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CONLANG  October 2004, Week 3

CONLANG October 2004, Week 3

Subject:

Re: Calénnawn and a manic digression on numbers

From:

"H. S. Teoh" <[log in to unmask]>

Reply-To:

Constructed Languages List <[log in to unmask]>

Date:

Mon, 18 Oct 2004 11:16:09 -0700

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (334 lines)

On Mon, Oct 18, 2004 at 03:26:41AM -0400, Sally Caves wrote:
[...]
> <MAJOR DIGRESSION!>
> In my salad days I experimented with a non-zero enumeration founded on a
> "sort of" base nine, call it a novenus or whatever, but it became massively
> unusable, partly because of my poor math skills.  That "zero" is essential,
> and a great gift from the Arabs, and I can't think numerically without it.
> Here were my early attempts:
[...]

It looks like a pretty good start, to me.

> I couldn't pursue it.  Can't still.  The Teonim adopted the zero.  Math
> experts out there of COURSE? Who knows what symmetries I may have discovered
> had I persisted?  What contributions to the mythology of number I may have
> created for an alien people fixated on three?  I'm SURE somebody else has
> tried this; but my knowledge of math problems is meager, and this system
> seems flawed.

It doesn't look flawed to me. Keep in mind that you're really counting
in base 9, and it may help to count base 9 using a zero-based system
(with which we are probably more familiar with than a zero-less
system), and then translate this into your zero-less notation. You're
probably having difficulties because you're trying to map base 10 onto
base 9.

Let's begin by counting in a base 9 system with a zero. In
mathematical lingo, that's "nonal", base 9 (as opposed to "decimal",
base 10).  First of all, just as in decimal, base 10, we count from 0
to 9 rather than 1 to 10 (since we have no single symbol for 10), in
nonal the digits we have are 0 through 8 (there is no "9" in nonal,
nine is written as "10"). Here then, is how one counts in nonal:

	decimal		nonal
	0		0
	1		1
	2		2
	...
	8		8
	9		10 <-- note this divergence
	10		11
	11		12
	...
	17		18
	18		20 <-- note this
	19		21
	...

Let's compare this system with decimal and see if we can discern its
underlying workings. In decimal, when we write 27, we mean 2*10 + 7,
since the 2 is in the tens' place, and 7 is in the ones' place. When
we write 123, we mean 1*100 + 2*10 + 3. This is because the 1 is in
the hundreds' place. Notice the sequence: ones, tens, hundreds. Ten is
simply 10 times 1, and a hundred is 10 times 10. A thousand, which is
represented by the 4th decimal place, is 10 times 100. Therefore, we
see that each position to the left represents the next power of 10. So
if we write 12345, that's 5 + 4*10 + 3*100 + 2*1000 + 1*10000.

In a similar way, in nonal we multiply not by 10, but by 9: when we
write _21_ in nonal, we mean 1 + 2*9 = 19. Similarly, _1234_ in nonal
is 4 + 3*9 + 2*18 + 1*81 = 922 (decimal).

Let's adopt a notation to keep track of which base we're counting in:
when I write 1234_10, it means "1234" in base 10, which is one
thousand, two hundred and thirty-four. When I write 1234_9, it means
"1234" in base 9, which is nine hundred twenty-two.

Now, let's do some arithmetic in base 9:
	  127_9
	+  32_9
	-------
	   ?

This is done in an analogous way to base 10. We first add 2 to 7,
which is 9. However, in base 9, there is no such digit "9"; hence, we
must carry the 9 to the next place. This is analogous to how in base
10, if adding two digits gives a number >= 10, we have to carry it to
the next place. So 2 + 7 = 10_9. So in the next place, we have 1 (the
carry) + 2 + 3 = 6. There is no carry from this place. The last place
is easy: 1 + 0 = 1. Hence, the result is:

	127_9 + 32_9 = 160_9.

Subtraction works in a similar way. Let's say we have:
	  362_9
	-  64_9
	-------
	   ?

We proceed in an analogous way as in decimal: subtract 4 from 2. Well,
2 is smaller than 4, so we need to borrow. Now, in decimal, we'd
borrow a 10 from the next place; but since this is nonal, we don't
borrow 10, but 9. So we have (9+2) - 4 = 7. This is the first digit of
the answer. In the next place, the 6 has become 5 because we borrowed
a 9 from it. So we need to subtract 6 from 5. But 5 is smaller than 6,
so we again need to borrow. Again, we borrow 9 rather than 10, since
we're counting in nonal. So we have (9+5) - 6 = 8. Finally, in the
last place, we have (3-1) - 0 (because we borrowed a 9 from the 3). So
the final answer is 287_9.

