Arithmetic: A Crash Review
Yes, this is still a useful skill to have facility with.
 Counting change without this is not recommended...if your business is to make
money. Cash registers that do this for the cashier are still outrageously expensive, even
in the U.S.
 Is what you are reading off the calculator (or slide rule) the answer to what you intended
to enter into the calculator? If you don't know arithmetic, you're clueless.
 Inaccurate arithmetic in formal writing is a sign of inattention to detail.
Unfortunately, the first impression may be the last impression.
Lack of skill in arithmetic will severely impair your appearance of having learned more intricate mathematics.
The worst part of learning this subject is that several key parts must be learned by rote memorization.
[Feel free to skip the following if it proves too difficult.]
Possibly the most confusing aspect of arithmetic is that it treats adjectives as nouns. E.g.:
 Consider the sentence "One orange plus one orange is two oranges." Here, the words "one" and "two" modify
the word "orange". Since "orange" is a noun, "one" and "two" are being used as adjectives. (For reference, the object of
this sentence is "two oranges". The subject of this sentence is "one orange plus one orange", so clearly
"plus" takes two nouns (one on each side) to construct a phrase which behaves as a noun.
 Now, consider the sentence "One plus one is two." [This is the arithmetic statement corresponding to
the first statement.] Here, the object of the sentence is "two"...so "two" must be
used as a noun. The subject of the sentence is "One plus one". While we are accustomed to such
a phrase acting as a noun, we know that we must put nouns in...so "one" is being used as a noun here. [English complicates things
further, since "one" is also usable as a thirdperson singular pronoun...a difficulty not present in some
other languages.]
 So, what is going on here? Apparently, arithmetic describes how to manipulate
certain adjectives that declare quantity. In doing so, arithmetic statements can be viewed as
abusing natural language by temporarily using adjectives as nouns.
The following presentation is not intended to be rigorous. It is intended to assume no significant prior algebraic
experience.
Arithmetic Survival 101
What are numerals (and numbers)?
What is the number line?
How do I know which Arabic numeral represents a larger number?
What is addition?
What is multiplication?
What is a negative sign?
What is the absolute value of a number?
What is subtraction?
What is division?
What is a fraction?
What is a decimal point?
Numerals physically represent numbers.
Here, I will review (naturallanguage) English numerals and (symbolic) Arabic numerals.
Numerals in other natural languages will have to be reviewed in those particular languages. While
quite a few other symbolic numeration systems are known to history, most of these have severe
computational disadvantages compared to Arabic numerals  to wit, they don't represent
arbitraryprecision addition and multiplication cleanly.
Arabic numerals are ultimately the invention of Hindu astrologers out to compute the north
and south nodes of the moon's orbit in the sky. They are known as Arabic because Renaissance
Europe learned about them from the Arabs, who had implemented a comprehensive library project
in the 12th century AD/CE of all classical literature they could find (including both Greek and
Hindu). The computation of the proper orientation of mosques (to face Mecca) is an
exercise in numerical spherical trigonometry, and would have been far more difficult without
this computationally simpler numeration system.
A Hindu horoscope cannot be completely computed without these: the points where the moon's
orbit (as viewed from earth) crosses the celestial equator from:
 North node: the celestial southern hemisphere to the celestial northern hemisphere.
 South node: the celestial northern hemisphere to the celestial southern hemisphere.
The other historical numeration system that is computationally competitive with Arabic numerals
is Mayan (symbolic) numerals (inherited by the Aztecs). Mayan numerals apparently are the
invention of astrologers out to compute such things as the exact period of Venus' orbit, and when
the December solstice position of the Sun on the ecliptic crosses from the Galactic celestial
northern hemisphere to the Galactic celestial southern hemisphere. [This date is astrologically important to the
Mayan, Aztec, and Toltec civilizations: "when the skydemons come down to destroy mankind".]
 My current understanding is that Dec. 21, 2012 is the first winter solstice (in the past few millenia)
when the Sun will be in the Galactic celestial southern hemisphere. Before then, the winter solstice
position of the Sun is/was in the Galactic celestial northern hemisphere (for roughly half of an
Earth axis precession period). (It is almost certainly not exactly half of an Earth axis precession period, because
of General Relativity. However, I have not worked this yet, or seen it worked.)
 The Mayan calendar "rolls over" [date 13.0.0.0.0 ] within a few days of Dec. 21, 2012 AD/CE.
There is some uncertainity due to revisions between the ninth century AD and when the Spanish
explorers arrived, but it is plausible to assume that the original calibration of the calendar
put 13.0.0.0.0 exactly on the winter solstice in 2012 AD/CE.
[Since reading Inca quipu is a thoroughly lost art due to efficient Spanish colonization
techniques, I cannot report on whether the Inca numeration system is computationally
competitive with Arabic numerals, or not.]
Babylonian numerals are almost competitive: finite addition and multiplication tables existed for
the fiftynine digits [corresponding to the Arabic numerals 1 through 59] used in commercial computations. [The tables are easier
to memorize than this would suggest: the digits explicitly denoted their construction in terms of one and ten.] Unfortunately,
Babylonian numerals did not use anything corresponding to the digit zero. This makes Babylonian numerals ambiguous outside of
their original context.
In the interest of good formal English style, here is a table of the English and Arabic numerals representing the numbers zero to
ninetynine.
English cardinal numeral  Arabic numeral 
zero  0 
one  1 
two  2 
three  3 
four  4 
five  5 
six  6 
seven  7 
eight  8 
nine  9 
ten  10 
eleven  11 
twelve  12 
thirteen  13 
fourteen  14 
fifteen  15 
sixteen  16 
seventeen  17 
eighteen  18 
nineteen  19 
twenty  20 
twentyone  21 
twentytwo  22 
twentythree  23 
twentyfour  24 
twentyfive  25 
twentysix  26 
twentyseven  27 
twentyeight  28 
twentynine  29 
thirty  30 
thirtyone  31 
thirtytwo  32 
thirtythree  33 
thirtyfour  34 
thirtyfive  35 
thirtysix  36 
thirtyseven  37 
thirtyeight  38 
thirtynine  39 
forty  40 
fortyone  41 
fortytwo  42 
fortythree  43 
fortyfour  44 
fortyfive  45 
fortysix  46 
fortyseven  47 
fortyeight  48 
fortynine  49 
fifty  50 
fiftyone  51 
fiftytwo  52 
fiftythree  53 
fiftyfour  54 
fiftyfive  55 
fiftysix  56 
fiftyseven  57 
fiftyeight  58 
fiftynine  59 
sixty  60 
sixtyone  61 
sixtytwo  62 
sixtythree  63 
sixtyfour  64 
sixtyfive  65 
sixtysix  66 
sixtyseven  67 
sixtyeight  68 
sixtynine  69 
seventy  70 
seventyone  71 
seventytwo  72 
seventythree  73 
seventyfour  74 
seventyfive  75 
seventysix  76 
seventyseven  77 
seventyeight  78 
seventynine  79 
eighty  80 
eightyone  81 
eightytwo  82 
eightythree  83 
eightyfour  84 
eightyfive  85 
eightysix  86 
eightyseven  87 
eightyeight  88 
eightynine  89 
ninety  90 
ninetyone  91 
ninetytwo  92 
ninetythree  93 
ninetyfour  94 
ninetyfive  95 
ninetysix  96 
ninetyseven  97 
ninetyeight  98 
ninetynine  99 
(Numbers with an Arabic numeral representation of one or two Arabic digits are usually written
as English numerals in formal English, while larger numbers are usually written as Arabic numerals
in formal English. The hyphens might disappear from the correct spellings in a century or so. Spellings are for American English. Cardinal numerals are those used to
denote quantity.)
First of all, notice that Arabic numerals use only ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The
English names for these digits are the same as the English numerals that these digits directly represent.
In contrast, the English numerals use noticeably more word roots. (There are a few more that I have not mentioned yet.)
When writing long Arabic numerals, I have seen at least three conventions for correcting the format to informal English (or other natural languages).
For instance, 123456 may be written as:
 Americanstyle [groups of three, separator is comma]: 123,456
 Europeanstyle [groups of three, separator is period]: 123.456
 Japanesestyle [groups of four, separator is comma]: 12,3456
The correct convention depends on the target audience, of course.
The base of a numeral system is defined when said system uses digits (including the digit representing
the number zero) to use finite tables to define arbitraryprecision addition and multiplication. Then the base
of the numeral system is the number of digits it uses. Once we see how Arabic numerals are used this way, we
will know that Arabic numerals are a base ten system of numerals. English
numerals do not permit such a procedure (and thus are inefficient for memorizing arithmetic
tables).
Incidentally, cardinal numerals refer to quantity. Ordinal numerals refer to relative position (most commonly in a list or enumeration).
The number line is the basis of Cartesian coordinate geometry. It also provides a method of
visualizing addition and subtraction, and helps physically motivate the concept of negative number.
(These will all be discussed later.)
The following ancient civilizations left little or no evidence of using (or knowing about)
negative numbers:
Number lines are also useful when visualizing basic algebraic inequalities and results.
For the moment, let's consider this for those numbers whose Arabic numerals are represented by writing a string of Arabic digits.
 Leading zeros don't count. The normal form for writing an Arabic numeral starts with 0 only for the number zero. For instance,
0 is the normal way to write the number zero as an Arabic numerals, while 02 is a nonstandard way to write the number two as an Arabic numeral.
[The correct way is in the prior table: 2. A typical motive for doing this is to be computerfriendly when entering data (such as credit card expiration dates).]
 If an Arabic numeral is longer than another Arabic numeral, the number represented by the first Arabic numeral
is greater than the number represented by the second Arabic numeral. Alternatively, we say that the number represented by
the second Arabic numeral is less than the number represented by the first Arabic numeral.
The symbolic notation for "greater than" is >. The symbolic notation for "less than" is <.
 If two Arabic numerals have the same sequence of digits (in normal form), the number represented by
the first Arabic numeral is equal to the number represented by the second Arabic numeral. The symbolic notation for "is equal to" is =. The symbolic notation for "is not equal to" is ¹. [NOTE: Programming languages
differ in their notation for denoting an equality test of two variables, in the conditional part of an ifthen clause. Both == and = are commonly used. Most programming languages use != or <> to denote "is not equal to".]
 The usual order for listing Arabic digits [0, 1, 2, 3, 4, 5, 6, 7, 8, and 9] lists them in increasing order. For instance,
0 is less than any other Arabic digit, while 9 is greater than any other Arabic digit.
 If the Arabic numerals representing two numbers have the same length, we compare their digits going
left to right. The first pair of nonequal digits causes the numbers to have the same relationship as those digits.
For instance:11=11 (1=1, 1=1)
12<13 [1=1, 2<3]
41>21 [4>2]
23<34 [2<3], 51>23 [5>2]
 For a horizontal number line in the usual orientation, all points to the left of a given point represent numbers less than the number represented by that given point.
All points to the right of that given point represent numbers greater than the number represented by that given point. [You should have noticed that the
number line does not have visible holes, whereas there are no Arabic numerals between 0 and 1. We will eventually discuss how to represent some of these numbers
by extending the Arabic numeration system. There also points that represent numbers less than the number zero. We will discuss how to represent these, also.]
 For a vertical number line in the usual orientation, all points below a given point represent numbers less than the number represented by that given point.
All points above that given point represent numbers greater than the number represented by that given point.
 For any two given numbers represented by Arabic numerals, one of the three relations mentioned above holds: either
the two numbers are equal, or the first number is less than the second number, or the first number is greater than the second number.
Some concrete examples (without solutions):
 If the red basket contains three apples and two oranges, and the blue basket contains four oranges:
 How many apples are in the two baskets?
 How many oranges are in the two baskets?
 How many fruits are in the two baskets?
 How would you measure four feet of string  with a twelveinch ruler?
The symbolic name for addition is + .
We use a matching pair of parentheses ( ) or brackets [ ] to indicate explicitly that the operation inside
the parentheses, or brackets, is to be done before using the result outside of the parentheses or brackets.
A number line can represent addition directly, by representing numbers by directed lengths. First, select a point on the number line to
represent the number zero. Then the first number to be added is represented by the point whose distance, from the point that represents zero, is that number. The number to be added is
represented by the point, whose distance from the first point, is the number to be added. These distances are to be measured in the same direction.
The result of the addition is the distance between the second point, and the point that represents zero. This technique will work even for the extensions of the
Arabic numeration system yet to be discussed.
+   0  1  2  3  4  5  6  7  8  9 

