2}}}}={{4{\sqrt {2}}} \over {4}}={\sqrt {2}}}$

Adding and subtracting radicals can only be done when the numbers inside the radicals are the same. For instance, consider the following expression:

${\displaystyle {{\sqrt {7}}+{\sqrt {28}}}}$

You can not add those two terms as they are. However, the following equation simplifies the second term, and you get this:

${\displaystyle {{\sqrt {7}}+{\sqrt {4*7}}}={{\sqrt {7}}+2{\sqrt {7}}}}$

These terms can be added (or subtracted, for that matter). Also, coefficients of 1 are usually not written, so:

${\displaystyle {{\sqrt {7}}+2{\sqrt {7}}}={1{\sqrt {7}}+2{\sqrt {7}}}={3{\sqrt {7}}}}$

The reason we can do this is the distributive property. Proving it is quite simple, and would look like this:

${\displaystyle {{\sqrt {7}}+2{\sqrt {7}}}}$

Remember that coefficients of 1 are not written, so this equation could also be written as:

${\displaystyle {1{\sqrt {7}}+2{\sqrt {7}}}}$

We then extract the term ${\displaystyle {\sqrt {7}}}$  from the expression:

${\displaystyle {{\sqrt {7}}*(1+2)}}$

We all know that ${\displaystyle {1+2}={3}}$ , so:

${\displaystyle {{\sqrt {7}}*(3)}={3{\sqrt {7}}}}$

### Multiplication and Division

Multiplying and dividing radicals is quite simple. Multiplication requires very little work. When you are given two radicals to multiply, all you do is multiply the numbers inside the radicals and put the product under a radical. Consider the following equation:

${\displaystyle {{\sqrt {11}}*{\sqrt {5}}}={\sqrt {55}}}$

Division is a bit different, however. One way to divide radicals that uses a concept mentioned earlier on this page is to set up the division problem as a fraction. The following equation illustrates this concept using constants (regular numbers):

${\displaystyle {{1}\div {2}}={{1} \over {2}}={0.5}}$

The same concept applies to the following equation:

${\displaystyle {{\sqrt {10}}\div {\sqrt {3}}}={{\sqrt {10}} \over {\sqrt {3}}}}$

You can then rationalize the expression ${\displaystyle {{\sqrt {10}} \over {\sqrt {3}}}}$  using the technique found earlier on the page under Rationalizing the Denominator

## Exercises

1

 ${\displaystyle {\sqrt {50}}=}$  ${\displaystyle \surd }$

2

 ${\displaystyle {\sqrt {392}}=}$  ${\displaystyle \surd }$

3

 ${\displaystyle {\sqrt {290}}=}$  ${\displaystyle \surd }$
Rationalize the Denominators

4

 ${\displaystyle {2 \over {\sqrt {4}}}=}$  ${\displaystyle \surd }$

5

 ${\displaystyle {3 \over {\sqrt {7}}}=}$  ${\displaystyle \surd }$

6

 ${\displaystyle {11 \over {\sqrt {72}}}=}$  ${\displaystyle \surd }$

7

 ${\displaystyle {396 \over {\sqrt {11}}}=}$  ${\displaystyle \surd }$
Calculate

8

 ${\displaystyle {{\sqrt {2}}+{\sqrt {8}}}=}$  ${\displaystyle \surd }$

9

 ${\displaystyle {{\sqrt {60}}-{\sqrt {15}}}=}$  ${\displaystyle \surd }$

10

 ${\displaystyle {{\sqrt {13}}\times {\sqrt {7}}}=}$  ${\displaystyle \surd }$

11

 ${\displaystyle {{\sqrt {17}}\div {\sqrt {5}}}=}$  ${\displaystyle \surd }$