## Archive for the ‘History’ Category

### Roman Numerals…not quite so simple

January 1, 2011

Happy New Year!  And since the New Year is all about numbers (especially if you have come to look forward to Denise’s annual January 1 post on Let’s Play Math:  form all the integers from 1 to 100 using (exactly) the digits 2, 0, 1, 1 and common mathematical symbols), here’s a picture of a number that I meant to post in October November December.

LIIII

Recognize this number?  Even though it’s not written as LIV?  This is from the 54th entryway to the Colosseum in Rome, which was built almost 2000 years ago when Roman numerals didn’t always use the subtraction property that we’re taught, where 4 is written as IV instead of IIII.

I found that to be interesting in and of itself, since I’d heard that the subtraction was a later addition but never witnessed it.  But what’s weird?  It wasn’t a sudden change.  Here’s the forty-fifth gate:

XLV

The subtraction principle was used with 40, just not with 4.  Which leads to a natural question:  what about gate 44?

XLIIII

I’m bummed that we didn’t get a better picture of this, but you can kind of see all four Is after the L.  Apparently, according to our usual font of knowledge, the reluctance to use IV is because that was the standard abbreviation for Jupiter’s name in Rome (IVPPTER), and this mixture of sometimes using four symbols in a row continued for more than a thousand years:  in the 1390 English cookbook The Forme of Cury (here on Project Gutenberg) the author still uses IIII [as in the Table of Contents, where Section IIII is rapes in potage] and there are also some IV for section numbers and references to Edward, though those might be later additions.

Published under GNU-FDL

And even 100 years ago [last year, in 1910], the Admiralty Arch in London uses MDCCCCX instead of MCMX in the inscription

ANNO : DECIMO : EDWARDI : SEPTIMI : REGIS :
: VICTORIÆ : REGINÆ : CIVES : GRATISSIMI : MDCCCCX :

(In the tenth year of King Edward VII, to Queen Victoria, from most grateful citizens, 1910).

So what does all this mean?  Nothing much, except that Roman Numeral Rules were maybe not quite as hard and fast as I once believed.

### The Fourth Bunch of Ways to Multiply

June 17, 2009

Just three more ways today, although with all the ones that have been suggested I think we’ll get to the 25!

(15) Multiplication with Log Tables.  People had already been using trig tables to multiply, but when logarithms were discovered they became THE way to multiply numbers.  The idea behind log tables is that logarithms turn multiplication (nasty) into addition (fun!) without having to derive a bunch off trig formulas.  In particular, we’re going to use the fact that log(x·y)=log(x)+log(y).

Here’s how we can find 875×978 with logarithm tables.  We’ll start by writing the two numbers in Scientific Notation:  875 would become 8.75×102 and 978 would become 9.78×102.   We’re going to multiply 8.75 and 9.78, and then adjust that product by the appropriate power of 10 (in this case, 104).

The next step for the multiplication is to look in your Table of Common [Base 10] Logarithms.  If you don’t have a copy handy, you can look here.  It turns out that log(8.75) is 0.9420081 [that’s our log(x)] and log(9.78) is 0.9903389 [that’s our log(y)].

Now we’ll add those together to get 1.9323470.  This must be the log of our product!  So we work backwards with the table, looking for the number whose log is 1.932347.  Unfortunately, the table only gives results between 0 and 1, so initially it seems like we’re stymied, but we can be sneaky and subtract 1.  We’ll come back to that in a moment.

So now we’re looking for a number whose log is 0.932347.

Looking back at our table, we see that log(8.55)=0.9319661 and log(8.56)=0.9324738, so the number we’re looking for must be between 8.55 and 8.56.  We can pick 8.56, which is the closer number, or do a little interpolation.  If we round, we’re looking for 0.9323 and instead we got 0.9320 and 0.9325.  The number we wanted was about 60% of the way from the smaller to the larger, and so instead of choosing 8.55 or 8.56 we could go about 60% of the distance between them, and guess that the product was 8.556.  Let’s do that.  [We could be even more accurate if we used a calculator to figure the exact percentage, but that seems to defeat the purpose of using the log table to multiply.]

So now we have a product of 8.556, but clearly that’s not exactly right.  We first have to account for the fact that we subtracted 1.  Notice that 1+log(blah) is the same as log(10)+log(blah) [because we’re using the common logarithm], and THAT is the same as log(10·blah).  Here log(blah) is 0.9323470, and we just saw that blah was approximately 8.556, so the product that we want — the product of 8.75 and 9.78 — is approximately 85.56.

We’re almost there!  Remember how originally we wanted the product of 875 and 978 but we first wrote those in Scientific Notation?  We need to adjust our answer of 85.56 by 104, leading us to the conclusion that 875×978 is approximately  (drum roll please) 855600.  The correct answer is 855750, so we’re certainly in the right ballpark and more accurate than we were with the trig tables, but, well, it’s a wonder to me that this method was so popular for so long when so many other ways are even more accurate.  Maybe I’m missing something.

(I’ve multiplied with log tables before, but refreshed my memory with this site on the Obsolete Skills Wiki, which also explains how to Get off the couch to Change Channels on the TV set and how to Make Change in [old] Shilling and Pence.)

(16) Slide Rule.  This takes the ideas of the log table, but bypasses the actual looking up.  Instead, the numbers are scaled on the slide rule in such a way that you don’t have to do much at all.  William Oughtred is credited in many places (online) with the slide rule, and the date 1622 shows up, so this happened just a few years after the invention of logarithms.  Pretty quick thinking!