0   0  1  2  3  4  5  6  7  8  9 
1   1  2  3  4  5  6  7  8  9  10 
2   2  3  4  5  6  7  8  9  10  11 
3   3  4  5  6  7  8  9  10  11  12 
4   4  5  6  7  8  9  10  11  12  13 
5   5  6  7  8  9  10  11  12  13  14 
6   6  7  8  9  10  11  12  13  14  15 
7   7  8  9  10  11  12  13  14  15  16 
8   8  9  10  11  12  13  14  15  16  17 
9   9  10  11  12  13  14  15  16  17  18 
Here's an addition table for the Arabic digits 0 through 9, i.e. the numbers zero through nine.
One key to Arabic numerals being a baseten numeration system is that this table, combined with
some rules regarding how to add multipledigit numerals, is sufficient to define addition of
(positive) numerals of arbitrary length in digits.
Notice that adding 0 to any Arabic digit yields that particular Arabic digit. The technical
name for this is that "0 is the identity element for addition". This works for any number represented by an Arabic numeral. It will show up again in elementary algebra.
One of the standard checks when defining the operation + on any system of mathematical
objects is that 0 (as named in that system) is the identity element for +.
This table is symmetric in the sense that reversing the order of adding two
digits gives the same result. For instance,
3+4 = 7 = 4+3 The technical name for this is "the commutative property of addition". It works for
any two numbers represented by Arabic numerals.
It will show up again in elementary algebra.
One of the standard checks when defining the operation + on any system of mathematical
objects is that the commutative property holds for +...regardless of whether it is
"easy" to evaluate the result.
Now, let us consider this expression:
1+2+3
A priori, this could be illdefined, because there are two possible ways to evaluate this:
(1+2)+3
1+(2+3)
However, these are equal:
(1+2)+3 = 3+3 = 6 = 1+5 = 1+(2+3)
The above equality is true for any three numbers represented by Arabic numerals. The technical name for this is "the associative property of addition".
It will show up again in elementary algebra.
One of the standard checks when defining the operation + on any system of mathematical
objects is that the associative property holds for +...regardless of whether it is "easy" to
evaluate the result.
Now, let us review how addition of numbers larger than one digit works. E.g., 16+5 :
 16 + 5 


