Monday, July 31, 2017

Pappas Computer Post: Number Bases

Table of Contents

1. A Minor Announcement
2. Pappas Page of the Day
3. Pappas and the Binary System
4. Divisibility Rules in Decimal
5. Divisibility by 7 in Decimal
6. Finding the Perfect Base
7. Divisibility by 5 in Dozenal
8. SPD in Other Bases
9. Other Uses of Dozenal
a. Lower Bases
b. Higher Bases
10. Conclusion

A Minor Announcement

Today I'm close to securing a substitute position for the fall. It's actually located at one of the charter schools to which I applied -- it turns out that a teacher who was expected to leave will be staying.

If I become a sub there, then I'll adapt its calendar as the official blog calendar and return to posting from the U of Chicago text. The calendar at this charter is similar to the LAUSD calendar, except that the holidays unique to the LAUSD (Admissions Day, Rosh Hashanah, etc.) are not observed -- instead there are more PD days. Also, the spring break at the charter may be different from LAUSD.

Pappas Page of the Day

This is what Theoni Pappas writes on page 212 of her Magic of Mathematics:

"Speculations on the fourth dimension surfaced in the 19th century when August Mobius noticed that the shadow of a right hand could be made into a left handed shadow simply by passing one's hand through the third dimension. Little did one realize that the term hypercube would lead to a generation of such terms as hyperspace, hyperbeing, hypercard, and now hypertext."

I've written about the fourth dimension several times on the blog before -- first when discussing Rudy Rucker's book The Fourth Dimension, and then in passing with Eugenia Cheng's Beyond Infinity, when she writes about infinitely many dimensions. Pappas writes about the fourth dimension as well, but we're already well past that chapter, as this was back on page 43. She only mentions 4D again here because she's trying to compare a hypercube (a 4D cube) to "Hypertext," which is the actual title of this new section.

Notice that Mobius, the 19th century German mathematician mentioned on this page, is better known as the creator of the Mobius strip. But he also worked in the fourth dimension -- his name appears in title of a Futurama episode about a four-dimensional whale, "Mobius Dick."

This is another section in Pappas which predicts the power of the internet, which was still in its infancy when Pappas wrote her 1994 book. Every time we use the internet, we use hypertext:

-- The web address of the very page that you're reading right now begins with https:// -- which stands for "hypertext transfer protocol (secure)."
-- The web address of the very page that you're reading right now ends with .html -- which stands for hypertext markup language.

Therefore you're reading hypertext this very moment! Pappas proceeds:

"While reading the story on the computer, you can choose the direction in which you want the story line to move."

This sounded very exciting back in 1994, but it seems quite ordinary now.

I've decided to use these pages in the Pappas computer chapter to discuss number bases. She writes a little about the binary system in this computer chapter. I'll borrow information about other bases from the Dozens Online Forum (written in hypertext, of course). As I promised earlier, this will become a traditionalist post where I compare the addition and multiplication tables in other bases to those in our current base ten system.

Pappas and the Binary System

Pappas writes about base two, or binary, due to its relationship with computers. In an earlier post, I wrote about the Scottish mathematician John Napier, who moved markers on a chessboard in order to add two numbers in binary.

But now suppose that we wish to multiply two numbers in binary instead. Pappas tells us how Napier would have performed the multiplication on the same chessboard:

"Suppose we want to multiply 15 * 11. One number is expressed with markers along the bottom row and the other along the vertical column at the right. Then a new marker is placed at the intersecting square where a row with a marker meets a column with a marker. After this is done, the multiplication process is simply done by sliding the markers from the bottom row diagonally to the vertical row. As with addition, any place two markers occupy the same square, they are removed and one marker is placed in the square above them."

It might seem amazing that this works, but there's a reason that it does work. As it turns out, this is identical to the lattice method for multiplication -- except in base two rather than ten!

Think about it -- the lattice method begins by placing the factors on the bottom (or top) and the side of the lattice, and Napier does the same on the chessboard. The statement:

"Then a new marker is placed at the intersecting square where a row with a marker meets a column with a marker."

takes advantage of the fact that 1 * 1 = 1 while 0 * 0, 0 * 1, and 1 * 0 are all 0. This multiplication step is more complicated in decimal, since the product of two digits could be as large as 81. And of course, the diagonal addition is the same in both methods.

As it turns out, Pappas mentions the lattice method on the page previous to the Napier page. She displays a page from a 15th century Italian book which includes four methods of multiplication. One of them is the lattice method, and the second is the lattice method again but with the numbers placed slightly differently. The third looks similar to the standard algorithm, and the last has multiplication similar to the standard algorithm but with the addition step diagonally as with the lattice methods.

The whole point of this is that many traditionalists hate the lattice method. They act as if the lattice method is something invented by the Common Core authors, who are more interested in controlling students than educating them. But as we see here, the lattice method existed more than 500 years before the Common Core.

I already mentioned the simple multiplication table in binary, and the addition table is also easy, since 0 + 0 = 0, 0 + 1 and 1 + 0 are both 1, and 1 + 1 = 10 in binary. I suspect that many students struggling with the tables wish that they only had to learn the binary tables instead of the decimal tables!

The main reason that binary isn't suitable as a general purpose base is that it's so small, so numbers in binary are longer than their decimal equivalents. Consider the binary multiplication:

111 * 111

By the time a binary user has found the answer as 110001, a decimal user -- provided he has learned the decimal times tables -- would have multiplied 7 * 7 = 49 in seconds (since after all, 111 in binary is just 7 in decimal). And two-digit numbers in decimal can contain anywhere from four to seven digits in binary. It's easier to multiply two-digit numbers than seven-digit numbers, even though the decimal tables are much larger than the binary tables.