Let’s do a simple example first:  2.5×3.   You can use an actual slide rule, or use the java version here.    The slide rule looks complicated because it can do a lot of things, but we’re mostly going to be looking at the bottom of the slidey part in the middle (C) and the fixed part at the bottom (D).

To multiply 2.5 by 3, you start with2.5  and align the 1 on C with the 2.5 on D.  Then look for 3 on C, and right below it will be the product (in this case, 7.5):

See how once you’ve aligned the 1 with the first factor 2.5, you can move the slider doohickey to the 3 on C so that you can see what’s right below it?

Now let’s look at a more complicated example:  back to 875×978.  As with log tables, we’ll start by writing the numbers in scientific notation (8.75×102 and 9.78×102) and then we’ll multiply 8.75 and 9.78 and adjust our final answer.

Normally we’d align the 1 of C with 8.75 beneath it on D, and then start looking to the right.  But 8.75 is so large, we quickly run out of room:

So we’re sneaky, and instead of aligning the 1 (on C) with the 8.75, we align 10 (on C) with 8.75.  We’ll have to adjust by multiplying by an extra power of 10 at the end, kind of like we did with the log tables before (and, really, it’s not a coincidence that this happened in both calculations).

So we’ve aligned the 10 on C with the 8.75 on the bottom.  Then we move the slider over to 9.78 on C, and look below to see the product!  The product looks like, ummm, it seems to be a bit over 8.55 but not yet at 8.56, so we’ll say 8.555.

As our final step, we need to multiply by 10 because we aligned with the 10 instead of with the 1 (as discussed above), and also by 104 because we’d had to write the numbers in scientific notation.  This means our final answer is 8.555×105, or 855500.  As we’ve seen before, it’s not exact but it is pretty close and it’s a lot faster than looking up in tables!

(17) The Gunter scale.  This is the precursor to the slide rule, invented by Edmund Gunter, and it didn’t slide at all but it was BIG:  two feet long was standard.  It was a big ole piece of wood with a logarithm scale on it, and if you wanted to measure a product like 2.5×3 you’d measure the physical distances to 2.5 and to 3, add them together, and see where you ended up on the scale. Essentially it was a Slide Rule that didn’t slide:  you had to do that part by hand.

(There’s a picture here and more information about Gunter here, including the fact that he coined the terms for Cosine and Cotangent, though Cosecant is a bit older.  I hadn’t known about this method until Pat Ballew brought it up in a recent comment:  thanks Pat!)

### The Third Bunch of Ways to Multiply

June 16, 2009

Let’s multiply!  (The First Bunch had Egyptian, Babylonian, lattice, and current methods, and the Second Bunch had Greek, Napier and Genaille-Lucas, plus links to Vertically and Crosswise Multiplication and finger multiplication.)

We’ll just do two more today, starting with one I forgot we’d already talked about:

(13) The Method of the Cups, which was described by Friar Juan Diez in 1556 (see  this post from last year).  It’s done digit by digit, but multi-digit numbers aren’t always written on a single line: if you want a glimpse of what it’s like, here’s the image for 875×978.

(14) Prosthaphaeresis, or Multiplication with Trig Tables. This method dates back to the late 1500s, although logarithms essentially rendered this method obsolete.

Here’s the basic idea:  remember the sum and difference formulas for sine and cosine?  We’ll use those.  For example, since:

$cos(a+b)=cos(a)cos(b)-sin(a)sin(b)$

and

$cos(a-b)=cos(a)cos(b)+sin(a)sin(b)$

then by adding those together, we get

$cos(a+b)+cos(a-b) = 2cos(a)cos(b)$

$cos(a)cos(b)=\frac{1}{2}cos(a+b)+\frac{1}{2}cos(a-b)$

And that’s the formula that we’ll use, although by also looking at sine we could just have easily come up with a different formula that had a mixture of sines and cosines.    When the last step of dividing by 2 is ignored, the resulting formulas are called the Werner formulas.

Here’s how we’ll do it.  Suppose we want to multiply 875 and 978, because those were the numbers that appeared in the previous example.  We’re going to be using a table of cosine values (like this one).  This table actually shows both sine and cosine, but cosine is read on the right.

Notice that 0.9997 is cos(1.4°), for example — we’re adding 0.2° to the angle for each column, going right to left.

Cosine values always fall between 0 and 1, so we need to scale our numbers to be between 0 and 1.  Thus 875 becomes 0.875 and 978 becomes 0.978, and we’ll have to multiply our final answer by 103·103=106.

Remember that our formula is $cos(a)cos(b)=\frac{1}{2}cos(a+b)+\frac{1}{2}cos(a-b)$.  In this case, $cos(a)$ is 0.875 and $cos(b)$ is 0.978.  We need to find a and b.

Remember that we read the angles on the right.  Since 0.875 is between 0.8746 and 0.8763, our angle a must be between 28.8° and 29°.  We’ll use 29°, since that gives the best estimate.

Likewise, we need to find b knowing that $cos(b)=0.978$.  From the chart we can see that b is between 12.0° and 12.2°, but basically it’s 12°.  Yay!  We’re halfway there!

We have a and b,  so the sum $a+b$ is 41° and the difference $a-b$ is 17°.  Now we need to know $cos(a+b)$ and $cos(a-b)$.