Multiplication proceeds in the same way. Say we have:
	  68
	* 14
	----
	  ?

First, we multiply 8 by 4, and get 32. Now, in base 10, we'd carry the
3 and keep the 2. But what we're *really* doing is, we're *dividing*
32 by 10, and carrying the quotient, keeping the remainder in the
first place. So in base 9, we divide 32 by 9, giving a quotient of 3
and a remainder of 5. Therefore, we carry 3 and write 5 in the first
position:
	  68
	* 14
	----
	   5

Next, we multiply 6 by 1, and add the carry, which is 3. This gives 9.
But since we're counting in nonal, we can't write _9_, so we have to
carry it. This is done by dividing 9 by 9, giving a quotient of 1 and
a remainder of 0. Therefore, the first row in the multiplication is:
	  68
	* 14
	----
	 105

Proceeding to the second row, we now multiply 1 by 8, giving 8:
	  68
	* 14
	----
	 105
	  8

Continuing, we multiply 6 by 1, giving 6:
	  68
	* 14
	----
	 105
	 68

Finally, we add them up:
	  68
	* 14
	----
	 105
	+68
	----
	 785

Therefore, 68_9 * 14_9 = 785_9.

I'd proceed with long division, but I think you get the idea already.
What we're *really* interested in is, how do we translate nonal into a
zero-less system? This is not too hard once you're familiar with
arithmetic in nonal.

First of all, there will be no 0, obviously. But if we're gonna count
in base 9, we must have 9 distinct symbols for each digit. This means
we add the digit "9" to the system. Let's adopt the convention that
when I write 123_N, it means the number "123" in novenus. Let's see
how to translate nonal to novenus:
	1_9 -> 1_N
	2_9 -> 2_N
	...
	8_9 -> 8_N

Here's where it gets interesting. The next number in nonal is 10_9
(which is nine). In novenus, however, we have no zero, so we can't
write 10_N. But we do have a "9", so we write 9_N instead:
	10_9 -> 9_N

Keep in mind that 9_N is already in the 2nd place in nonal. So the
next numbers would be:
	11_9 -> 11_N
	12_9 -> 12_N
	...
	18_9 -> 18_N
	20_9 -> 19_N	<-- note this transition
	21_9 -> 21_N
	22_9 -> 22_N
	...

Notice how, except for the powers of 9, the representation of nonal is
identical to the representation in novenus. As long as there are no
0's in the nonal representation of a number, the novenus number is
exactly the same. Now, what if there *are* zeroes in the nonal? For
example:
	103_9 -> ?_N

Now, note that 103_9 = 100_9 + 3_9. 100_9 is simply 9*9, which is
eighty-one (81_10). In novenus, this is simply 90_N. So 103_9 = 90_N +
3_N (because 3_9 = 3_N). This gives us:

	  90_N
	+  3_N
	------

Let's see how addition is done in novenus. Adding 3 to 0 gives 3:
	  90_N
	+  3_N
	------
	   3_N

Next, adding 9 to 0 gives 9:
	  90_N
	+  3_N
	------
	  93_N

Therefore, 103_9 = 93_N. Notice that we've essentially just collapsed
the "10" in the nonal to "9" in the novenus.

Let's see if we can convert novenus back to nonal:
	99_N = ?_9

To convert it, we simply note that 9_N = 10_9. So the rightmost "9" is
represented by 0 in the nonal, carry 1. Adding this 1 to the leftmost
"9" gives ten. But ten is larger than 9, so we need to carry it as
well.  We do this by dividing by 9, giving us a quotient of 1 and a
remainder of 1. Therefore:
	99_N = 110_9

Note that the next number in the sequence is:
	111_N = 111_9.

(Because adding 1 to 99_N causes the first "9" to become ten, which
requires a carry. Dividing by 9 gives a quotient of 1 and a remainder
of 1. The quotient is carried to the next digit where the same thing
occurs: 9+1 = ten, which needs to be carried; carrying turns it into a
quotient of 1 and a remainder of 1. Hence, 99_N + 1 = 111_N.)

Let's do one last example: novenus subtraction.
	  3121_N
	- 1129_N
	--------
	   ?