Suppose our goal is to find the perfect base. We see that there is a trade-off -- smaller bases with fewer digits have easier tables, but with fewer digits, the numbers must be longer. Computers can easily handle longer binary numbers, since having a simple table is more important for computers (where 1 represents "on" and 0 represents "off"). But for humans, we'd prefer to have a slightly larger table if it shortens our numbers. The question is, what size base is suitable for human use?

Divisibility Rules in Decimal

I've mentioned the Dozens Online Forum earlier in this post. As its name implies, that forum draws users who believe that the perfect base is dozenal, or base 12.

But before we try to discover the perfect base ourselves, note that there are other considerations when determining the perfect base between the size of the base. For example, since our current base is 10 and the dozenalists advocate base 12, why don't we compromise and just use base 11. After all, if 10 and 12 are both human-scale bases, then surely base 11 must be human-scale as well.

At Dozens Online, there are many threads devoted to finding the perfect base. The dozenalists have developed a special jargon for describing number bases, which include:

-- alpha and omega
-- log wheel
-- opaque
-- sevenite
-- SPD
-- Stevinian algorithm
-- tuning
-- twelfty
-- uncial and unqual

Don't worry if you don't understand any of these words now -- by the end of this post, I'll have described all of them. The concepts represented by these terms are used to determine which base is the best.

Let's begin with "alpha" and "omega." These are used for the various divisibility rules. The following is a link to a typical list of divisibility rules:

We notice that many of these divisibility rules are different depending on the factor. For example, divisibility by two only requires looking at the last digit:

A number is divisible by 5 if its last digit is a 0 or 5.

But divisibility by four requires looking at the last two digits:

A number is divisible by 4 if the number's  last two digits are divisible by 4.

And divisibility by three requires looking at all the digits:

A number is divisible by 3 if the sum of its digits  is divisible by 3.

Why do some factors require one digit, others two digits, and others all digits? Well, let's look at the most straightforward divisibility rule:

A number passes the test for 10 if its final digit is 0

If we think about this for a moment, it should be obvious why this works -- a multiple of ten should have a whole number of tens, with no ones left over, so the digit in the ones place should be 0. Of course, this is all because we're working in base 10. In ternary (base 3), a number ending in 0 is a multiple of three, in quaternary a number ending in 0 is a multiple of four, and so on.

This strongly suggests that divisibility rules are base-dependent. The divisibility rules for a base like dozenal are different from the corresponding rules in decimal.

We notice that the divisibility rules requiring only the last digit are 2, 5, and 10 -- and it doesn't escape our notice that 2 * 5 = 10. In general, all factors or divisors of the base have this simplest possible divisibility rule.

The divisibility rule for 9 should sound familiar:

A number is divisible by 9 if the sum of the digits  are evenly divisible by 9.

This rule is mentioned in the Square One TV song "Nine, Nine, Nine." At the Dozens Online Forum, the number one less than the base is called the omega of the base. All factors of the omega have the same divisibility rule, so the digit-sum works for 3 as well as for 9.

On the other hand, 11 is one more than the base:

A number passes the test for 11 if the difference of the sums of alternating digits is divisible by 11.

At the Dozens Online Form, the number one more than the base is called the alpha of the base. All factors of the alpha have the same divisibility rule, but unfortunately 11 is prime, so no factors inherit the alpha rule.

There are other divisibility rules mentioned at the link above. Let's look at six:

Since 6 is a multiple of 2 and 3, the rules for divisibility by 6 are a combination of the rule for 2 and the rule for 3.

This is called a compound test. In this case, it combines the divisor rule for the factor 2 with the omega rule for the factor 3.

Earlier, we mentioned that for 4, we must look at the last two digits. This works because square of the base, 100, is divisible by 4. For 8, we must look at the last three digits. This works because the cube of the base, 1000, is divisible by 8. At Dozens Online, these are called regular tests.

(By the way, another word on the list is "log wheel." Apparently, "log wheel" refers to writing the regular numbers in a base on a wheel. Here "log" means logarithmic -- the regular numbers are spaced logarithmically. In a base like hexadecimal, the regulars are 1, 2, 4, and 8 spaced equally around the wheel, but in bases that aren't powers this is a little more complex.)

Divisibility by Seven

We notice that the factor 7 is missing. This is because seven is a divisor of neither the base, nor the omega, nor the alpha, nor the square of the base, nor the cube of the base. At Dozens Online, the factor seven is considered to be opaque in base 10.

When I was young, I once saw the following divisibility rule for 7:

We wish to test the number 742 (N1) for divisibility by 7. We get to the smaller number (N2) by chopping off the units digit, multiplying it by 5 and adding it to the number of tens in the orginal number (N1):
742 -> 74 + (2 x 5) = 84, which is clearly a multiple of 7
Therefore 742 is also a multiple of 7.
Notice that seven is opaque in our base 10, but this process involves multiplying by 5. If we multiply 10 * 5 we get 50. Now let's look at base 50 -- suddenly seven is no longer opaque. Instead, we see that the omega, 49, is a multiple of 7.

The above link also gives an alternate rule for 7:

Take the last digit, double it, and subtract it from the rest of the number; 
if the answer is divisible by 7 (including 0), then the number is also.

If we double the base 10, we obtain base 20. Again, while 7 is opaque in base 10, it is not opaque in base 20, since the alpha, 21, is a multiple of 7. Since this is an alpha, we must subtract the last digit rather than add it, just as we're required to subtract for alpha divisibility (11) in the original base.