This part is easy!  The cosine of 41° is just 0.7547

and the cosine of 17° is 0.9563.

Now we can plug all this into our formula!  Remember how 0.875 was $cos(a)$ and 0.978 was $cos(b)$?  Using:

$cos(a)cos(b)=\frac{1}{2}cos(a+b)+\frac{1}{2}cos(a-b)$

we get

$0.875 \cdot 0.978 = \frac{1}{2}\cdot 0.7547 + \frac{1}{2} \cdot 0.9563$

which simplifies down to 0.8555.  Multiplying that by the 106 from earlier gives 875×978 as approximately 855500.   We’re only off by 250, which feels like a lot but it’s less than ½% of the answer, so in the grand scheme of things it’s pretty good, and with more accurate tables we could have done even better.

Next up:  Log tables and slide rules!

### The Second Bunch of Ways to Multiply

June 11, 2009

Hey, let’s do some more multiplication!   (See Ways (1)(6) here.)  I promised some that wouldn’t be taught in school (at least not as a practical way to multiply), so that’s where I’ll start.

(7) Greek Multiplication.  This is really different from other kinds of multiplication because it’s based on Geometry.  The Ancient Greeks certainly had the concept of discrete quantities, but in a lot of their mathematics numbers were interpreted as lengths; to multiply a and b like an Ancient Greek, you start with the quantities a and b and also the reference point 1.

Draw two intersecting lines and, from the point of intersection mark lengths 1 and b on one line and length a on the other line, and then draw a line between 1 and a.

Starting from b, draw a line that is parallel to the line you just drew.  (I could really have said “construct” because all of this can be done with a straightedge and compass.)

See how the length x is where that line you just drew intersects the line with a?  It turns out that x is the length of the product ab!    It’s actually pretty easy to show this:  the triangle with sides x and b is similar to the triangle with sides a and 1 [they have one angle in common, and because their third side are parallel the remaining pairs of angles are equal as well].  Since they are similar, x/b must equal a/1, and cross multiplication gives x equal to ab.

One nice feature of this is that you can actually see how if b<1 then the positions of 1 and b will be switched, and the product ab will be less than a.

(8) Napier Rods.  These were first published in 1617 in the Rabdologia by John Napier, a Scottish mathematician who is more famous for introducing the idea of logarithms.  Napier Rods, which I just realized are supposed to be called Napier‘s Rods (or Napier’s Bones) are like a portable version of Grid Multiplication, so if they were allowed on Standardized Testing it might take care of that whole time issue.  (Then again, one could ask why these would be allowed and not, say, a calculator.  So I guess this also falls under the cool just because it’s cool category of multiplication.)

Godzilla is here to show how they work.  He’s going to demonstrate how to multiply 3558  by 274.

Start by getting yourself  some Napier Rods.

Do you need to see those up close?  Here they are:

Those are still hard to see aren’t they?  That’s because I don’t know how to focus my camera.  And THAT’S why I love Wikipedia, folks.  Here’s a drawing that (someone?  I can’t tell who) posted on Wiki under GNU-FDL.

So each rod (or bone) has a digit on top, and the multiples of that digit are written underneath.

Since Godzilla wants to multiply 3558 by 274, we’ll start by picking up rods for 3, 5, 5, and 8 and lining them up.  [I printed out two copies of this set of Napier rods since the digit 5 appeared more than once.]  There’s a rod with nothing at the top:  this is called the Index Rod, and it can be put at the right or left.  It just helps you keep track of the rows.

We’re going to multiply this number by 274, so we’ll start by looking in Row 2.

See how that looks like a little tiny grid from lattice multiplication?  If you add along the diagonals, it shows that 3558×2=07116.

Now let’s multiply by the 7 of 274.  We’ll look in Row 7.

This shows us (after adding along the diagonals) that 3558×7=24906.  We had to do a little carrying here.

Finally, we’ll multiply 3558 by the 4 of 274.  Look in Row 4:

So 3558×4=14232.

Now that we have all the pieces we need, we add, keeping track of the place value (by staggering on the left):

Yay!  We have our answer!  If you prefer to play with a java version that allows you to switch the base, you can find it here at Cut The Knot.

(9) Genaille-Lucas Rulers.  These take the basic concept of the Napier Rods and modify it using shaded triangles in a way that completely eliminates adding and carrying within each row (though you’ll still have to add to get the total at the very end).  They were invented by Frenchmen Henri Genaille and Edouaird Lucas just over a century ago.  Godzilla is using paper versions that Brian Borchers created.

He’s already lined up the 3558 to do the same multiplication (3558×274) as before.  Let’s take a closer look.  Notice that with this set, the Index piece is aligned on the left.

OK, we’re going to use the same basic idea as before, where we’ll look in rows 2, 7, and 4.  Let’s start with Row 2:

This row should give us 3558×2.  To read it, start on the RIGHT with the number on the very top.  That’s a 6.  Then you read towards the LEFT, following the little gray triangles.  The final number is 7116.

Now we’ll look at 3558×7:

Again, you start with the number in the TOP of the right-hand column (which is a 6).  Follow the little gray triangles to the next number, to get that 3558×7=24906.

Finally, here is Row 4:

As before when we start from the top of the rightmost column and read to the left we see that 3558×4=14232.