Recall: 6+5=11  1 16 + 5  1 
The 1 on top is called a carry. With practice, this can be tracked in shortterm memory, rather than explicitly. 
Recall: 1+1=2  1 16 + 5  21 

Summary:  16 + 5  21 
Clearly, any significant facility with addition will require the rote memorization of the addition table
of the digits 0 through 9.
 Rightjustify the Arabic numerals to be added.
 Recall/look up the sum of the rightmost digits from the table. If the result is
two Arabic digits, the leading 1 becomes the carry. Copy the rightmost Arabic digit of this result
below the rightmost digits of the two Arabic numerals.
 For each succeeding column (going right to left) where more than one digit is in the column,
apply the previously mentioned step to that column.
 When all columns left have only one digit, copy those digits straight down.

(If you don't like theoretical quibbles, skip what's next.)
If this is the first time you have read this, you should be skeptical of the way I worded the
second stage. Why is it correct to say "If the result is two Arabic digits, the leading 1 becomes the carry."?
Consider the most extreme case: the two Arabic numerals both have digit 9 in the column.
If this is the rightmost column (i.e, first column from the right), "there is no carry".
9+9 = 18 (from the table). So the second column (right to left) will have either no carry
[alternatively, 0 as carry], or 1 as carry.
If this is the second column from the right, the worst possible case is 1+9+9. There are two
possible ways to work this explicitly: (1+9)+9 and 1+(9+9).
In practice, work the first one that occurs to you. You should get 19 as the result.
So the third column from the right will have either no carry [alternatively, 0 as carry], or
1 as carry.
At this point, one might suspect that the reasoning would repeat itself at each stage indefinitely. That is:
 We start the first column from the right with either no carry [alternatively, 0 as carry], or 1 as carry.
 For each column (from the right) that we start with either no carry [alternatively, 0 as carry], or 1 as carry: the next column starts with either no carry [alternatively, 0 as carry], or 1 as carry.
 JUMP TO CONCLUSION???: all columns with either no carry [alternatively, 0 as carry], or 1 as carry.
Is the above outline convincing...or is it handwaving?
This is an example of "proof by (weak) natural induction". The last stage (labeled "JUMP TO CONCLUSION???")
may be considered "proven by (weak) natural induction", rather than a nonsequitur.
The validity of "proof by (weak) natural induction" is assumed in at least one method of formally describing those numbers
whose Arabic numeral representations do not require a decimal point. Methods of formally describing these numbers,
that do not assume the validity of "proof by (weak) natural induction", are required to have "proof by (weak) natural induction" valid in
order to be credible.
"Thus", the largest possible carry when adding two Arabic numerals is the Arabic digit 1.
EXERCISE: Reword the addition table for Arabic numerals in English. [You are only allowed to use as many columns and rows as the original addition table.]
EXERCISE: Explain why the direct translation of the Arabic numeral addition table into English cannot be used to work "eleven plus fiftytwo".
EXERCISE: Use the addition procedure for Arabic numerals, and the addition table for Arabic numerals, to work 11+52. (11+52 is the translation of "eleven plus fiftytwo" into Arabic numerals).
The usual procedure, when adding two English numerals, is to:
 Translate the English numeral sum into its corresponding Arabic numeral sum.
 Work the Arabic numeral sum.
 Translate the Arabic numeral result into its corresponding English numeral result.
The numbers represented by numerals (English or Arabic) are unchanged by translation.
We translate from English numerals to Arabic numerals in order to do the computation easily.
We translate from Arabic numerals to English numerals in order to write down how one would say
it in English.
Some concrete examples (without solutions):
 How much money is in a roll of forty U.S. quarters?
 What is the area of a rectangular field with sides one hundred yards and two hundred yards?
The usual symbolic name for multiplication is · . Some elementary arithmetic textbooks use
× . Unfortunately, the latter notation becomes explicitly confusing in an entire category
of engineering calculations...so I will concentrate on the · notation.
A number line can represent the multiplication of a number to be multiplied (representable on the number line by any technique yet to be discussed),
by a number representable by a given Arabic numeral as defined prior to this point in the review.
First, select a point on the number line to represent the number zero. If zero is the number multiplied by, we are done (see below for why). Otherwise,
select the point whose distance from the point that represents zero is the number to be multiplied. Do this as many times as the number represented by the given Arabic numeral. The result of
the multiplication is the distance between the last point selected, and the point that represents zero. This technique will work even for the extensions of the
Arabic numeration system yet to be discussed  for the number to be multiplied.
Generalizing for the number multiplied by is problematic. These issues were responsible for
ancient Greek mathematics fascination with how to implement geometric constructions without measuring lengths.
The Grecians who developed ancient Greek mathematics were trying to describe space
(two and threedimensional) in ways that were not subject to measurement error. They were
willing to accept a straightedge as physically constructable (to within sensory limits), and
similarly for a compass  but measuring lengths was obviously subject to error, so Euclid
(and later ancient Greek mathematicians) disallowed rulers for ideal geometric constructions.
This spawned several classical problems: the cubing of the sphere, the squaring of the circle,
and the trisection of the angle. These problems' insolubility under ancient Greek ideal
geometric construction rules had to wait until the development of "Galois theory" (1800's),
applied to algebraic geometry (1700's). That is a delay of about two millenia between these
problems' statement, and their answer (in the negative).
·   0  1  2  3  4  5  6  7  8  9 