And so we see that these rules are just omega and alpha rules in higher bases -- specifically those that are multiples of the original base. The divisibility rule for 13 given at the above link is merely the omega rule for base 40. And the divisibility rule for 17 given at the above link is merely the alpha rule for base 50. And the divisibility rule for 19 given at the above link is merely the omega rule for base 20, and so on.

I also saw the following divisibility rule for 7 in the book Quick Arithmetic (the same book in which I found the "sexy six"):

-- Divide the number in question by 50. Add the quotient and remainder. If the result is divisible by seven, then so is the original number.

Notice that this directly converts the number to base 50 (the remainders are actually the base 50 digits written in reverse). But I don't like having to divide in a divisibility rule. Otherwise, if we have to divide, we might as well have written the rule as:

-- Divide the number in question by 7. If the remainder is 0, then the number is divisible by seven.

The multiply by 5 rule takes advantage of base 50 without having to divide by 50. To see why, let's look at the number 742 again -- this number has 74 tens and 2 ones. But the omega rule in base 50 means that we must add the number of fifties and ones, not the tens and ones. Unfortunately, 74 tens is the same as 74/5 fifties, and 74/5 isn't a whole number.

Now if 742 is divisible by seven, then so is 742 * 5. This new number has 74 fifties and 2 * 5 ones -- and now we can add the number of fifties and ones, to obtain 84, a multiple of seven. So this explains why the multiply by 5 rule works, and it allows us to avoid division in a divisibility rule.

Then again, we might want to avoid multiplication in our divisibility rules as well. In this case, seven is truly opaque in base 10. If we include only divisor, omega, and alpha tests, then base 10 provides six divisibility rules -- 2, 3, 5, 9, 10, and 11.

Finding the Perfect Base

It's now time to look at the divisibility rules in other bases. The following link gives a list of bases that have many divisibility rules relative to the size of the base:

The list of good bases according to this link is:

2, 3, 4, 6, 9, 12, 16, 21, 25, 36, 60, 81, 85, 120, 225, 240

Notice that the link provides the following disclaimer:

21 is the lowest base with 'easy' divisibility tests for 8 different numbers, assuming that the casting out 11's method is not considered 'easy'.

The "casting out 11's" method is what Dozens Online calls the "alpha" method. Surely we want to count the "alpha" method, since Dozens Online counts it and even gives it a special name.

But as it turns out, even though alpha counts, the list of "best bases" including alpha doesn't provide as good a list as the previous list:

2, 3, 4, 5, 7, 9, 11, 15, 19, 25, 29, 35, 41, 49, 55, 71, 119

We notice that most of these bases are odd -- and in fact, many of them are exactly one less than the bases from the first list. For example, we see that base 11 appears on the second list, while base 12 (dozenal) appears on the first list.

At Dozens Online, there is a strong preference for even bases over odd bases. We can look at the multiplication table for decimal to see why -- half of our base is 5. And the 5's times tables are very easy -- 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50. In even bases, the times tables for half of the base are just as easy. So the 4's in octal, the 6's in dozenal, and the 7's in base 14 are just as easy as the 5's are in decimal.

Notice that 9 makes both lists -- now let's look at the times tables in base 9. This base has one fewer digit than decimal, and since it's odd, it doesn't have an easy row in the times table similar to the 5's in decimal. So base 9 is like taking base 10 and eliminating the 5's row -- that is, we're "simplifying" the table by removing one of its easiest rows!

At Dozens Online, odd bases cause a "tuning problem." The user who created this term (icarus) compares this to an old-fashioned radio. We solve the tuning problem by tuning an odd base either up to the next even base or down to the previous even base.

Let's return to bases 11 and 12. Base 11 has divisibility rules for 2, 3, 4, 5, 6, 10, 11, and 12. When we tune this base up to the next even base of 12, the old divisor (11) becomes an omega, while the factors of the old alpha (2, 3, 4, 6, 12) become divisors.

Thus in determining which base has more divisibility rules, we must compare the omega of the odd base to the alpha of the even base. And base 11 has divisibility rules for both 5 and 10, while dozenal has a divisibility rule for 13. Base 11 has one more divisibility rule than 12, and therefore 11 makes the list while 12 doesn't.

This pattern occurs with many of the odd bases on the second list -- the odd base has an even omega while the even base has an odd alpha, and so tuning up from odd base to even base renders both the old omega and its half opaque, while rendering the new alpha (often a useless prime) transparent. So tuning up from base 9 to decimal gives us 11 at the expense of 4 and 8, and tuning up from base 15 to hexadecimal gives us 17 at the expense of 7 and 14. Some cases are even worse -- tuning up from base 29 to base 30 (trigesimal) gives us 31 at the expense of 7, 14, and 28, and so on. This is why mainly odd bases appear on the second list.

As it turns out, the second list overvalues the alpha. Given a choice among the divisor, omega, and alpha rules, we prefer the divisor rule as it's the simplest. We avoid overvaluing the alpha by choosing the first list, where only the divisor and omega rules are considered. Even though this list contains odd bases, there are a healthy number of even bases as well.

Let's look at this first list again:

2, 3, 4, 6, 9, 12, 16, 21, 25, 36, 60, 81, 85, 120, 225, 240

We assume that our ideal base appears somewhere on this list -- and now, finally, we can consider the size of the base. We see that of the smaller bases, of course binary makes the list as it's the smallest possible base, ternary makes the list (as it adds a divisibility rule for 3 to binary), and quaternary makes the list (as it adds a divisibility rule for 4 to ternary). It doesn't take much for small bases to make the list -- and so we consider the smallest nontrivial base on the list to be six. Senary is the first base that makes the list after a base that misses it (base five, or quinary). And so we consider all smaller bases to be below human scale. Like binary, bases 3 and 4 are more suitable for computers (which can handle numbers with many digits) than for humans.

