Arithmetic: A Crash Review

• 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.

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 third-person 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.

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?

What are numerals (and numbers)?

Here, I will review (natural-language) 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 arbitrary-precision 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.

• 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.

Babylonian numerals are almost competitive: finite addition and multiplication tables existed for the fifty-nine 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 ninety-nine.

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
twenty-one	21
twenty-two	22
twenty-three	23
twenty-four	24
twenty-five	25
twenty-six	26
twenty-seven	27
twenty-eight	28
twenty-nine	29
thirty	30
thirty-one	31
thirty-two	32
thirty-three	33
thirty-four	34
thirty-five	35
thirty-six	36
thirty-seven	37
thirty-eight	38
thirty-nine	39
forty	40
forty-one	41
forty-two	42
forty-three	43
forty-four	44
forty-five	45
forty-six	46
forty-seven	47
forty-eight	48
forty-nine	49
fifty	50
fifty-one	51
fifty-two	52
fifty-three	53
fifty-four	54
fifty-five	55
fifty-six	56
fifty-seven	57
fifty-eight	58
fifty-nine	59
sixty	60
sixty-one	61
sixty-two	62
sixty-three	63
sixty-four	64
sixty-five	65
sixty-six	66
sixty-seven	67
sixty-eight	68
sixty-nine	69
seventy	70
seventy-one	71
seventy-two	72
seventy-three	73
seventy-four	74
seventy-five	75
seventy-six	76
seventy-seven	77
seventy-eight	78
seventy-nine	79
eighty	80
eighty-one	81
eighty-two	82
eighty-three	83
eighty-four	84
eighty-five	85
eighty-six	86
eighty-seven	87
eighty-eight	88
eighty-nine	89
ninety	90
ninety-one	91
ninety-two	92
ninety-three	93
ninety-four	94
ninety-five	95
ninety-six	96
ninety-seven	97
ninety-eight	98
ninety-nine	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:

• American-style [groups of three, separator is comma]: 123,456
• European-style [groups of three, separator is period]: 123.456
• Japanese-style [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 arbitrary-precision 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).

What is the number line?

The following ancient civilizations left little or no evidence of using (or knowing about) negative numbers:

• Babylonia
• Egypt
• Greece

How do I know which Arabic numeral represents a larger number?

• 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 computer-friendly 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 if-then 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 non-equal 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.

What is addition?

• 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 twelve-inch 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 base-ten numeration system is that this table, combined with some rules regarding how to add multiple-digit 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,

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:

A priori, this could be ill-defined, because there are two possible ways to evaluate this:

However, these are equal:

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 short-term 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.

Right-justify 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.)

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.

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 non-sequitur.

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 fifty-two".

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 fifty-two" into Arabic numerals). The usual procedure, when adding two English numerals, is to:

1. Translate the English numeral sum into its corresponding Arabic numeral sum.
2. Work the Arabic numeral sum.
3. Translate the Arabic numeral result into its corresponding English numeral result.

What is multiplication?

• 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 three-dimensional) 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 base-ten numeration system is that this table, combined with some rules regarding how to add multiple-digit 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,

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:

A priori, this could be ill-defined, because there are two possible ways to evaluate this:

However, these are equal:

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:

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 short-term 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 short-term memory.

1. Right-justify the Arabic numerals to be multiplied.
2. 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.
3. 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
4. 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.
5. Repeat the sequence in 2-4 for each digit in the bottom numeral (right to left). Start recording the results, for each digit in the bottom numeral, under that digit.
6. Add the intermediate results (below the line) as if the empty columns were zero.

Now, this expression looks ambiguous:

Do we do the addition first?

Or, do we do the multiplication first?

EXERCISE: verify that 2+15=17

The standard convention is that, in the absence of parentheses, multiplication is done first. That is,

What is a negative sign?

• 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 possibly-oriented quantities are actually measured in an unoriented way. For instance, the length of a fence is unoriented, as are physical areas and volumes.

• - 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.

What is the absolute value of a number?

• itself, when the number is positive or zero.
• its additive inverse, when the number is negative.

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.

What is subtraction?

• Upon arriving at the eighty-first floor of Malaysia's tallest skyscraper, you found out that your meeting is on the sixty-seventh 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 twenty-eight 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 correct (unambiguous) way to write 1+(-1) concisely is 1-1 : "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., 2-1 = 1, while 1-2 = -1. These two equations are related this way:

This also means that |2-1|=|1-2| : "the procedure of first subtracting, then taking absolute value, is commutative".

	223 - 72 ----
Recall: 3-2=1.	223 - 72 ----    1
2 is less than 7. Recall that 12-7=5. Recall that 2-1=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 223-72, i.e. 223+(-72). The answer to 72-223 would be -151.

1. Right-justify 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.
2. • 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.
3. For each column from the right, repeat the above step with the bottom digit of the column on the top digit of that column.
4. When there are no more bottom digits, copy the digits on the top straight down.
5. 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.]

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.

What is division?