This is a completely mindless way to multiply until you get to the end, where you align your answers (staggered on the right):

And we have the same answer as before!  And right here is a picture of some of the rods from 1885 in France.  (They and Napier’s Bones were actually more like square prisms, with a different Ruler printed on each side so that each Rod could show one of four different sides depending on what digits you need.)

This post is already rather long, so I’ll end by briefly mentioning three final methods:

(10) Vertically and Crosswise Multiplication (which TwoPi wrote about just over a year ago).  TwoPi tells me that it should probably be called Trachtenberg multiplication, since it was created by the Ukrainian engineer Jakow Trachtenberg while he was imprisioned in a concentration camp in WWII.  (There are widespread claims that this technique is of Vedic origins, but there is little historical evidence to support that.)

Edited to add: Whoops, Trachtenberg might have come up with it on his own, but he wasn’t the first.  It was published in Italy over 500 years ago and was known as Crocetta.  (See Pat’s comment below.)

Edited again (7/8) to add: Looking in “Capitalism and Arithmetic”, I saw that this is attributed even further back, to the Indian Lilavati of around 1150.

(11) and (12) Speaking of multiplication we’ve already talked about, in this post from New Year’s Eve 2007 I wrote about multiplying by 9 on your fingers (I’ll call that #11) and also two Medieval ways to multiply on your fingers.  They’re pretty similar, but one works for multiplying any two of {5, 6, 7, 8, 9, 10} and the other works for multiplying any two of {10, 11, 12, 13, 14, 15}.  Together I’ll count those as #12.

That’s it for now!  I can think of three more ways offhand (abacus, slide rule, and trig functions) so I’ll write up something about those in the next couple days.  This is fun!

### The First Bunch of Ways to Multiply

June 10, 2009

I blithely mentioned in yesterday’s post that I only knew about 13 ways to multiply [“only” because it would be great to write a book called Twenty-five ways to Multiply], and then Jason asked me to list them.  I was originally going to list them all, but then I started describing them which is taking a lot longer and I won’t have time to watch the next episode of Heroes on Netflix prepare for a committee meeting if I do that all today so I’ll do it in steps and you can read all about multiplication for a few days!   I’ll see if I can get Godzilla do to some demonstrations of the more complicated methods tomorrow.

These are carefully ordering according to The Order In Which I Thought of Them.

(1) Doubling and Halving, as described yesterday (used in Egypt, Ethiopia, and presumably Russia).  This is one of my favorite methods, because it’s surprising that it works.

(2) Duplation, which is a variant on the above method, in which you start with 1 and one of the the numbers and double both.  For example, to multiply 14 and 12 you’d start with 1 & 12, and double both until the left-hand column was going to be bigger than the other number, in this case 14.  That’s not at all clear, is it?  Here’s what I mean for 14×12:

1 & 12
2 & 24
4 & 48
8 & 96
16 & <— Oh, I can stop because 16 is bigger than 14.

Since 14 can be written as 8+4+2, put a little mark by those rows:
1 & 12
*2 & 24
*4 & 48
*8 & 96

and add up the corresponding numbers on the right:  96+48+24=168.  And there’s your product!  What you’re really doing is adding the appropriate doubles (96 is 8 12s, 48 is 4 12s, and 24 is 2 12s so when you add them you get 14 12s, as you wanted).  This was also used by the Ancient Egyptians, and it’s referred to on the video from yesterday.

One thing that neat about these methods is that all they use is adding, subtracting, doubling, and/or halving.   The folk who used them had to remember the process, but they didn’t have to memorize 45 separate single digit multiplication facts.

(3) and (4) Mesopotamian Multiplication.  Like the Egyptians, the Babylonians broke down multiplication into addition, subtraction, and halving.  They had one more trick, though:  they had tables that contained the squares of all the numbers from 1 to 60 [which was all they really needed, since they used a Base 60 system].   So to multiply a and b they used one of two formulas:

$\frac{\left( a+b \right) ^2 - a^2-b^2}{2}$ or $\frac{\left( a + b \right) ^2 - \left( a - b \right) ^2}{4}$

In other words, to find 14×12, to use the first formula you’d look up 262, 142, and 122 in your table, subtract the last two numbers (196 and 144)  from the first one (676), and then cut that answer in half.  With the second formula you’d look up 262 and 22 in your table, take the difference, and cut that in half twice.

This method totally amuses me because although you can check the algebra to make sure it works (indeed, I usually introduce this by asking students to come up with a formula for ab that uses only addition, subtraction, doubling, halving, and a2, b2, (a+b)2, and/or (ab)2), it’s really an ancient plug-and-chug method.  You don’t have to think at all about why it works, you just do it.

Incidentally, if you walpha a number, Wolfram Alpha will give it to you in Babylonian symbols.  Here’s  34.  But watch out: it’s sometimes wrong.

Edited 7/8 to add: Although I read in one book (I’m not sure which now; it’s been several years) that this was done by the Babylonians, Victor J. Katz’s A History of Mathematics doesn’t match that.   I’m not quite sure what to believe now.

(5) Grid or lattice multiplication.  A lot of schools are teaching this today because, although it takes a little while to draw the boxes, you do all your single digit multiplication first and then all your addition.  I’ve heard people complain about it being a new-fangled method, and I like to point out that it’s actually about a thousand years old (Doesn’t that sound snarky?  I try to say it in a friendly way.  I probably fail.).

It’s possible that it originated in India (I’ve seen sources claim this, but I think it’s not considered absolutely certain); it certainly appeared in Arabic books before it made its way to Western Europe.