Now we need an upper limit to the human scale. We notice that since we are skipping odd bases, there is a huge gap between hexadecimal and the next even base, 36. This increases the size of the table greatly, and so we consider base 36 to be above the human scale. Base 36 is interesting in its own right, if only because its 36 digits could be the decimal digits 0-9 plus 26 letters a-z.

The larger even bases on the list, 60, 120, and 240, are definitely on the superhuman scale. Of these, 60 and 240 have primes for both alpha and omega, while 120 has these as composites. Because of this, one Dozens Online user, Wendy Krieger, actually advocates 120 as the perfect base. She often refers to this base as "twelfty."

And indeed, base 120 almost made the second list despite it being even. Tuning from base 119 to twelfty gives us 11 and 121 at the expense of 59 and 118. Yet 120 fails to make the list because it merely ties the record set by 119, but only outright record breakers make the list.

Anyway, our human scale runs from base 6 to base 16. In this range, there are four bases that make the list, namely 6, 9, 12, and 16. But base 6 is borderline human scale -- many consider it to be too low of a base (like bases 2-4). And base 16 is also borderline -- many consider it to be too high of a base (like base 36). With base 9 already eliminated as an odd base, this leaves only one base -- the perfect base.

And this is why the Dozens Online Forum was created -- its users view it as the perfect base.

For the rest of this post, I actually want to focus not as much on dozenal as on the borderline cases, and whether we should adjust the borders up or down. This is when we must consider how easy it is to learn arithmetic in the various bases.

By the way, before we leave the list of bases, I point out that the odd bases in the first list can be tuned down to the previous even base -- then the old base's omegas become divisors and the old base's divisors become alphas (thereby restoring the importance of alpha). This means that we could add bases 8, 20, 24, 80, 84, and 224 to the list. (Notice that our usual base, decimal, doesn't make any version of list at all.)

The official name of base 12 is duodecimal, just as base 16 is hexadecimal. Yet advocates of base 12 like to refer to their chosen base as dozenal. This is because "duo" means two, so "duodecimal" actually means two beyond "decimal" (base 10). But base 12 advocates prefer complete independence from decimal, so they don't want a base with "decimal" in its name. Since many people are already familiar with groups of 12 via the concept of a "dozen," the preferred name of the base is "dozenal."

Some base 12 advocates want to go a step further. Even "dozenal" is unacceptable because "dozen" comes from the French word douze, which is the same as the Latin duodecim. So even "dozen" indirectly refers to "decimal" in its name.

There is another Latin word, uncia, that actually means "twelfth." It appears in the English words "inch" (1/12 of a foot) and "ounce" (1/2 of a troy pound). So some Dozens Onlune users prefer using "uncial" as the name of base 12. The name "unqual" is a modification of "uncial" -- the thought being that "uncial" refers to the fraction 1/12 while "unqual" refers to the whole number 12.

We normally write numbers in other bases by writing the name of the base as a subscript (which doesn't show up well in ASCII). So, for example, 43_7 means 43 (base 7). Sometimes Pappas even writes the base out in letters, as in 43_seven. The problem with writing 27_12 to mean 27 (dozenal) is that the base 12 is itself written in decimal -- so again we're favoring decimal.

On the Dozens Online Forum, a special notation is used to name bases:

{a} (decimal)
{c} (dozenal)
{e} (14-mal)
{g} (hexadecimal)

and so on. The idea is that in hexadecimal, the digits are 0-9 and a-f, and so the first letter that isn't used in the base is g, and so {g} is the name of the base. Dozenal requires 0-9 and a-b, and so the next letter {c} is the name of the base. Decimal uses only 0-9, and so the first letter of the alphabet, {a}, can name the base.

To me, it might have been interesting to use {a} for base 9 rather than 10. Then {d} works out to be base 12 (as in dozenal), {f} would be base 14 (as in fourteen-mal), and {h} would be base 16 (as in hexadecimal), and so on. But actually, no one on Dozens Online ever uses "d" for "dozenal." If they do use the subscript"d" to stand for the first letter of a base, it instead means decimal. Instead, the letter "z" is used for dozenal ("dozen" is often shortened to "zen"), "o" is used for octal, and so on. Again, these are subscripts and are separate from the braces, where {a} is decimal and {c} is dozenal.

Here's a link to a Dozens Online post, by the user Double Sharp, which uses the symbols {a} and {c} in order to switch from decimal to dozenal in a single post:

{a} (default decimal)

Indeed some numbers don't have ten or eleven in them, but most do. The reason why separate-identity is a flawed idea is because it presupposes that ten-ness is inherent in the 1 of "10". Indeed ten-ness is inherent somewhere in that numeral, but it is not to be found in that one. It is to be found in our mental assumption that any number with multiple digits is decimal. If we just overturn that and say that any number on a particular page or post is dozenal, then "10" would instantly switch to having inherent twelve-ness, and it would once again not be found in the "1" but in our assumption.

{c} (default dozenal)

And to me, it is as simple as flicking a mental switch, like I just did, upon which I can say that 7*5=2b or 9*8=60 without batting an eyelid. Yes, in fact, I am fluent in dozenal and know its multiplication tables. I also know them in senary, octal, tetradecimal, and hexadecimal, and hope to soon add octodecimal and vigesimal to that list. I just find it a little uncomfortably constraining in having only two transparent important primes rather than three or four.