First, we subtract 9 from 1. Since 1 is too small, we borrow a 9 from
the next place. (This is exactly the same as in nonal.) So we have
(9+1) - 9 = 1:
	  3121_N
	- 1129_N
	--------
	     1_N

Now, in the next place, the 2 has become a 1 because of the borrow.
Now we need to subtract 2 from 1. But 1 is less than 2, so we borrow 9
from the next digit. This gives us (9+1)-2 = 8.
	  3121_N
	- 1129_N
	--------
	    81_N

Now, in the third place, the 1 has become a zero because of the
borrow. But we can't have zeroes in novenus, so we must borrow a 9
from the next place to avoid having to write a zero. We then subtract
9 from this 9... but we still get a zero. Since we can't write zero in
novenus, we must borrow a *second* time to get another 9:
	  3121_N
	- 1929_N
	--------
	   981_N

So in the final place, 3 has become 1 because we've borrowed twice
from it. Subtracting 1 makes it vanish; so:
	  3121_N
	- 1929_N
	--------
	   981_N

There is a tricky special case which I should mention, and that is
when a second borrow would make the last digit smaller than what is
being subtracted from it. For example:
	  211_N
	- 199_N
	-------
	   ?

We proceed as usual: we can't subtract 9 from 1, so we borrow a 9 from
the next place. This gives us (9+1)-9 = 1:
	  211_N
	- 199_N
	-------
	    1_N

In the second place, the 1 has vanished, because of the borrow; but we
can't write zero in novenus so we must borrow from the 2. This gives
us 9 - 9. But that is still zero, so we borrow again from 2, so the
second digit is now 9:
	  211_N
	- 199_N
	-------
	   91_N

But now we have a problem. Having borrowed twice from 2, it is now
zero. We can't subtract 1 from 0, and there's nothing left to borrow
from. What do we do now?

One way to resolve this is to notice that if we *hadn't* borrowed the
second time, we would have ended up with zero in the 2nd place, and 1
in the third place. Subtracting 1 from 1 makes it zero, so BOTH digits
vanish. Hence, the correct answer is actually:
	  211_N
	- 199_N
	-------
	    1_N

because both the 2nd and 3rd digits have vanished. And indeed, if we
check using novenus addition, we see that 199_N + 1 = 211_N.

The same thing may happen with larger numbers, for example:
	  21113_N
	- 19921_N
	---------
	     82_N

The rightmost 3 digits all vanish.


> Are there any natural numeric systems that do this kind of thing?
[...]

I don't know if there are any *natural* systems that do this, but I
seem to recall some time ago on the list where someone (was it David
Peterson?) devised a zero-less base 8 system along exactly the same
lines. Addition/subtraction in that system works exactly in an
analogous way with what I've described above.


T

--
Real Programmers use "cat > a.out".