• 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 twenty-four ounce sausage for U.S.$4.29, but the sale brand (imported from Wisconsin) is a sixteen-ounce 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.]

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 quotient-remainder form. [Special case: zero divided by a non-zero 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.

What is 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 one-gallon jug of water holds one-sixth the water that a six-gallon jug holds.

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
twenty-first	one twenty-first	1/21
twenty-second	one twenty-second	1/22
twenty-third	one twenty-third	1/23
twenty-fourth	one twenth-fourth	1/24
twenty-fifth	one twenty-fifth	1/25
twenty-sixth	one twenty-sixth	1/26
twenty-seventh	one twenty-seventh	1/27
twenty-eighth	one twenty-eighth	1/28
twenty-ninth	one twenty-ninth	1/29
thirtieth	one thirtieth	1/30
thirty-first	one thirty-first	1/31
thirty-second	one thirty-second	1/32
thirty-third	one thirty-third	1/33
thirty-fourth	one thirty-fourth	1/34
thirty-fifth	one thirty-fifth	1/35
thirty-sixth	one thirty-sixth	1/36
thirty-seventh	one thirty-seventh	1/37
thirty-eighth	one thirty-eighth	1/38
thirty-ninth	one thirty-ninth	1/39
fortieth	one fortieth	1/40
forty-first	one forty-first	1/41
forty-second	one forty-second	1/42
forty-third	one forty-third	1/43
forty-fourth	one forty-fourth	1/44
forty-fifth	one forty-fifth	1/45
forty-sixth	one forty-sixth	1/46
forty-seventh	one forty-seventh	1/47
forty-eighth	one forty-eighth	1/48
forty-ninth	one forty-ninth	1/49
fiftieth	one fiftieth	1/50
fifty-first	one fifty-first	1/51
fifty-second	one fifty-second	1/52
fifty-third	one fifty-third	1/53
fifty-fourth	one fifty-fourth	1/54
fifty-fifth	one fifty-fifth	1/55
fifty-sixth	one fifty-sixth	1/56
fifty-seventh	one fifty-seventh	1/57
fifty-eighth	one fifty-eighth	1/58
fifty-ninth	one fifty-ninth	1/59
sixtieth	one sixtieth	1/60
sixty-first	one sixty-first	1/61
sixty-second	one sixty-second	1/62
sixty-third	one sixty-third	1/63
sixty-fourth	one sixty-fourth	1/64
sixty-fifth	one sixty-fifth	1/65
sixty-sixth	one sixty-sixth	1/66
sixty-seventh	one sixty-seventh	1/67
sixty-eighth	one sixty-eighth	1/68
sixty-ninth	one sixty-ninth	1/69
seventieth	one seventieth	1/70
seventy-first	one seventy-first	1/71
seventy-second	one seventy-second	1/72
seventy-third	one seventy-third	1/73
seventy-fourth	one seventy-fourth	1/74
seventy-fifth	one seventy-fifth	1/75
seventy-sixth	one seventy-sixth	1/76
seventy-seventh	one seventy-seventh	1/77
seventy-eighth	one seventy-eighth	1/78
seventy-ninth	one seventy-ninth	1/79
eightieth	one eightieth	1/80
eighty-first	one eighty-first	1/81
eighty-second	one eighty-second	1/82
eighty-third	one eighty-third	1/83
eighty-fourth	one eighty-fourth	1/84
eighty-fifth	one eighty-fifth	1/85
eighty-sixth	one eighty-sixth	1/86
eighty-seventh	one eighty-seventh	1/87
eighty-eighth	one eighty-eighth	1/88
eighty-ninth	one eighty-ninth	1/89
ninetieth	one ninetieth	1/90
ninety-first	one ninety-first	1/91
ninety-second	one ninety-second	1/92
ninety-third	one ninety-third	1/93
ninety-fourth	one ninety-fourth	1/94
ninety-fifth	one ninety-fifth	1/95
ninety-sixth	one ninety-sixth	1/96
ninety-seventh	one ninety-seventh	1/97
ninety-eighth	one ninety-eighth	1/98
ninety-ninth	one ninety-ninth	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 non-zero number times one over that number is one. E.g.,

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

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(9-7,7) =GCF(2,7) =GCF(2,7-2) =GCF(2,5) =GCF(2,5-2) =GCF(2,3) =GCF(2,3-2) =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(23-20,2)=GCF(3,2)=...=1 [see above]
• EXAMPLE: GCF(9,2)=GCF(9-8,2)=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: 2/3-5/7=(2·7-5·3)/(3·7)=(14-15)/21=-1/21
• EXAMPLE: -2/3-5/7=(-2·7-5·3)/(3·7)=(-14-15)/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(2,4-2)=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(10-8,8)=GCF(2,8)=GCF(2,8-6)=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 5/7-2/3=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 quotient-remainder 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.

What is a decimal point?

What the decimal point is, depends on the numeration system in effect.

• American-style and Japanese-style numerals use the period (.) as the decimal point.
• European-style 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 as-is. 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. .

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.)

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