Here’s how it works.  Suppose you want to multiply 345 and 12, which are conveniently the numbers that I found in this Wikipedia example.  You have a 3-digit number and a 2-digit number, so you make a 3×2 array of boxes.  Put the first number (345) on the top, and the second number (12) on the side.  One of those is the multiplier and one is the multiplicand, but I can never remember which is which.

The picture above also shows the next step:  divide each little square in half, and then multiply each pair of digits.  For example, 5×2 is 10, so you write the 1 above the diagonal and the 0 below.

For the next step, you add along the diagonals, carrying as necessary.

The reason this works is that the digaonals automatially take care of place value.  For example, look at the third diagonal, with the purple arrow (that ends up in the hundreds place).  It has the 4 of 4×1, which was really 40×10=400.  It has the 0 in the tens place of 4×2, which really would stand for the hundreds place of 40×2.  It has the 6 of 3×2, which really comes from 300×2=600.  Plus it has anything that was carried from the previous diagonal.

Writing the total flat along the bottom makes more sense if you’re teaching it to kids, but in some older books people wrote the answer on the bottom and left-hand side, like on this example from page 23 of The Treviso Arithmetic (Arte dell’Abbaco), an Italian textbook [published in the town of Treviso in Northeastern Italy] from 1478.  This example shows that 934×314=293,276.

When I do this in class I often decorate the sides when I’m done, because I’ve seen that done before.  Then it looks like this:

(5½) There’s a variation of the above method in which the diagonals go in the other direction but you write the number on the right “upside down”.   Here’s an example from the same page of the Trevisio, again showing 934×314=293,276.

I didn’t want to count this as a separate method, because it has all of the same principles as the previous grid.  But I wanted to mention it because if you look at it, it’s a small step to getting the next version of 934×314, shown below, in which the boxes above have had the diagonals removed but you still add along the diagonal (although the total is written at the bottom):

This is actually a little bit confusing because with the diagonals removed, the numbers don’t quite line up the way they’re supposed to.   It’d be clearer if the rows were shifted a bit, like this:

And THAT looks an awful lot like “traditional” multiplication, where in this case “traditional” means “the way I learned multiplication in in 1978”.  But this isn’t a new-fangled method  either:  this “traditional” multiplication ALSO appeared in the Trevisio!  And that leads us to….

(6) Multiplication like I first learned it, like this:

Phew!  That was more lattice multiplication than you ever wanted to see, wasn’t it?  But isn’t it neat how all those pictures (all on the same two pages) just lead into one another, even though they actually were in the reverse order in the original book?

Incidentally, while we’re on the subject of The Trevisio it might amuse you to learn that this 500 year old book also contains the problem “If 17 men build 2 houses in 9 days, how many days will it take 20 men to build 5 houses?”.  If you’re wanting to get yourself a full copy, here’s a link to all 24 Mb of it!  Or, if you don’t read Italian, you can learn a little more about it in this 1996 column by Ivars Peterson.

Next up, some different ways that they don’t teach in school.

Edited to add: Here is The Second Bunch of Ways!

### Ethiopian Multiplication

June 9, 2009

One of our recent (oh my goodness has it really been seven years???) grads just sent me this Youtube video of Ethiopian Multiplication, with a note that this reminded her of History of Mathematics.  Which, of course, made me totally happy.

This method of multiplication is also called Egyptian Multiplication (because it was done in Egypt) and Russian Peasant Multiplication (although the Peasant part might be intended as a bit of a pejorative).

Here’s the basic idea:  Suppose you want to multiply two numbers like 14 and 12.  You could use your fingers, of course, but here’s another way:

Start with the two numbers on top.  Halve one, ignoring any remainders or fractions, and double the other, stopping when you get to 1.

14 & 12

7 & 24

3 & 48  [See how I ignored the fact that halving 7 leaves 1 left over?]

1 & 96  <— Stop here.

Now look at the numbers on the right.  Some are across from an even number: in this case, 12 is across from the original 14.  Ignore those, and add the rest.  So we’ll add 24, 48, and 96, which were across from odd numbers, and get 168.  And that’s the product!  Isn’t that cool?

(I think it would be fantastic to write a book called 25 ways to multiply.  I only have about 13 at the moment, though.)

Here’s the video!

Number photo from gokuro.

### Longitude, Part II

June 1, 2009

As mentioned in the last post about longitude, while one group of people were charting stars and hoping to use tables to help out with the determination, others were working the time angle (so to speak).  What those folk needed was a good clock, one that would keep time even if it got bounced around a bit, like on a ship, because people on the ocean were in especial need of figuring out where they were.

So people worked on it.  And worked on it.  And then worked some more.  Prizes were offered, and went unclaimed.  Then a famous shipwreck in 1707 (involving HMS Association, HMS Eagle, HMS Romney, and HM Fireship Firebrand) took the life of 1500 sailors, apparently because they miscalculated longitude, and Britain was all, “Enough of this!” and 1714 formed the Commission for the Discovery of the Longitude at Sea, which was a mouthful to say so everyone just called it the Board of Longitude.  They offered a prize for calculating longitude and didn’t even insist that the longitude be exact, just within 60 nautical miles for a prize, or within 40 or 30 nautical miles for better prizes.