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June 2009, Week 4
June 2009, Week 3
June 2009, Week 2
June 2009, Week 1
May 2009, Week 5
May 2009, Week 4
May 2009, Week 3
May 2009, Week 2
May 2009, Week 1
April 2009, Week 5
April 2009, Week 4
April 2009, Week 3
April 2009, Week 2
April 2009, Week 1
March 2009, Week 5
March 2009, Week 4
March 2009, Week 3
March 2009, Week 2
March 2009, Week 1
February 2009, Week 4
February 2009, Week 3
February 2009, Week 2
February 2009, Week 1
January 2009, Week 5
January 2009, Week 4
January 2009, Week 3
January 2009, Week 2
January 2009, Week 1
December 2008, Week 5
December 2008, Week 4
December 2008, Week 3
December 2008, Week 2
December 2008, Week 1
November 2008, Week 5
November 2008, Week 4
November 2008, Week 3
November 2008, Week 2
November 2008, Week 1
October 2008, Week 5
October 2008, Week 4
October 2008, Week 3
October 2008, Week 2
October 2008, Week 1
September 2008, Week 5
September 2008, Week 4
September 2008, Week 3
September 2008, Week 2
September 2008, Week 1
August 2008, Week 5
August 2008, Week 4
August 2008, Week 3
August 2008, Week 2
August 2008, Week 1
July 2008, Week 5
July 2008, Week 4
July 2008, Week 3
July 2008, Week 2
July 2008, Week 1
June 2008, Week 5
June 2008, Week 4
June 2008, Week 3
June 2008, Week 2
June 2008, Week 1
May 2008, Week 5
May 2008, Week 4
May 2008, Week 3
May 2008, Week 2
May 2008, Week 1
April 2008, Week 5
April 2008, Week 4
April 2008, Week 3
April 2008, Week 2
April 2008, Week 1
March 2008, Week 5
March 2008, Week 4
March 2008, Week 3
March 2008, Week 2
March 2008, Week 1
February 2008, Week 5
February 2008, Week 4
February 2008, Week 3
February 2008, Week 2
February 2008, Week 1
January 2008, Week 5
January 2008, Week 4
January 2008, Week 3
January 2008, Week 2
January 2008, Week 1
December 2007, Week 5
December 2007, Week 4
December 2007, Week 3
December 2007, Week 2
December 2007, Week 1
November 2007, Week 5
November 2007, Week 4
November 2007, Week 3
November 2007, Week 2
November 2007, Week 1
October 2007, Week 5
October 2007, Week 4
October 2007, Week 3
October 2007, Week 2
October 2007, Week 1
September 2007, Week 5
September 2007, Week 4
September 2007, Week 3
September 2007, Week 2
September 2007, Week 1
August 2007, Week 5
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August 2007, Week 3
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August 2007, Week 1
July 2007, Week 5
July 2007, Week 4
July 2007, Week 3
July 2007, Week 2
July 2007, Week 1
June 2007, Week 5
June 2007, Week 4
June 2007, Week 3
June 2007, Week 2
June 2007, Week 1
May 2007, Week 5
May 2007, Week 4
May 2007, Week 3
May 2007, Week 2
May 2007, Week 1
April 2007, Week 5
April 2007, Week 4
April 2007, Week 3
April 2007, Week 2
April 2007, Week 1
March 2007, Week 5
March 2007, Week 4
March 2007, Week 3
March 2007, Week 2
March 2007, Week 1
February 2007, Week 4
February 2007, Week 3
February 2007, Week 2
February 2007, Week 1
January 2007, Week 5
January 2007, Week 4
January 2007, Week 3
January 2007, Week 2
January 2007, Week 1
December 2006, Week 5
December 2006, Week 4
December 2006, Week 3
December 2006, Week 2
December 2006, Week 1
November 2006, Week 5
November 2006, Week 4
November 2006, Week 3
November 2006, Week 2
November 2006, Week 1
October 2006, Week 5
October 2006, Week 4
October 2006, Week 3
October 2006, Week 2
October 2006, Week 1
September 2006, Week 5
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September 2006, Week 3
September 2006, Week 2
September 2006, Week 1
August 2006, Week 5
August 2006, Week 4
August 2006, Week 3
August 2006, Week 2
August 2006, Week 1
July 2006, Week 5
July 2006, Week 4
July 2006, Week 3
July 2006, Week 2
July 2006, Week 1
June 2006, Week 5
June 2006, Week 4
June 2006, Week 3
June 2006, Week 2
June 2006, Week 1
May 2006, Week 5
May 2006, Week 4
May 2006, Week 3
May 2006, Week 2
May 2006, Week 1
April 2006, Week 5
April 2006, Week 4
April 2006, Week 3
April 2006, Week 2
April 2006, Week 1
March 2006, Week 5
March 2006, Week 4
March 2006, Week 3
March 2006, Week 2
March 2006, Week 1
February 2006, Week 4
February 2006, Week 3
February 2006, Week 2
February 2006, Week 1
January 2006, Week 5
January 2006, Week 4
January 2006, Week 3
January 2006, Week 2
January 2006, Week 1
December 2005, Week 5
December 2005, Week 4