By the way, notice that not every dozenalist uses "a" and "b" for digits ten and eleven. Many prefer some version of "X" for ten and "E" for eleven. These many have been influenced by Schoolhouse Rock, which uses them in its multiplication video "Little Twelvetoes." That's right -- in teaching students how to multiply by twelve, the video actually introduces dozenal numerals. A version of "X" is "dek," a version of "E" is "el," and 10, the dozen, is "doh."

In this post, I will stick to "a" for ten and "b" for eleven, especially if I use the marker {c} to denote the dozenal base.

In this above post, Double Sharp is writing about the divisibility rules for dozenal. We've seen that dozenal made the earlier list due to having so many divisibility rules. In this base, the factors which are transparent in dozenal are:

-- 2 (divisor)
-- 3 (divisor)
-- 4 (divisor)
-- 6 (divisor)
-- 8 (regular)
-- 9 (regular)
-- b (omega)
-- 10 (divisor)
-- 11 (alpha)

But Double Sharp laments that only the first two primes are transparent (2 and 3), while the next two primes (5 and 7) are opaque. Double Sharp is willing to skip 5 in favor of 7 (as in octal, where 7 is the omega), but skipping both 5 and 7 in favor of the large primes b and 11 is undesirable.

Divisibility by 5 in Dozenal

It's possible to find rules for 5 and 7 similar to the rule for 7 in decimal. We see that 5 * 7 = 2b, and so 30 is a dozenal multiple with both 5 and 7 in the omega. Therefore both 5 and 7 can be resolved by tripling the last digit and adding to the rest of the number. Five also has another rule based on the alpha of 20 -- double the last digit and subtract. But I'd prefer tripling and adding, since a single rule tests for both five and seven.

But this isn't the usual rule for 5 on Dozens Online. Instead, another rule that avoids multiplication is used for 5 -- and this rule is known as SPD.

We notice that not only does 20 have 5 in its alpha, but so does 100, the square of the base (also known as "gross"). So according to the pattern, we can take the last digit, multiply it by 10 (dozen), and then subtract it from the remaining digits. In other words, we take the last digit and subtract it from the third-to-last digit (rather than the second-to-last digit for ordinary alpha). We can actually perform two steps at once -- take the last two digits and subtract that two-digit number from the remaining digits.

There's another way to see why this works. The square of the base, 100, has 5 in its alpha. But the digits of the number in base gross are the same as those in dozenal, taken two at a time (just as the digits of binary taken two at a time produce quaternary and the digits of quaternary taken two at a time produce hexadecimal). So we take the dozenal digits in pairs and perform the alpha test (by subtracting them in pairs).

The usual method for divisibility by 5 at Dozens Online is called SPD, for "split, promote, discard," and it is based on this square alpha test. Here is a link to the SPD test, by the user treisaran:

(Note: At the above link, an upside-down 2 and 3 are used for ten and eleven, respectively.)

SPD simplifies the square alpha test even more. The "split" part is the same as for square alpha -- split off the last two digits. But we don't merely subtract these digits. First of all, it's pointed out that if the last two digits are already a multiple of five, we can just "discard" them -- since after all the sum or difference of two multiples of five is a multiple of five.

If the last two digits aren't a multiple of five, then we really only need to subtract the remainder those digits leave when divided by five, not the entire two-digit number. So if the last two digits are 1 more than a multiple of 5, then subtract 1, if 2 more than a multiple of 5, then subtract 2, and so on. At the link, we "promote" by subtracting/adding the same number to both the two-digit number and the remaining digits until the former is a multiple of 5, and then we discard them. At the link, 1 is added to both the last two digits and the remaining digits to produce a multiple of 5.

This is worth looking at some more examples:

Example: Is 1985 divisible by 5?

-- Split: 19, 85
-- Promote: 85 is one more than 84, a multiple of 5, so we subtract 1 from both:
             18, 84
-- Discard: 18

And 18 is a multiple of 5, hence so is 1985.

Example: Is 39266 divisible by 5?

-- Split: 392, 66
-- Promote: 66 is two less than 68, a multiple of 5, so we add 2 to both:
             394, 68
-- Discard: 394
-- Split: 3, 94
-- Promote: 94 is two more than 92, a multiple of 5, so we subtract 2 from both:
             1, 92
-- Discard: 1

And 1 is not a multiple of 5, hence neither is 39266.

One thing about SPD is that it requires knowing all the two-digit multiples of five -- beyond those that appear in the 5's times table (up to 100, not just 50).

SPD in Other Bases

If we count SPD, then 5 is no longer opaque in any base. We can show that in any base, we can resolve 5 using the divisor, omega, alpha, or SPD rules. Indeed, SPD for 5 generally works in base 2, 3, 7, 8, twelve, and also thirteen. The next bases with an SPD for 5 are seventeen and eighteen, but in those bases require memorizing too many multiples of 5. Indeed, the square of seventeen is about twice as much as the square of a dozen, so twice as many multiples of five are needed.

It's also possible to use SPD for factors other than 5, but this is tricky. It can be shown that SPD based on square alpha never works -- instead we need either cube alpha or cube omega. This means that we must divide the digits into groups of three rather than two. For bases like decimal and dozenal, 7 is cube alpha, which is similar to square alpha. But for bases 9 and eleven, 7 is cube omega, which means that we must do the opposite at the "promote" step (that is, if we add 1 to the last three digits, we must subtract 1 from the remaining digits, and vice versa).

Square omega rarely appears as a divisibility rule. This is because b^2 - 1 factors as (b + 1)(b - 1), so almost any factor of square omega is already a factor of alpha or omega. The only time we'd need square omega is for odd bases. In odd bases, either alpha or omega has the factor 4, while the other is also even and thus has the factor 2. Then 8 can be tested with SPD for square omega. If 8 already appears in the alpha or omega, then sixteen is testable with SPD, and so on. Of course, Dozens Online focuses on even bases, where square omega is useless.