0   0  0  0  0  0  0  0  0  0  0 
1   0  1  2  3  4  5  6  7  8  9 
2   0  2  4  6  8  10  12  14  16  18 
3   0  3  6  9  12  15  18  21  24  27 
4   0  4  8  12  16  20  24  28  32  36 
5   0  5  10  15  20  25  30  35  40  45 
6   0  6  12  18  24  30  36  42  48  54 
7   0  7  14  21  28  35  42  49  56  63 
8   0  8  16  24  32  40  48  56  64  72 
9   0  9  18  27  36  45  54  63  72  81 
Here's a multiplication table for the Arabic digits 0 through 9, i.e. the numbers zero through nine.
One key to Arabic numerals being a baseten numeration system is that this table, combined with
some rules regarding how to add multipledigit numerals, is sufficient to define multiplication of
(positive) numerals of arbitrary length in digits.
Multiplication has some properties in common with addition.
Notice that multiplying 1 by any Arabic digit yields that particular Arabic digit. The technical
name for this is that "1 is the identity element for multiplication". As with addition, this works for any
number represented by an Arabic numeral. It will show up again in elementary algebra.
One of the standard checks when defining the operation · on any system of mathematical
objects is that 1 (as named in that system) is the identity element for ·.
This table is symmetric in the sense that reversing the order of multiplying two
digits gives the same result. For instance,
3·4 = 12 = 4·3 The technical name for this is "the commutative property of multiplication".
This works for any two numbers represented by Arabic numerals. It will show up again in elementary algebra.
It is true that the commutative property holds for · over the numbers that Arabic numerals represent,
and several important generalizations. Unfortunately, there are important mathematical systems
for which the commutative property does not work.
Now, let us consider this expression:
1·2·3
A priori, this could be illdefined, because there are two possible ways to evaluate this:
(1·2)·3
1·(2·3)
However, these are equal:
(1·2)·3 = 2·3 = 6 = 1·(2·3)
The above equality is true for any three numbers represented by Arabic numerals. The technical name for this is "the associative property of addition".
It will show up again in elementary algebra.
One of the standard checks when defining the operation · on any system of mathematical
objects is that the associative property holds for ·...regardless of whether it is "easy" to
evaluate the result.
Now, for a property multiplication has, that addition does not. Note that 0 times any Arabic digit is 0.
This is actually true for any number represented by Arabic numerals. A technical name for this is "0 is the annihilator for ·".
One of the standard checks when defining both of the operations + and · on any system of mathematical
objects is that 0 (for that system) is the annihilator for · . [Yes, this is the same 0 that is the identity for +].
Then, consider this:
2·(1+3) = 2·1+2·3(1+3)·2 = 1·2+3·2
This works for any three numbers represented by Arabic numerals. The technical name for this is "· distributes over +".
One of the standard checks when defining both of the operations + and · on any system of mathematical
objects is that · distributes over +.
 36 · 5 


Recall: 6·5=30  3 36 · 5  0 
The 3 on top is called a carry. With practice, this can be tracked in shortterm memory, rather than explicitly. 
Recall: 5·3=15. Recall: 5+3=8  13 36 · 5  80 
The final carry (1) is beyond the leftmost digit of either numeral. Copy it.  13 36 · 5  180 

Summary:  36 · 5  180 
Now, let's see how multiplication of larger Arabic numerals works. An example of multiplying a large Arabic numeral
by an Arabic digit is to the right.
Any practical skill with multiplication will require memorizing the multiplication table for the Arabic digits 0 through 9.
When multiplying by an Arabic numeral with more than one digit, it is more practical to track the multiplicative carry in shortterm memory.
 Rightjustify the Arabic numerals to be multiplied.
 Recall/look up the product of the rightmost digit of the top numeral by the rightmost digit of the bottom numeral. If the result has two
digits, the leading digit becomes the carry. Copy the rightmost Arabic digit of this result below the rightmost digit of the
bottom Arabic numeral.
 For each succeeding column (going right to left) where the topmost Arabic numeral has a digit: ecall/look up the product of the rightmost digit of the top numeral by the rightmost digit of the bottom numeral. Add the carry to this intermediate result.
If the second result has two Arabic digits, the leftmost digit is the new carry. Copy the rightmost digit under the column
 When the topmost numeral does not have a digit in the column, check to see if there is a carry. If so, copy it downwards.
 Repeat the sequence in 24 for each digit in the bottom numeral (right to left). Start recording the results, for each digit in
the bottom numeral, under that digit.
 Add the intermediate results (below the line) as if the empty columns were zero.
 36 ·15 


Recall: 6·5=30  3 36 ·15  0 
Recall: 5·3=15. Recall: 5+3=8  13 36 ·15  80 
The final carry (1) is beyond the leftmost digit of either numeral. Copy it.  13 36 ·15  180 
Recall: 1·6=6.  13 36 ·15  180 6 
Recall: 1·3=3.  13 36 ·15  180 36 
Add the numerals under  as if the second line's empty space was 0  13 36 ·15  180 36  540 

Summary:  36 ·15  540 
Here's another example multiplication.
Now, this expression looks ambiguous:
2+3·5
Do we do the addition first?
(2+3)·5=5·5=25
Or, do we do the multiplication first?
2+(3·5)=2+15=17
EXERCISE: verify that 2+15=17
The standard convention is that, in the absence of parentheses, multiplication is done first.
That is,
2+3·5=2+(3·5)
The negative sign  is used to indicate the following (among other possibilities):
 Measurement of a length, or angle, against the usual orientation. For instance,
the water level of the Dead Sea is over a thousand feet below standard sea level. If
an arbitary landmark near the Dead Sea was 1000 feet below sea level, this could be rewritten
as 1000 feet above sea level. (This is not optimal English style, but might help
when compiling a table). This does not mean that a distance of 1000 feet
can physically exist! It simply indicates that we are measuring in a specific
orientation (e.g., above), and this particular measurement is against that orientation
(e.g., below).
 English word pairs declaring opposite orientation include (but are not limited to) above/below (relative to gravity, usually), left/right (relative to where one looks),
clockwise/counterclockwise, north/south, and east/west.
 Some quantities represented by a single numeral have no reasonable way to "reverse orientation" : mass is an example.
 Context may dictate that some possiblyoriented quantities are actually measured in an unoriented way. For instance, the
length of a fence is unoriented, as are physical areas and volumes.
Some important properties of , for arithmetic, are:
  is used to construct the additive inverse of a number  the number, that when added to a given number, yields for the sum the number 0 (the identity for +).
For instance:
 0+0 = 0, so 0=0
 Other Arabic numerals, that we have been using so far, require  to denote their additive inverse.
For instance, (1)+1 = 0 = 1+(1). (1)+1 may be written as 1+1 without loss of clarity : the  only modifies the numeral to its
right, and does not propagate further. Due to other notational considerations (not yet reviewed), 1+1 is ambiguous.
 The English version of (1)+1 is "negative one plus one".
 The English version of 1+(1) is "one plus negative one".
 Two negative signs cancel: e.g., 3=3. (This makes sense physically. If you reverse orientation twice, you should get your original orientation.) English version of this: "negative negative three equals three".
 The above comments work in more general mathematical systems.
 We say that numbers (other than the number zero) that do not require  to denote them in Arabic numerals are positive.
Numbers whose Arabic numeral representations require  to denote are called negative.
The number zero is neither positive or negative.
 If it is necessary to emphasize that a given number is positive, the Arabic numeral representing it may
have a + in front of it. For instance, to emphasize that 1 is positive, we could
write +1. (English version: "positive one".) Most contexts do not benefit from this.
 The positive/negative distinction may (or may not) be meaningful in other mathematical systems. It is meaningful
for numbers representable by the number line.
 +   
+  add  subtract 
  subtract  add 
We have already discussed adding two Arabic numerals representing positive numbers.
Adding two Arabic numerals representing negative numbers uses a similar procedure:
do the addition for the Arabic numerals as if they were positive, then prefix the  to
the result.
 A procedure, called subtraction, is often used to compute the result of adding a positive number
and a negative number represented by Arabic numerals.
 We have already discussed multiplying two Arabic numerals representing positive numbers. When multiplying Arabic numerals,
defer deciding the sign of the result until after the multiplication is done.
 Zero is zero. Sign is irrelevant.
 If both numbers have the same sign (both positive or both negative), the answer is
positive.
 If the numbers have different signs (one positive, one negative), the answer is
negative.
We say that the absolute value of a number is:
 itself, when the number is positive or zero.
 its additive inverse, when the number is negative.
That is, absolute value measures how far away a number is from zero, on the number line. It is
used to guarantee that some computational procedures always yield nonnegative numbers.
We denote taking the absolute value of a number by putting the numeral representing that number between two  characters.
For instance, 3=3=3. In Arabic numerals, computing absolute value simply strips off the negative sign.
An alternative notation, to be aware of, is using two  characters on each side, in close succession: 3=3=3. This is
used in formal contexts (not physical problems!) where it is useful to emphasize that absolute value gives length.
When trying to reverse the effects of taking absolute value, we usually get two distinct possibilities.
For instance, if we know that the absolute value of a number is 3, that number may be either 3, or 3. A shorthand for
this is ±3.
It is usually ambiguous, by itself, to use two or more ± characters in one expression. Other commentary
should be used to explain how (or if) the various ± characters are related.
Some concrete examples (without solutions):
 Upon arriving at the eightyfirst floor of Malaysia's tallest skyscraper, you found out
that your meeting is on the sixtyseventh floor. How many floors do you need to go down to
reach the correct floor? [Whether the stairs, or the elevator, appear more effective as a
remedy, is not addressed here.]
 [Convenience store] Yesterday morning, you (the manager) recorded twelve cartons of Marlboro Red, and sixteen
packs in the display. Yesterday evening, the closing shift reported eight cartons of Marlboro Red, and nineteen packs in
the display. [Note: there are 10 packs of cigarettes in a full carton of cigarettes.] The register tapes indicate that 1 carton and twentyeight packs
were scanned yesterday. When you (the manager) did the count this morning, you found 8 cartons of Marlboro Red and
eighteen packs in the display. What kind of error has happened?
The historical origins of subtraction defined it for positive numbers only  in fact, the number to be subtracted had
to be smaller than the number subtracted from, to avoid creating a negative number. [As mentioned before,
several ancient civilizations did not use negative numbers. These all used subtraction. They had to
arrange calculations to avoid creating negative numbers at all stages.]
The correct (unambiguous) way to write 1+(1) concisely is 11 : "one minus one", in English.
This is often used to define subtraction, for mathematical systems which have + defined.
   0  1  2  3  4  5  6  7  8  9 