December 2005, Week 3
December 2005, Week 2
December 2005, Week 1
November 2005, Week 5
November 2005, Week 4
November 2005, Week 3
November 2005, Week 2
November 2005, Week 1
October 2005, Week 5
October 2005, Week 4
October 2005, Week 3
October 2005, Week 2
October 2005, Week 1
September 2005, Week 5
September 2005, Week 4
September 2005, Week 3
September 2005, Week 2
September 2005, Week 1
August 2005, Week 5
August 2005, Week 4
August 2005, Week 3
August 2005, Week 2
August 2005, Week 1
July 2005, Week 5
July 2005, Week 4
July 2005, Week 3
July 2005, Week 2
July 2005, Week 1
June 2005, Week 5
June 2005, Week 4
June 2005, Week 3
June 2005, Week 2
June 2005, Week 1
May 2005, Week 5
May 2005, Week 4
May 2005, Week 3
May 2005, Week 2
May 2005, Week 1
April 2005, Week 5
April 2005, Week 4
April 2005, Week 3
April 2005, Week 2
April 2005, Week 1
March 2005, Week 5
March 2005, Week 4
March 2005, Week 3
March 2005, Week 2
March 2005, Week 1
February 2005, Week 4
February 2005, Week 3
February 2005, Week 2
February 2005, Week 1
January 2005, Week 5
January 2005, Week 4
January 2005, Week 3
January 2005, Week 2
January 2005, Week 1
December 2004, Week 5
December 2004, Week 4
December 2004, Week 3
December 2004, Week 2
December 2004, Week 1
November 2004, Week 5
November 2004, Week 4
November 2004, Week 3
November 2004, Week 2
November 2004, Week 1
October 2004, Week 5
October 2004, Week 4
October 2004, Week 3
October 2004, Week 2
October 2004, Week 1
September 2004, Week 5
September 2004, Week 4
September 2004, Week 3
September 2004, Week 2
September 2004, Week 1
August 2004, Week 5
August 2004, Week 4
August 2004, Week 3
August 2004, Week 2
August 2004, Week 1
July 2004, Week 5
July 2004, Week 4
July 2004, Week 3
July 2004, Week 2
July 2004, Week 1
June 2004, Week 5
June 2004, Week 4
June 2004, Week 3
June 2004, Week 2
June 2004, Week 1
May 2004, Week 5
May 2004, Week 4
May 2004, Week 3
May 2004, Week 2
May 2004, Week 1
April 2004, Week 5
April 2004, Week 4
April 2004, Week 3
April 2004, Week 2
April 2004, Week 1
March 2004, Week 5
March 2004, Week 4
March 2004, Week 3
March 2004, Week 2
March 2004, Week 1
February 2004, Week 5
February 2004, Week 4
February 2004, Week 3
February 2004, Week 2
February 2004, Week 1
January 2004, Week 5
January 2004, Week 4
January 2004, Week 3
January 2004, Week 2
January 2004, Week 1
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December 2003, Week 3
December 2003, Week 2
December 2003, Week 1
November 2003, Week 5
November 2003, Week 4
November 2003, Week 3
November 2003, Week 2
November 2003, Week 1
October 2003, Week 5
October 2003, Week 4
October 2003, Week 3
October 2003, Week 2
October 2003, Week 1
September 2003, Week 5
September 2003, Week 4
September 2003, Week 3
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September 2003, Week 1
August 2003, Week 5
August 2003, Week 4
August 2003, Week 3
August 2003, Week 2
August 2003, Week 1
July 2003, Week 5
July 2003, Week 4
July 2003, Week 3
July 2003, Week 2
July 2003, Week 1
June 2003, Week 5
June 2003, Week 4
June 2003, Week 3
June 2003, Week 2
June 2003, Week 1
May 2003, Week 5
May 2003, Week 4
May 2003, Week 3
May 2003, Week 2
May 2003, Week 1
April 2003, Week 5
April 2003, Week 4
April 2003, Week 3
April 2003, Week 2
April 2003, Week 1
March 2003, Week 5
March 2003, Week 4
March 2003, Week 3
March 2003, Week 2
March 2003, Week 1
February 2003, Week 4
February 2003, Week 3
February 2003, Week 2
February 2003, Week 1
January 2003, Week 5
January 2003, Week 4
January 2003, Week 3
January 2003, Week 2
January 2003, Week 1
December 2002, Week 5
December 2002, Week 4
December 2002, Week 3
December 2002, Week 2
December 2002, Week 1
November 2002, Week 5
November 2002, Week 4
November 2002, Week 3
November 2002, Week 2
November 2002, Week 1
October 2002, Week 5
October 2002, Week 4
October 2002, Week 3
October 2002, Week 2
October 2002, Week 1
September 2002, Week 5
September 2002, Week 4
September 2002, Week 3
September 2002, Week 2
September 2002, Week 1
August 2002, Week 5
August 2002, Week 4
August 2002, Week 3
August 2002, Week 2
August 2002, Week 1
July 2002, Week 5
July 2002, Week 4
July 2002, Week 3
July 2002, Week 2
July 2002, Week 1
June 2002, Week 5
June 2002, Week 4
June 2002, Week 3
June 2002, Week 2
June 2002, Week 1
May 2002, Week 5
May 2002, Week 4