Notice that the following link (from Art of Problem Solving, which I mentioned last December) gives us divisibility rules (in decimal) for 7, thirteen, and other primes:

Both 7 and thirteen are listed with cube alpha rules (but these aren't true SPD). Of course, using any cube alpha rule entails knowing the multiples of the factor up to the cube of the base.

The above link also mentions a "tail-end" divisibility rule that works for any factor that is coprime to the base (that is, it ends in 1, 3, 7, 9 decimal, or 1, 5, 7, b dozenal). I've used this rule myself when playing "the factor game" with myself -- take a stopwatch, stop it at any random time, and then mentally factor the number that appears. It's stated that this rule works great in binary, where every odd number ends in 1.

There are several bases where thirteen is testable via cube alpha, cube omega, or square alpha. But eleven is never testable with any of these rules. It can be shown that unless it appears as a divisor, alpha, or omega, eleven is testable only via fifth power alpha or omega. Fifth power alpha is basically useless unless we're in binary, where there are only three multiples of eleven to memorize. Notice that eleven could be testable in bases 4, 8, or sixteen by converting to binary first.

Other Uses of Dozenal

I remember reading a story about a college student who earned grades of B+ and B- in his two college classes, yet he was considered to have only a B- average for the semester. Here is the reason:

{a} (default decimal -- that is, all numbers are in decimal from this point on)

A = 4.000
A- = 3.666
B+ = 3.333
B = 3.000
B- = 2.666
C+ = 2.333
C = 2.000
C- = 1.666
D+ = 1.333
D = 1.000
D- = 0.666
F = 0.000

B+ = 3.333
B- = 2.666
Average = 2.999 = B-

This was posted to Math Forum as part of an argument that 0.999... is not 1. Notice that I actually taught my eighth graders that 0.999... = 1 -- see my September 9th post for more information. (Yes, I taught them about 0.999... on 9/9 -- that was intentional as it was back when I used the Pappas trick of using the date in my math problems.) The poster argued that if 0.999... = 1, then 2.999... = 3, and his average would be B, not B- as his college insists.

There are several issues here. First of all, notice that if the average falls between two grades, then the GPA is considered to be the lower grade, no matter how close to the higher grade it may be. Thus only straight-A students are considered to have an A average. A student who takes 100 classes and earns 99 A's and one A- has a GPA of 3.99666, which isn't quite 4.0, so it's an A- average. (As Eugenia Cheng would say, it's 4 - epsilon for some small epsilon.)

In a recent Simpsons episode, Bart jokes about Lisa having a 3.9999 average -- and I forget how many 9's Bart said. Notice that if there were infinitely many 9's there, then Lisa would have an A average since 3.999... is exactly 4. Of course Bart can't keep saying 9's forever. Thus there are only finitely many 9's and Lisa is stuck with the A- average -- assuming, of course, that she attends the college where grades are calculated this way.

Let's return to the original poster. Note that his B+ is worth 3.333, in other words, 3 + 333/1000. (Do you recall the special cousin from my last post, with her mixed number calculator?) There are only finitely many 3's there, not infinitely many. His B- is worth 2.666, in other words, 2 + 666/1000. And again, there are only finitely many 6's there, not infinitely many.

The average of 3 + 333/1000 and 2 + 666/1000 is 2 + 9995/10000 or 2.9995. There are only finitely many 9's there, not infinitely many. The last 5 didn't appear on the transcript, and so the original poster only saw 2.999.

Of course, we can argue all day over whether colleges should always round grades down. (Notice that a student with grades of A- and B really does have a B+ average, since the average of 3.666 and 3.000 is exactly 3.333.) The real question is, why did the original poster make the unwarranted assumption that there were infinitely many 9's after the decimal point? Why did he assume that a B+ is worth 3.333..., with infinitely many 3's after the decimal point, when only finitely many 3's actually appear on the transcript? And indeed, why would the college choose such a strange number of points as 3 + 333/1000 for a grade of B+?

Well, the answers to all of these questions is obvious. The intended number of points for a grade of B+ is actually 3 + 1/3, and all the grades are supposed to be 1/3 point apart. But the values on the transcript are calculated in decimal, so it uses 3.333 = 3 + 333/1000 for 3 + 1/3. The original poster instinctively knew that 0.333... = 1/3, and so he assumed that there were infinitely many digits after the decimal point. In other words, the source of the problem is that 1/3 doesn't terminate in decimal.

This whole problem goes away if we use another base that has three as a factor. By now, you should already know what base we're going to use:

{c} (default dozenal)

A = 4.0
A- = 3.8
B+ = 3.4
B = 3.0
B- = 2.8
C+ = 2.4
C = 2.0
C- = 1.8
D+ = 1.4
D = 1.0
D- = 0.8
F = 0.0

B+ = 3.4
B- = 2.8
Average = 3.0 = B

Many dozenalists stress the importance of three as a divisor of the base. On the other hand, five doesn't show up as much. Notice that if there were such grades as double-plus and double-minus, then decimal, with its divisor of five, works perfectly:

{a} (default decimal)

A = 4.0
A- = 3.8
A-- = 3.6
B++ = 3.4
B+ = 3.2
B = 3.0
B- = 2.8
B-- = 2.6
C++ = 2.4
C+ = 2.2
C = 2.0
C- = 1.8

and so on. But in reality, double-plus and double-minus aren't actual grades.