Uh, nautical miles?  One nautical mile is 1 minute of an arc of latitude, so 60 nautical miles would be 1º and 30 nautical miles would be ½°.  Of latitude.  It translates to just over 1 regular mile.

Where were we?  Oh yes, in England.  Which is also where John Harrison was.   He was born in 1693, and made clocks out of wood with his younger brother.   One of his great achievements was to design the parts so that they had almost no friction, and therefore didn’t need any oil.  This was a big improvement because 18th century oil quite frankly stunk as far as clocks were concerned.

Harrison decided to make a clock good enough to win the prize.  His first clock, conveniently called H1, was made when he was about 40 years old.  And it worked well during the Official Testing on board a couple ships (because you didn’t think a prize would be awarded without checking how the clock did at sea, did you?) but Harrison wasn’t completely happy with it so instead of the full prize he asked for money to make a second version.  He worked on the next clock (H2) from 1737 to 1740, then decided that was all wrong and began work on H3.  This took 19 years — our man Harrison was nothing if not thorough.  But sadly, H3 wasn’t good enough to win the prize, and meanwhile he began working on — hold your breath everyone — H4.

Incidentally, one of the neat things about Harrison’s clocks is that they weren’t just different versions of the same thing.  It’s not like he said, “Hey, I have a new edition out — no, really, the fact that I changed one tiny thing makes it completely different.”  His clocks really were different, and H4 was down to being a pocket watch, which is mighty convenient for being on board a ship.

Here the story gets complicated.  H4 kept really good time, losing less than a second a day, but the Board of Longitude was all, “Well, maybe, maybe not” and Harrison had to make more copies, and blah blah and yadda yadda yadda and the end result was that he also made a new clock H5, plus his buddy Larcum Kendall made a copy (called K1), but the Board was still, “Umm, well” and people  — by people I mean King George III — got all upset and finally in 1773 the Board said, “OK, you win.”

Interestingly, this recent article from New Scientist says that when they opened up H1 to re-fix something (it had been in disrepair and was fixed more than 40 years ago), the way the parts were manufactured suggested that he had some help with some of the chains and whatnot inside.  It’s a pretty interesting article, and the comments are fun to read (mostly saying things like “Of course he had a bit of help!  He didn’t smelt his own metal, now did he?”) but the best part is the gallery of pictures here.

The photo of H5 is published on wikimedia by racklever under GNU-FDL.  A lot of the information about Harrison is from this site.

### Longitude, Part I

May 28, 2009

So yesterday we talked about how to find your latitude if you’re in the Northern Hemisphere (and I’m pretty sure there are ways to find it in the southern hemisphere with an astrolabe, but I don’t think they’re as straightforward).  Longitude, however, turns out to be tough no matter where you are.

For one thing, although the Earth spins so there are two natural poles, and therefore you can measure Latitude in relation to those poles, there isn’t any natural Longitude:  it’s all in relation to an arbitrary point.  According to this official site at Greenwich, Hipparchos was the first guy to use longitude formally, about 150 BCE,  and he used Rhodes in southern Greece as the key spot.  About 250 or 300 years later Ptolemy joined the fun, but he used the Canary Islands in Spain as the starting point.    Mecca was used as a Prime Meridian for many mathematicians.  France had one, as did the US and Canada.  Altogether there were lots and lots of 0°, which could get a little confusing, and so about 130 years ago a bunch of countries got together and decided that Greenwich could be #1 Number 0, and everyone was happy.  Except France.  France didn’t want to give up its own Prime Meridian through Paris, so they refused to vote for Greenwich and kept using their own meridian until…until… actually, I don’t know if they ever formally adopted the Greenwich one.  Here’s a picture of the Paris one:

FredA took this picture (published under GNU-FDL).

So the first problem is deciding on a reference point, and clearly that was a big one.  The next problem is figuring out how far away from that reference point you are, and that’s hard too because the earth keeps spinning.  The simplest way is actually to use time: the Earth makes one complete rotation in 24 hours 23 hours and 56 minutes, so the Earth turns 15° in just under an hour.  Which means that if your buddy calls you and says, “Dude, did you see that amazing eclipse that happened at 1:23?” and you say, “Yes I did, but here it happened at 2:15!” then it means that you are 52 minutes ahead, so that’s (52 minutes)*(1 hour/60 minutes)*(15°/1 hour)= 13° longitude further east than your friend (assuming the AM/PM match up).    You could also set a clock to noon, and then if your clock keeps working  you could compare it to local noon (when the sun is at the highest point in the sky) as you travel away from your starting point.

You don’t HAVE to use time to determine longitude, and in fact some of the great minds did look into ways to use the stars to find a way, but it did turn out that clocks were the key (as we’ll see in part II).

### Latitude

May 27, 2009

Suppose you’re on the open seas, and your GPS falls overboard.  Or maybe it’s 400 years ago.  How can you tell what your latitude and longitude are?

We’ll start with latitude, because it’s easier.  A LOT easier.  If it’s a nice night, and you’re somewhere in the Northern Hemisphere, the easiest way is to take a gander at the North Star using an astrolabe.  Some astrolabes are simple, others are complicated, but they can all be used to find the North Star.

Hey, how about if Godzilla shows us how to use a homemade one!

This model was originally designed by…someone.  I don’t actually know who came up with it.  But all you need is a protractor, a straw, tape, string, a weight, and someone to drill the hole in the protractor so you can attach the string [or you can use a paper protractor].