0   0  1  2  3  4  5  6  7  8  9 
1   1  0  1  2  3  4  5  6  7  8 
2   2  1  0  1  2  3  4  5  6  7 
3   3  2  1  0  1  2  3  4  5  6 
4   4  3  2  1  0  1  2  3  4  5 
5   5  4  3  2  1  0  1  2  3  4 
6   6  5  4  3  2  1  0  1  2  3 
7   7  6  5  4  3  2  1  0  1  2 
8   8  7  6  5  4  3  2  1  0  1 
9   9  8  7  6  5  4  3  2  1  0 
   0  1  2  3  4  5  6  7  8  9  10 

0   0  1  2  3  4  5  6  7  8  9  
1    0  1  2  3  4  5  6  7  8  9 
2     0  1  2  3  4  5  6  7  8 
3      0  1  2  3  4  5  6  7 
4       0  1  2  3  4  5  6 
5        0  1  2  3  4  5 
6         0  1  2  3  4 
7          0  1  2  3 
8           0  1  2 
9            0  1 
Here are two subtraction tables for subtracting the Arabic digits 0 through 9, i.e. the numbers zero through nine. The table on the right displays all of the results.
The table on the left displays only the nonnegative results, and should be extended to describe subtraction of an Arabic digit from the numbers 10 through 18 as needed to complete the subtraction algorithm. Both tables are oriented: the digit subtracted from is on the top row, while the digit subtracted is on the left
column.
EXERCISE: complete extending the subtraction table on the left. [Do not list results using two digits.]
The diagonal of both tables has all entries 0. (This makes sense. If we take all of the apples out of the basket, it doesn't matter how many there were initially.)
From the table on the right, we see that reversing the order of subtraction does not leave the answer unchanged. Rather, the answer is the negative of the original answer.
(We have already mentioned that 0 = 0, so the diagonal is unchanged.) This is true for any pair of numbers represented by Arabic numerals.
The technical name for the above is "subtraction is anticommutative". E.g., 21 = 1, while 12 = 1. These two equations are related this way:
21=2+(1)=(1)+2=1·1·((1)+2)=1·(1+(2))=1·(12)=(12)
This also means that 21=12 : "the procedure of first subtracting, then taking absolute value, is commutative".
 223  72  
Recall: 32=1.  223  72  1 
2 is less than 7. Recall that 127=5. Recall that 21=1.  1 223  72  51 
The 1 on top is what is left after taking the carry from 2. With practice, it can be tracked in memory. 
The lower number has no more digits to subtract.  1 223  72  151 
Summary:  223  72  151 
To the right is an example subtraction. This gives the answer to 22372, i.e. 223+(72). The answer to 72223 would be 151.
 Rightjustify the numerals. If the number to be subtracted from is less than the number being subtracted, reverse the order of subtraction  and remember to apply a negative sign to the answer, at the end.
 If the result of subtracting the rightmost bottom digit from the rightmost top digit is nonnegative, copy the result in the rightmost column.
 Otherwise, work the subtraction for 10+the intended digit, and copy that result down. If the top digit in the second column from the right is nonzero, subtract 1 from it and copy the result above. If the top second digit from the right is zero,
copy a 9 above it and work rightwards until a nonzero digit occurs. Then subtract one from that digit and write the result above it.
 For each column from the right, repeat the above step with the bottom digit of the column on the top digit of that column.
 When there are no more bottom digits, copy the digits on the top straight down.
 If we reversed the order of the subtraction, prepend  to the result to get the final answer. [We do this because: let both the number subtracted from, and the number subtracted, be positive. Then
the result is positive if the number subtracted from is greater than the number subtracted. The result is negative if the number subtracted from is less than the number subtracted.]
Since subtraction can be thought of as an alternate notation for addition, it has the same priority as addition. E.g., 53·6=5(3·6).
A number line can represent subtraction directly, by representing numbers by directed lengths. First, select a point on the number line to
represent the number zero. Then the number to be subtracted is represented by the point whose distance, from the point that represents zero, is that number. The number to be subtracted is
represented by the point, whose distance from the first point, is the number to be subtracted  measured in the opposite direction from what you would expect.
The fast overview: College Algebra 
Name of property  Notation  Valid for anything where operations are properly defined, not just numbers? 
Addition 
Existence of additive identity 0  Yes, may be slightly different notation 
Identity  0+x = x = x+0  Yes 
Commutativity  x+y = y+x  Yes 
Associativity  (x+y)+z = x+(y+z)  Yes 
Inverse  x+(x) = xx = 0  Yes 
Multiplication 
Existence of multiplicative identity 1  Yes, may be slightly different notation 
Identity  1·x = x = x·1  Yes 
Commutativity  x·y = y·x  No 
Associativity  (x·y)·z = x·(y·z)  Yes 
Inverse  x·(1/x) = x/x = 1 = (1/x)·x  Yes 
Addition and Multiplication 
(left) distributive law  x·(y+z) = x·y+x·z  Yes 
(right) distributive law  (x+y)·z = x·z+y·z  Yes 
Precedence  x+y·z = x+(y·z)  Yes 
Multiplicative annihilation  0·x = 0 = x·0  Yes 
No zero divisors  0 = x·y implies (x = 0 or y = 0)  No 
Some concrete examples (without solutions):
 A mother is used to preparing sack lunches for her four children in elementary school. Last night, she bought (at the grocery store)
two variety packs of potato chips (24 little bags in all). At one bag per lunch per child, how many school days will her purchase last?
 Summer sausage is on sale at the supermarket...you found out after arriving. You had an upper bound of U.S.$6.00 to spend on summer sausage. You had planned to
buy one twentyfour ounce sausage for U.S.$4.29, but the sale brand (imported from Wisconsin) is a sixteenounce sausage for U.S.$2.95 . Which brand is a
better value per ounce? [The other question, how to buy as much sausage as possible within your budget, has several other methods for answering. All figures quoted (including the budget) are before taxes.]
I will cover three ways to describe the results of a division of one number represented by an Arabic numeral (the dividend) by another number represented by an Arabic numeral (the divisor).
The first method to be described will give the result as a quotient and a remainder. It relies on both subtraction and multiplication.
We use the same sign prediction rules for division as for multiplication. [This will make sense
once we review fractions.] Thus, we need only know how to divide a positive number by a positive number,
plus a few special cases.
NOTE: division by the number zero is always undefined. This is reasonable,
since multiplication by the number zero always has, as its result, the number zero. This is
true in any reasonable context which has both multiplication and zero defined.
If the remainder is zero, we say that the quotient is the result of dividing the dividend by the divisor. We also say that the dividend is divisible by the divisor.
If the dividend is smaller than the divisor, the quotient is zero and the remainder is the dividend.
We use this to know when we are finished giving an answer to a division in quotientremainder form. [Special case: zero divided by a nonzero number is zero.]
If the dividend and the divisor are equal, the result is the number one  unless the dividend (and divisor) are equal to the number zero.
Zero divided by zero is undefined. [Calculus has methods of attempting to find a reasonable substitute answer in place of zero divided by zero.
If this substitute answer exists, it need not be the number one. That's not arithmetic.] E.g.: 123456 divided by 123456 is 1.
If the divisor is the number one, the quotient is the dividend. E.g.: the quotient of 3 and 1 is 3.
If the divisor has, as an Arabic numeral representation, the digit 1 followed only by the digit 0 (for a certain length),