By the way, another term from our Dozens Online vocab list is "sevenite." This is related to repeating "decimals" in other number bases. A "sevenite" in is a prime p such that both 1/p and 1/p^2 repeat the same number of digits. In decimal, the smallest "sevenite" is 3:

1/3 = 0.333... (one digit repeats, 3)
1/9 = 0.111... (one digit repeats, 1)

Dozenal has no small sevenites -- its smallest sevenite is more than 12^3. The only other bases in the (near-)human range with single digit sevenites are 5 (in base 7) and 3 (in base 17), according to the official list of sevenites:

For example, in base 17, we have:

1/3 = 0.5b5b5b... (two digits repeat, 5b)
1/9 = 0.1f1f1f... (two digits repeat, 1f)

The name "sevenite" comes from the fact that 7 is a sevenite in both bases 18 and 19. Moreover, not only do 1/7 and 1/7^2 repeat the same number of digits in those bases, but so does 1/7^3.

Dozenalism often crosses over with music theory. Recall that on Pi Approximation Day, I posted some musicians who used the digits of pi in dozenal to create a song.

Here, dozenal is chosen because our current music scale is 12EDO. In previous posts, I wrote about other scales such as 17EDO, 19EDO, and 31EDO. Musicians of those scales would then end up using pi in base 17, 19, or 31. Neither 17 nor 31 make any version of the list of best bases. Meanwhile, base 19 does make the best bases list when alpha rules are given importance. (But base 19 could be tuned up instead to vigesimal, where all of those alpha rules become divisor rules.)

Dozenalism also crosses over with calendar reform. I've written before about Timothy Travis, who once created a calendar in dozenal. His calendar contained only six days per week.

By the way, I noticed that today, the famous MTBoS blogger Sarah Carter posted her weekly "Monday Must Reads" series, and I couldn't help notice the following:

There are 6 days in a Zen week.
The first day of the Zen week is called Aquaday.
The second day of the Zen week is called Dolphinday.

Recall that some members of Dozens Online use "zen" as short for "dozen." So if I didn't know any better, I'd thought that this was a link to a dozenal calendar! In reality, it's a link to a complicated math puzzle -- in decimal. (By the way, earlier I wrote about a calendar based on 11 and 33. The original idea for this calendar ultimately goes back to our twelftyist, Wendy Krieger.)

Some dozenalists are also tauists. It appears that dozenalism draws people who wish to come up with alternatives to the standard -- whether its the base, musical scale, calendar, or circle constant.

Before we leave dozenalism, here's a link to a relevant Numberphile video. It was first posted on December 12th, 2012 -- that is, 12/12/12:

Lower Bases

Now let's get to the smaller bases and the lower range of the human scale. As we've said before, the problem with binary is that the numbers grow too long, too fast. Recall that this post is labeled as "traditionalists" because we'll learn more about arithmetic in decimal by imagining how we would learn it in other number bases.

Back in the days of the old California State Standards, first graders were expected to learn about numbers up to 100, second graders to 1000, and third graders to 10,000. (This is why Day 100 was traditionally a first grade "holiday.")

We have an obvious pattern here:

1st grade: 10^2
2nd grade: 10^3
3rd grade: 10^4

To make things simple here, let's assume that 4th graders learn up to 10^5. (In reality, I think at some point students learned about numbers "in the millions.")

Notice that this doesn't quite match up with the new Common Core Standards. Even though the limit for second grade remained at 1000, suddenly 100 became the kindergarten limit. (This is why Day 100 is now a kindergarten "holiday.") For some reason, the limit for first graders isn't something like 200 -- instead it's 120. I'm not sure where 120 came from -- unless Wendy Krieger somehow sneaked twelfty into the Common Core Standards (all the way from her home in Australia)!

By the way, before we look at the smaller bases, let's look at Krieger's favorite base of twelfty. She doesn't use 120 different symbols. Instead, she uses something she calls "dozen-dicker" -- she alternates between bases 12 and 10. (The old Babylonian system would be a "half dozen-dicker" system to represent sexagesimal.)

Here's how we count by tens in Krieger's base twelfty:

10, 20, 30, 40, 50, 60, 70, 80, 90 (ninety), V0 (teenty), E0 (elefty), 100

According to Krieger, a native twelftyist wouldn't call 100 "twelfty." Instead, she'd call it "one hundred," since that's what it looks like. But for some reason, Krieger uses "one thousand" to refer to the square of the hundred -- that is, 10000. One hundred thousand is one "cention," and one thousand thousand is one "million." To keep track, Krieger usually writes her million as, since it's important to keep track of the pairs. The symbol for Krieger's base twelfty is:

{ca} (default twelfty)

If we wrote this as {ac}, it would become "dicker-dozen." Krieger writes that she often uses a "dicker-dozen" table in order to multiply in base twelfty.

{a} (default decimal)

But let's get back to small bases. To keep things simple, we cast away "twelfty" and just use rising powers of ten for the grades:

1st grade: 10^2
2nd grade: 10^3
3rd grade: 10^4
4th grade: 10^5
5th grade: 10^6

According to the pattern, kindergartners only have to count to 10^1, or ten itself. In practice, they were often taught to count to 20. The California Standards also mention 31 -- probably because of the calendar, but notice that it's approximately 10^1.5, to continue the pattern with powers of ten.