In theory, the bottom of the straw should be held up to the eye.  Godzilla’s short arms aren’t really conducive to holding things up (he’s had to survive using his brains as much as his brawn), so here’s a drawing of what it would look like:

When you look through the straw at an object like the top of a tree or the North Star (Hmm, why didn’t I draw a star?  Because Godzilla was actually facing tree in our yard at the time, so I got distracted).  Anyway, when you look through the straw, the weight causes the string to fall right at the angle that the object is above the horizontal!

Here’s why:

So sailors just grab their handy dandy astrolabe, measure the distance of the North Star above the horizon, and that’s the latitude!  [I always have to think that one through.  On the equator, the North Star would pretty much be on the horizon, so that’s 0° above the horizon, and at the North Pole the North Star would be directly overhead, which is 90° above the horizon.]

Next up:  Longitude (Part I and Part II).

### Memorial Day 2009

May 25, 2009

Last year, we posted a brief discussion of the history of Memorial Day, a US holiday of remembrance of Americans who died in military service for their country.

Following up on last year’s post:   Frank Buckles is now 108, and [naturally] is still the last known surviving American veteran of World War One.  Last year’s NPR interview with him is still available on-line.

Photo taken by TwoPi in January 2008 at Fort Rosencrans National Cemetery, on Point Loma, San Diego,  CA.

### The Actual Dresden Codex

May 5, 2009

Happy Cinco de Mayo!    A year ago we celebrated the day here by talking about some Cinco de May math, branching out into Spanish Colonial Mathematics [including a really cool multiplication technique called The Method of the Cup].

But that was a pretty broad geographical region, so today it’s back to Mexico, with Guatemala, Belize, and maybe a little bit of Honduras and El Salvador thrown in.  (Aside:  I often make students learn where countries are when I teach The History of Mathematics, and a few times I’ve had them memorize all the countries of Central America for the first test.   One year I had the great idea that I’d test them on a map with no political boundaries, reasoning that there is some value in being able to identify countries by other landmarks.   Although I’d warned them about this, it turned out to be a truly awful test to grade — some people started off wrong, so while the countries were right in relation to each other they were all either too far north or too far south, and figuring out partial credit was a bear. I’ve never given a test like that again.)

So anyway, here’s the region we’re talking about:

Yep, it’s the Mayas!  I’ve seen a lot of math books that talk about Mayan mathematics, because it’s pretty straightforward (until it isn’t).

A dot means 1:  •
A line means 5:  ______
And a shell means zero.

The Mayans used a modified Base 20 system, so that the group of numbers at the bottom counted the 1s, the next group up counted the 20s, the next group up counted the 360s [and yes, it should really be 400 because twenty 20s is 400, but for some reason they used 360 here.  I’ve seen speculation as to why — it might have something to do with the fact that there are just over 360 days in a year — but no one knows for sure], the next group up counted the 7200 [because that is twenty 360s], the next group up counted the 144000 [because that is twenty 7200s], etc.

But the point of this post is that because the Internet is such an amazing place, you can see images of the actual original sources that use this!  Sadly, there aren’t that many:  most of the writings were destroyed, and only about four codices remain.  Codices are folded books, like this one (from the St. Andrew’s web site).

This is the Dresden Codex, and it’s about a thousand years old.  It was made of Amatl paper [fig bark covered with a lime paste], and although the picture above is in black and white it was actually done in color.  Indeed, when it came to numbers the groups (1, 20s, 360s, etc) were written in alternating black and red so that it was easy to tell where one group ended and other began.

Look at that number in the lower left-hand corner.   There are 7 (1s), then 1 (20), 4 (360s), 6 (7,200s), 3 (144,000s), 11 (2,880,000), 3 (57,600,000s), and 6 (1,152,000,000s), giving a sum of over 7 billion.  It’s a pretty big number, and only uses 2 symbols [since the zero wasn’t  necessary in this case].

When I started teaching about Mayan numbers, there weren’t many pictures of the Dresden Codex available so I made do with a photocopy of the one of the pages.  But the Internet is an amazing place, and now you can actually see the entire thing in color in not one but two places!

The first place I found it is this site (archaeoastronomie).  It has the best pictures; my favorite for looking at numbers is page 51 and the pages around it [the zooming is really impressive, although it doesn’t necessarily zoom to the page you want].

### Things that equal Pi

March 13, 2009

So you want to make a pie for Pi Day, but you don’t want to decorate it with the traditional symbol $\pi$.  What other expressions could you use that are equivalent?

You could go with the elegant:  a picture of a circle and the ratio of the circumfirence to the diameter

$\frac{C}{d}$

In a similar vein, you could move up a dimension to area

$\frac{A}{r^2}$

or volume $\left(\frac{3V}{4r^3}\right)$, although in this case you’d have to draw a sphere and I can tell you right now that I’d lose points for clarity.

If geometry isn’t your thing, you could decorate your confection with an infinite sum, perhaps the Madhava-Gregory-Leibniz series (discovered by Madhava of Sangamagram, India about 600 years ago, and then rediscovered by James Gregory of Scotland and Gottfried Wilhelm Leibniz of Germany 200 years later)

$\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\cdots$

or the slightly more complicated

$\sqrt{\frac{6}{1}+\frac{6}{4}+\frac{6}{9}+\frac{6}{16}+\cdots}$

found by Leonard Euler of Switzerland in 1735.  Or even the Bailey-Borwein-Plouffe formula (which is, face it, kind of fun to say) that was discovered only 14 years ago(!) by Simon Plouffe of Quebec, Canada:

$\displaystyle\sum_{k=0}^{\infty}\frac{1}{16^k}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6}\right)$

Incidentally, Simon Plouffe and Neil Sloane are the authors of the Encyclopedia of Integer Sequences, which gave rise to the online version.