May 2002, Week 3
May 2002, Week 2
May 2002, Week 1
April 2002, Week 5
April 2002, Week 4
April 2002, Week 3
April 2002, Week 2
April 2002, Week 1
March 2002, Week 5
March 2002, Week 4
March 2002, Week 3
March 2002, Week 2
March 2002, Week 1
February 2002, Week 4
February 2002, Week 3
February 2002, Week 2
February 2002, Week 1
January 2002, Week 5
January 2002, Week 4
January 2002, Week 3
January 2002, Week 2
January 2002, Week 1
December 2001, Week 5
December 2001, Week 4
December 2001, Week 3
December 2001, Week 2
December 2001, Week 1
November 2001, Week 5
November 2001, Week 4
November 2001, Week 3
November 2001, Week 2
November 2001, Week 1
October 2001, Week 5
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October 2001, Week 3
October 2001, Week 2
October 2001, Week 1
September 2001, Week 5
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July 2001, Week 1
June 2001, Week 5
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June 2001, Week 1
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May 2001, Week 1
April 2001, Week 5
April 2001, Week 4
April 2001, Week 3
April 2001, Week 2
April 2001, Week 1
March 2001, Week 5
March 2001, Week 4
March 2001, Week 3
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March 2001, Week 1
February 2001, Week 4
February 2001, Week 3
February 2001, Week 2
February 2001, Week 1
January 2001, Week 5
January 2001, Week 4
January 2001, Week 3
January 2001, Week 2
January 2001, Week 1
December 2000, Week 5
December 2000, Week 4
December 2000, Week 3
December 2000, Week 2
December 2000, Week 1
November 2000, Week 5
November 2000, Week 4
November 2000, Week 3
November 2000, Week 2
November 2000, Week 1
October 2000, Week 5
October 2000, Week 4
October 2000, Week 3
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October 2000, Week 1
September 2000, Week 5
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September 2000, Week 1
August 2000, Week 5
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June 2000, Week 5
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June 2000, Week 3
June 2000, Week 2
June 2000, Week 1
May 2000, Week 5
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May 2000, Week 1
April 2000, Week 5
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April 2000, Week 3
April 2000, Week 2
April 2000, Week 1
March 2000, Week 5
March 2000, Week 4
March 2000, Week 3
March 2000, Week 2
March 2000, Week 1
February 2000, Week 5
February 2000, Week 4
February 2000, Week 3
February 2000, Week 2
February 2000, Week 1
January 2000, Week 5
January 2000, Week 4
January 2000, Week 3
January 2000, Week 2
January 2000, Week 1
December 1999, Week 5
December 1999, Week 4
December 1999, Week 3
December 1999, Week 2
December 1999, Week 1
November 1999, Week 5
November 1999, Week 4
November 1999, Week 3
November 1999, Week 2
November 1999, Week 1
October 1999, Week 5
October 1999, Week 4
October 1999, Week 3
October 1999, Week 2
October 1999, Week 1
September 1999, Week 5
September 1999, Week 4
September 1999, Week 3
September 1999, Week 2
September 1999, Week 1
August 1999, Week 5
August 1999, Week 4
August 1999, Week 3
August 1999, Week 2
August 1999, Week 1
July 1999, Week 5
July 1999, Week 4
July 1999, Week 3
July 1999, Week 2
July 1999, Week 1
June 1999, Week 5
June 1999, Week 4
June 1999, Week 3
June 1999, Week 2
June 1999, Week 1
May 1999, Week 5
May 1999, Week 4
May 1999, Week 3
May 1999, Week 2
May 1999, Week 1
April 1999, Week 5
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April 1999, Week 3
April 1999, Week 2
April 1999, Week 1
March 1999, Week 5
March 1999, Week 4
March 1999, Week 3
March 1999, Week 2
March 1999, Week 1
February 1999, Week 5
February 1999, Week 4
February 1999, Week 3
February 1999, Week 2
February 1999, Week 1
January 1999, Week 5
January 1999, Week 4
January 1999, Week 3
January 1999, Week 2
January 1999, Week 1
December 1998, Week 5
December 1998, Week 4
December 1998, Week 3
December 1998, Week 2
December 1998, Week 1
November 1998, Week 5
November 1998, Week 4
November 1998, Week 3
November 1998, Week 2
November 1998, Week 1
October 1998, Week 5
October 1998, Week 4
October 1998, Week 3
October 1998, Week 2
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September 1998, Week 3

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