But to keep things simple, let's just write:

Kindergarten: 10^1
1st grade: 10^2
2nd grade: 10^3
3rd grade: 10^4
4th grade: 10^5
5th grade: 10^6

The whole point of this exercise is -- let's assume that in other number bases, students also learn to count to the corresponding powers of their base. So in senary (the lower end of the human scale), this gives us:

Kindergarten: 6^1 = 6
1st grade: 6^2 = 36
2nd grade: 6^3 = 216
3rd grade: 6^4 = 1296
4th grade: 6^5 = 7776
5th grade: 6^6 = 46656

We see here that senary fourth graders can only count to 7776, while decimal third graders can count all the way to 10,000. Now let's try octal:

Kindergarten: 8^1 = 8
1st grade: 8^2 = 64
2nd grade: 8^3 = 512
3rd grade: 8^4 = 4096
4th grade: 8^5 = 32768
5th grade: 8^6 = 262144

In no case is an octal student in one grade beaten by a younger decimal student. This is why senary is sometimes considered too low, with octal as the smallest human-scale base. (Base 7 fifth graders just barely beat decimal fourth graders, so base 7 is definitely a human scale, albeit an odd, base.)

Higher Bases

In higher bases, the main concern is whether we can memorize the tables well enough to use the Stevinian algorithms.

I can already hear traditionalists asking, what exactly are these "Stevinian algorithm" -- is it something that traditionalists want us to learn instead of the standard algorithms? Well actually, the Stevinian algorithms are the standard algorithms! They are named for Simon Stevin, a Flemish mathematician who first came up with these algorithms. (This was about a century after the lattice method appeared, according to Pappas.)

Here are a few links to threads discussing the multiplication table in different bases:

I wish to pay special attention to bases 14 and 16, as these are near the upper human scale limit. We see that some posters consider 14 human scale, but not 16:

Bases 8-15: «Natural-scale» bases, and I would say 10, 12 and 14 stand out as particularly practical bases for civilization.
Bases 16-30: «Higher natural-scale» bases. There is a transition here...

Double Sharp:
2. Human-scale bases {7, 8, 9, 10, 11, 12, 14, 15}, from septenary to pentadecimal, excluding the very resistive tridecimal {13}. These are the bases that can be wielded with current algorithms, and would take less than twice the time it does currently to teach multiplication. At my most pessimistic the upper limit shrinks down to duodecimal, but usually I think {14, 15}'s close kinship should help a little. The best bases here are probably {10, 12}.

In fact, here are more threads discussing the multiplication tables in bases 14 and 16:

Double Sharp:
Here is a line-by-line approach to tetradecimal multiplication:

The 1 and 10 times tables
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, 10
10, 20, 30, 40, 50, 60, 70, 80, 90, a0, b0, c0, d0, 100

These are very, very trivial.

The 2 and 7 times tables
2, 4, 6, 8, a, c, 10, 12, 14, 16, 18, 1a, 1c, 20
7, 10, 17, 20, 27, 30, 37, 40, 47, 50, 57, 60, 67, 70

These are divisor rows and are thus very simple. The first is in a sense already covered in the doubles in addition. A low example is 6+6, because the natural way to express c on the fingers is to show 6 on each side. A high example is b+b. Normally in decimal you could guess based on the high evens from 12 to 20, but perhaps our heptadactyls from the initialisation would see that the obvious way to express b on just one side is 7 toes plus 4 fingers, leading to 8 on the hands plus 10 on the feet giving 18.

The opening of the sevens row then follows from the limbs easily till 20, and it is obvious how it must continue.

And let's skip to the bottom of this post:

The squares
1, 4, 9, 12, 1b, 28, 37, 48, 5b, 72, 89, a4, c1, 100

These usually come easily.

The 9 and b times tables
9, 14, 1d, 28, 33, 3c, 47, 52, 5b, 66, 71, 7a, 85, 90
b, 18, 25, 32, 3d, 4a, 57, 64, 71, 7c, 89, 96, a3, b0

These rows are very difficult, but almost everything in them has already been covered, so the only difficult fact becomes 9*b=71.

Here's a similar thread for hexadecimal:

We notice that Double Sharp first lists divisors of the base as easy, and then later on mentions squares as easy. In an earlier link, icarus gives "easily memorized product maps" -- these are rows which include multiples of the base.

We see that in bases like 14 and 16, neither Double Sharp nor icarus learned the entire table at once, but focuses on the easier products first. Common Core recommends the same in decimal -- which is why there are so many references to tens, doubles, and so on.


I wish to end this traditionalist post by linking to another heated Barry Garelick thread:

It's one of the few threads where one commenter actually stands up to the traditionalists:

Aaron Blackwelder:

You are critical of the Edutopia article that promotes progressive trends in education but do not offer any research-based counter arguments to substantiate your claim that traditional drill and kill, sit in rows, do worksheets, compliance-based education is superior to 21st Century models that are based on the work of Piaget, Montessori, and Gardner.
I’m wondering if you’re expecting your work to be taken seriously.
Here's a response by a traditionalist:

Tara Houle:

Furthermore, it is not Barry’s responsibility to defend what he’s written here. Rather, the onus is on those proposing these changes to offer up successful examples of why these changes ought to be made. If you could please provide examples of any jurisdiction that has had more success with pbl/inquiry based learning, especially with arithmetic, over conventional methods, I’d be happy to read it. But please…if we are to consider that education is a profession, I would ask the courtesy of providing empirical data that supports these illustrations. Nothing less will do.

Today's post is all about learning arithmetic. The original Edutropia article doesn't mention a grade level, but if it's suggesting project-based learning in the primary years, then I must actually side with the traditionalists.

Plus, I notice that the original article mentions controversial classroom management methods. Since I've been struggling with management during my first year, the last thing I need to read are such controversial methods.

I have yet to decide when my next post will be, but hopefully I'll wrap up one of the many loose ends that I've set up in my summer blogging!

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