the remainder is simply as many of the rightmost digits of the dividend as the divisor has zeros. The quotient is the other digits, or the number zero if there
are no other digits. E.g.: 119 divided by 10 has quotient 11, remainder 9. 210 divided by 10 is 21.
To visualize a fraction:
 Take an onion, and cut it into two (visually) equal pieces. Each of these pieces is approximately
one half of the original onion.
 A onegallon jug of water holds onesixth the water that a sixgallon jug holds.
We write fractions with Arabic numerals, and a fraction bar. [Again, this notation is really from Hindu arithmetic.] When writing inline with text, this bar is a forward slash: one half, written in
Arabic numerals, is 1/2. For the numeral 1/2: 1 is the numerator, and 2 is the denominator. When not writing inline, the fraction bar is horizontal, with the
numerator above the fraction bar, and the denominator below the fraction bar. Thus,
1
1/2=
2
English ordinal numeral  English fraction  Arabic fraction 
second  one half  1/2 
third  one third  1/3 
fourth  one fourth  1/4 
fifth  one fifth  1/5 
sixth  one sixth  1/6 
seventh  one seventh  1/7 
eighth  one eighth  1/8 
ninth  one ninth  1/9 
tenth  one tenth  1/10 
eleventh  one eleventh  1/11 
twelvth  one twelvth  1/12 
thirteenth  one thirteenth  1/13 
fourteenth  one fourteenth  1/14 
fifteenth  one fifteenth  1/15 
sixteenth  one sixteenth  1/16 
seventeenth  one seventeenth  1/17 
eighteenth  one eighteenth  1/18 
nineteenth  one nineteenth  1/19 
twentieth  one twentieth  1/20 
twentyfirst  one twentyfirst  1/21 
twentysecond  one twentysecond  1/22 
twentythird  one twentythird  1/23 
twentyfourth  one twenthfourth  1/24 
twentyfifth  one twentyfifth  1/25 
twentysixth  one twentysixth  1/26 
twentyseventh  one twentyseventh  1/27 
twentyeighth  one twentyeighth  1/28 
twentyninth  one twentyninth  1/29 
thirtieth  one thirtieth  1/30 
thirtyfirst  one thirtyfirst  1/31 
thirtysecond  one thirtysecond  1/32 
thirtythird  one thirtythird  1/33 
thirtyfourth  one thirtyfourth  1/34 
thirtyfifth  one thirtyfifth  1/35 
thirtysixth  one thirtysixth  1/36 
thirtyseventh  one thirtyseventh  1/37 
thirtyeighth  one thirtyeighth  1/38 
thirtyninth  one thirtyninth  1/39 
fortieth  one fortieth  1/40 
fortyfirst  one fortyfirst  1/41 
fortysecond  one fortysecond  1/42 
fortythird  one fortythird  1/43 
fortyfourth  one fortyfourth  1/44 
fortyfifth  one fortyfifth  1/45 
fortysixth  one fortysixth  1/46 
fortyseventh  one fortyseventh  1/47 
fortyeighth  one fortyeighth  1/48 
fortyninth  one fortyninth  1/49 
fiftieth  one fiftieth  1/50 
fiftyfirst  one fiftyfirst  1/51 
fiftysecond  one fiftysecond  1/52 
fiftythird  one fiftythird  1/53 
fiftyfourth  one fiftyfourth  1/54 
fiftyfifth  one fiftyfifth  1/55 
fiftysixth  one fiftysixth  1/56 
fiftyseventh  one fiftyseventh  1/57 
fiftyeighth  one fiftyeighth  1/58 
fiftyninth  one fiftyninth  1/59 
sixtieth  one sixtieth  1/60 
sixtyfirst  one sixtyfirst  1/61 
sixtysecond  one sixtysecond  1/62 
sixtythird  one sixtythird  1/63 
sixtyfourth  one sixtyfourth  1/64 
sixtyfifth  one sixtyfifth  1/65 
sixtysixth  one sixtysixth  1/66 
sixtyseventh  one sixtyseventh  1/67 
sixtyeighth  one sixtyeighth  1/68 
sixtyninth  one sixtyninth  1/69 
seventieth  one seventieth  1/70 
seventyfirst  one seventyfirst  1/71 
seventysecond  one seventysecond  1/72 
seventythird  one seventythird  1/73 
seventyfourth  one seventyfourth  1/74 
seventyfifth  one seventyfifth  1/75 
seventysixth  one seventysixth  1/76 
seventyseventh  one seventyseventh  1/77 
seventyeighth  one seventyeighth  1/78 
seventyninth  one seventyninth  1/79 
eightieth  one eightieth  1/80 
eightyfirst  one eightyfirst  1/81 
eightysecond  one eightysecond  1/82 
eightythird  one eightythird  1/83 
eightyfourth  one eightyfourth  1/84 
eightyfifth  one eightyfifth  1/85 
eightysixth  one eightysixth  1/86 
eightyseventh  one eightyseventh  1/87 
eightyeighth  one eightyeighth  1/88 
eightyninth  one eightyninth  1/89 
ninetieth  one ninetieth  1/90 
ninetyfirst  one ninetyfirst  1/91 
ninetysecond  one ninetysecond  1/92 
ninetythird  one ninetythird  1/93 
ninetyfourth  one ninetyfourth  1/94 
ninetyfifth  one ninetyfifth  1/95 
ninetysixth  one ninetysixth  1/96 
ninetyseventh  one ninetyseventh  1/97 
ninetyeighth  one ninetyeighth  1/98 
ninetyninth  one ninetyninth  1/99 
The general English wording for the Arabic fraction 1/2 is "one over two". [The table lists idiomatic wordings. Spellings are American English. Ordinal numerals
are the numerals used to denote positions in a list.]
In general, the numerator is said to be over the denominator. As an abuse of notation, I will refer to "Arabic fraction" as "fraction" for the rest of this crash review.
AGAIN: division of an Arabic numeral by the number zero is always undefined. In fraction notation, this
means that a denominator of zero causes the attempted fraction notation to be undefined.
Since we use the sign rules for multiplication when dividing, a fraction with negative numbers for both numerator and
denominator is equal to the fraction whose numerator is the absolute value of the original numerator, and whose
denominator is the absolute value of the original denominator. If only one of these is negative, we reduce by putting the negative sign
outside the fraction, and using absolute values as above. E.g.: (3)/(2)=3/2, 2/(4)=2/4=(2)/4 (to be evaluated further).
By definition, any nonzero number times one over that number is one. E.g.,
1
3·(1/3)=1=3·
3
Ancient Egyptian arithmetic used only fractions whose Arabic representation was one over a positive Arabic numeral.
Obviously, adding these fractions was somewhat complicated (even considering that Egyptian arithmetic did not
use digits).
We use the sign rules for multiplication when dividing.
The absolute value of a number represented by a fraction, is represented by the fraction whose numerator is the absolute
value of the original numerator, and whose denominator is the absolute value of the original denominator.
Multiplication by a fraction of the form one over an Arabic numeral is related to division. [At this point, I will lapse into
using variables (as in algebra) by their full English names.] Let the dividend and the divisor be Arabic numerals,
and the quotient and remainder be computed the usual way. Then
dividend·(1/divisor)=quotient+(remainder/divisor)
E.g., 7·(1/3)=2+(1/3)=2+1/3. [We assume that the fraction bar (denoting division) has the same priority as multiplication.]
More generally, we say that the product of an Arabic numeral and a fraction (numerator not necessarily one) is computed by:
 Multiply the Arabic numeral by the denominator of the fraction
 Multiply the resulting product by one over the numerator of the fraction.
To evaluate the product of two fractions:
 The numerator of the result is the product of the numerators of the two fractions being multiplied.
 The denominator of the result is the product of the denominators of the two fractions being multiplied.
None of the above comments are concerned with simplifying the fractions before multiplication, or after. This should be done when planning to continue the calculation in
fraction notation. To test whether simplification is possible, take the "greatest common factor" of the numerator and denominator of the fraction:
 The greatest common factor [GCF] is defined for pairs of numbers represented by Arabic numerals.
 The greatest common factor is unchanged by taking the absolute value of the given numbers. E.g.: GCF(2,6)=GCF(2,6)=GCF(2,6)=GCF(2,6).
 The greatest common factor is unchanged by changing the order of its given numbers. E.g.: GCF(2,6)=GCF(6,2). [This could be viewed as commutativity, just like addition and multiplication.]
 The greatest common factor is not defined if one of the given numbers is zero. E.g.: GCF(2,0) and GCF(0,0) may be writable, but they're undefined. If the numerator is the number zero, the fraction evaluates to zero (unless the denominator is also the number zero).
 The greatest common factor is the number one if at least one of the given numbers is the number one, and neither is the number zero. E.g.: GCF(1, 1001001001)=1 If the denominator is the number one, the fraction is equal to the numerator: evaluate it that much. If the numerator is the number one while the denominator is not the number one, the fraction does not simplify.
 The greatest common factor of two equal numbers represented by (the same) Arabic numerals is that number. E.g.: GCF(4,4)=4. [There is a technical name for this: "idempotence".]
 There is a way to determine the greatest common factor, in one of the last two forms, that is equal to a greatest common factor of two given arbitrary numbers represented by positive Arabic numerals,
neither of which is zero or one, and which are not equal to each other.
 Determine which number is greater than the other one. Replace that number (the larger one) with the difference of the larger number and the smaller number. Keep the smaller number.
 Repeat until a standard form (above) happens. Evaluate that to get the answer you want.
 EXAMPLE: GCF(9,7) =GCF(97,7) =GCF(2,7) =GCF(2,72) =GCF(2,5) =GCF(2,52) =GCF(2,3) =GCF(2,32) =GCF(2,1) =1
 It is legitimate to multiply the smaller number by a number representable by an Arabic numeral, as long as the product is still smaller
than the larger number. Whether this is useful is another question. For hand calculation, the
most practical example of this is adding zeros to the right of the smaller number before doing the
subtraction. [This multiplies by some factor of the number ten]. Another useful case is when the product is recalled from the multiplication table.
 EXAMPLE: GCF(23,2)=GCF(2320,2)=GCF(3,2)=...=1 [see above]
 EXAMPLE: GCF(9,2)=GCF(98,2)=GCF(1,2)=1 . This uses the entry 2·4=8 in the multiplication table.
 If the greatest common factor, of the numerator and denominator of a given fraction, is defined and not the number one, a reduced fraction equal to the original fraction may be
constructed.
 The numerator of the reduced fraction is the quotient of the original fraction's numerator, divided by the greatest common factor of the numerator and denominator.
 The denominator of the reduced fraction is the quotient of the original fraction's denominator, divided by the greatest common factor of the numerator and denominator.
 There are a few more properties of the greatest common factor that more properly belong to a review of elementary number theory. Two are important enough to mention here, as special cases motivating fraction reductions before attempting to take the GCF.
 Numbers whose Arabic numeral representation end in the digit 0 are divisible by the number ten.
 Numbers whose Arabic numeral representation end in the digits 0 or 5 are divisible by the number five.
 Numbers whose Arabic numeral representation end in the digits 0, 2, 4, 6, or 8 are divisible by the number two.
 If both the numerator, and denominator, of a given fraction are divisible by a known number, a partial reduction of the fraction may be done
by using that number in the reduction procedure (above). E.g: 210/180=21/18, 12/4=6/2=3/1=3, 35/25=7/5 . [The first fraction can be reduced further.]
 Depending on circumstances, it may be easier to attempt (partial) reduction before doing multiplication, rather than after. For instance,
(22/5)·(15/8)=(22/1)·(3/8)=22·(3/8)=11·(3/4)=33/4
To add two fractions:
 The denominator of the sum is the product of the denominators of the two given fractions.
 The numerator of the sum is the sum of the product of the denominator of the first fraction with the numerator of the second fraction, and the product of the denominator of the second fraction with the numerator of the first fraction.
 EXAMPLE: 2/3+5/7=(2·7+5·3)/(3·7)=(14+15)/21=29/21
 If a fraction is negative, assume its negative sign applies to the numerator.
 EXAMPLE: 2/3+5/7=(2·7+5·3)/(3·7)=(14+15)/21=1/21
 EXAMPLE: 2/35/7=(2·75·3)/(3·7)=(1415)/21=1/21
 EXAMPLE: 2/35/7=(2·75·3)/(3·7)=(1415)/21=29/21
 If the answer is to be used in further computation as a fraction, it should be
reduced as per the rules for reducing the results of multiplication. If the greatest common factor of the
two denominators is not the number one, this greatest common factor may be divided out once in each product that
has to be computed.
 EXAMPLE: GCF(2,4)=GCF(2,42)=GCF(2,2)=2, so 1/2+3/4=(1·(4/2)+3·(2/2))/((2/2)·4)=(1·2+3·1)/(1·4)=(2+3)/4=5/4 . Alternatively,
1/2+3/4=(1·4+3·2)/(2·4)=(4+6)/8=10/8 ; then compute GCF(10,8)=GCF(108,8)=GCF(2,8)=GCF(2,86)=GCF(2,2)=2. Then reduce: 10/8=(10/2)/(8/2)=5/4 .
To compare two fractions:
 Subtract the second fraction from the first fraction. [We would expect this to work by considering the number line.]
 Alternatively, simply compute the numerator of the aforementioned difference. Don't bother with preliminary simplification (unnecessary for a simple comparison). Only one of these
numbers is necessary.
 If the test number is negative, the first fraction is less than the second fraction.
 If the test number is zero, the first fraction is equal to the second fraction.
 If the test number is positive, the first fraction is greater than the second fraction.
 EXAMPLE: Since 5/72/3=1/21 (see above), 5/7 > 2/3.
If the absolute value of the numerator of a given fraction is less than that fraction's denominator, we say that the fraction is a
proper fraction. Otherwise, the fraction is an improper fraction.
Whether an improper fraction should be rewritten as the sum of an Arabic numeral and a proper fraction depends on context. Consider the numerator as the dividend, and the
denominator as the divisor, for a quotientremainder division. The quotient becomes the Arabic numeral in the rewrite, and the remainder becomes the numerator of the proper fraction in the rewrite.
[We use the same denominator before, and after.]
 If the improper fraction is to be used in a formal English report as a numeral, yes (after simplification). Furthermore, the + sign should be suppressed(?!?) after doing so. The resulting notation is called a mixed fraction. For instance, 3/2 can be rewritten as
1+1/2 (remember, / has the same precedence as ·), which would then have the + suppressed, leaving 1 1/2 in the final report.
 NOTE: The + sign must be made explicit in order to do any significant computation with a mixed fraction.
 If the improper fraction is to be used in multiplication, probably not. If it can be simplified, that may reduce the length of the numerals in the following calculation. [Don't count on the fraction simplifying to an Arabic numeral...but if it does so, take advantage of it.]
 If the improper fraction is to be used in addition, yes. Unlike a formal English report, the + sign must be kept explicit.
 NOTE: When adding a collection of Arabic numerals and fractions, feel free to add them in the easiest order that occurs quickly.
The decimal point (along with trailing digits) is used to denote the (usually partial) result
of continuing a quotientremainder division after the standard algorithm says to stop.
What the decimal point is, depends on the numeration system in effect.
 Americanstyle and Japanesestyle numerals use the period (.) as the decimal point.
 Europeanstyle numerals use the comma (,) as the decimal point.
At this point, a serious confusion awaits us. The term "decimal numerals" (often shortened to "decimals") is used both
to refer to "Arabic decimal numerals" [a string of Arabic digits, a decimal point, and then another string of Arabic digits],
and as another way of stating the technical fact that there are ten digits in a numeration system. (E.g. Arabic numerals in any of
the varieties we have mentioned so far.)
The latter usage is for discussing the theoretical basis of numeration systems. For the rest of
this crash review, I will use "decimals" to refer to "Arabic decimal numerals".
 1.6 +0.5 


Recall: 6+5=11  1 1.6 +0.5  .1 
Recall: 1+1=2  1 1.6 +0.5  2.1 

Summary:  1.6 +0.5  2.1 
Provided that the decimal point is aligned, and zeros used to temporarily fill in the blanks for
calculational purposes, the procedures discussed for addition and subtraction work asis.
Multiplication and division are more problematic. Division, in particular, often can be
continued indefinitely. Practical considerations must dictate how many decimal places can be
used from a decimal division result.
All of the sign rules we have been using for addition, subtraction, multiplication, and
division still work. The rule for taking absolute value still works.
NOTE: When writing a decimal in normal form, the trailing zeros to the right of the decimal
point are written only when they are "meaningful". For instance, 10.0 is an alternate way of
writing the number 10.
 The most usual motive, in English or scientific writing, would be to emphasize that
the first digit after the decimal point was physically measured.
 In some programming languages, it may be
necessary to influence the default internal representation of a number by using 10.0 (to
inform the compiler or interpreter that it is a "float") rather than 10 (to inform the compiler
or interpreter that it is an "integer"). A similar emphasis may be obtained for the first
digit to the left of the decimal point by writing 10. .
 3.6 ·1.5 


Recall: 6·5=30  3 3.6 ·1.5  0 
Recall: 5·3=15. Recall: 5+3=8  13 3.6 ·1.5  .80 
The final carry (1) is beyond the leftmost digit of either numeral. Copy it.  13 3.6 ·1.5  1.80 
Recall: 1·6=6.  13 3.6 ·1.5  1.80 .6 
Recall: 1·3=3.  13 3.6 ·1.5  1.80 3.6 
Add the numerals under  as if the second line's empty space was 0  13 3.6 ·1.5  1.80 3.6  5.40 

Summary:  3.6 ·1.5  5.40 
In the example decimal multiplication, you will notice that there are two decimal
places in the answer (including the explicit rightmost zero digit), while there is one decimal
place in both factors being multiplied. This is deliberate: 1+1=2.
Multiplication of two decimals is done by:
 doing the multiplication for the corresponding integers,
 and then locating where the decimal point should be.
The location of the decimal point is done by:
 adding the number of decimal places in each of the factors,
 then putting the decimal point in the correct location in the result.

It is possible to align the decimal point for both the factors and the result. (In fact,
this is a normal way to do it on paper.)
Opinions, comments, criticism, etc.? Let me know about it.
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