But back to $\pi$.  Do you prefer products?  Then maybe you’d want to turn to Wallis’s product, discovered by John Wallis of England in 1655:

$2\cdot\left(\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\frac{8}{9}\cdot\cdots\right)$

We’ll end on a more radical note:  the Viète formula, which was named after François Viète of France, but actually found by Euler.

$2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\sqrt{2+\sqrt{2}}}\cdot\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdot\cdots$

### Oh Christmas Tree!

December 25, 2008

If you have a Christmas Tree, you might be wondering where it came from.  Well, not the tree itself (though come to think of it, I’m not at all sure where ours came from other than a store on the highway, but it’s big and fabulous and hasn’t dropped too many needles.) but the custom of bringing a tree into the house and decorating it.

My research first led to the story that the Christmas tree dated back to a 7th century monk who used the triangular shape of the tree to represent the Holy Trinity.   This story is false, which should come as no surprise to anyone who has actually looked at an evergreen and realized that they are in fact three-dimensional and not flat triangles (and it’s hard to see how a cone could represent a trinity, although that does lead to interesting speculation about just what it would represent).

According to History.com, the Christmas tree actually has a much longer tradition, dating all the way back to the Roman Winter Solstice holiday of Saturnalia.  Saturnalia sounds a bit like baccanalia, and indeed both involved enjoying food and drink.  There were plenty of other feasts at this same time, including Juvenalia (honoring children) and the December 25 birthday of the infant god Mithra, which was viewed by some devout Romans as the most sacred day of the year.  During this season of celebrations, the Romans would bring in evergreen boughs to represent that the days were starting to get longer.

But the Romans weren’t the only ones who used evergreens in this way.  Egyptians and Druids also decorated with evergreens, and the Vikings thought this was the special plant of their Sun God Balder [no relation to the word balderdash].

But still, bringing in tree branches is a little different than bringing in the entire tree.  Although this site says that in the middle ages  Paradise Trees (evergreens decorated with apples) were used to celebrate/symbolize the Feast of Adam and Even on December 24, it’s the Germans who were primarily responsible some 500 years ago for the modern Christmas tree.  The Germans are also the ones who decorated the trees, and Martin Luther apparently is the first person to put candles on the tree.   (I haven’t seen too many candle-lit trees in the United States, but my German roommate in grad school said that her family still put little candles all over the tree and and lit them on Christmas Eve.)

Folk in the Colonies were much more skeptical of this whole tree thing and its pagan origins.    Many Puritan folk viewed decorated trees, Christmas carols, and the like as a distraction from the sacred aspect of the day, and a 1659 Massachusetts Law made it illegal to decorate or do anything to celebrate Christmas other than go to church.  Bah, humbug!  But the Irish and particularly the Germans were still all rah rah in favor of really celebrating Christmas, and in the mid 1700s the  German folk in Pennsylvania still had community trees.  Of course, many of those were actually wooden pyramids decorated with branches, making artificial trees more traditional to the US than actual trees.

However, the biggest influence on the acceptability of Christmas Trees in the United States was England’s Queen Victoria.  In 1846 she tacitly gave her approval to the tradition by being featured with her family in front of a big tree on the cover of the  Illustrated London News.

That made it The Thing to Do, and so the concerns about the Pagan Origins of the Christmas tree were put aside.   Ironically, it was about this time that the Germans started to get really concerned about conserving the fir forests, and encouraging people to use artificial trees.  Plastic wasn’t readily available, so the trees of those days used feathers, dyed for a more realistic look.  And from there, whether artificial or real, aluminum, wooden, or living, Christmas trees stuck with the mainstream Christmas celebrations of many countries.

### Happy Thanksgiving!

November 27, 2008

While you’re baking your turkey and pondering why the temperature continues to rise even after the bird comes out of the oven or thinking about how Game Theory showed that tryptophan affects trust and cooperation, you might also be wondering, “What kind of math did the folk at the first Thanksgiving study?”

The answer is…I don’t know. So I looked around at what textbooks would have been available. I couldn’t find any math texts of the Wampanoags from around 1621, and while I was enticed by a reference to “our earliest native American arithmetic, the Greenwood book of 1729” from David Smith’s History of Mathematics Vol.2, in looking further it seems that the phrase “native American” was used only in contrast to having been originally published in England, Spain, or other non-American country. So as far as the Wampanoags are concerned, I’ll admit to remaining in the dark about what specific kinds of math they would have been learning (formally or informally).

Then I looked into the Pilgrims. As they hailed from England, it seems likely that they’d be using a an English text. And for this they had few choices: there was a 1537 book entitled An Introduction for to : Lerne to Reckon with the Pen and with the Counters after the True Cast of Arsmetyke or Awgrym [that last word is Algorithm], which was a translation of a book in Latin by Luca Pacioli. Wikipedia mentions that another textbook was published in 1539, but I couldn’t find reference to the title or author (though I suppose I could add one myself).

A more likely candidate would be Arithmetick, or, The ground of arts by Robert Recorde, first published in 1543 and listed in several places (e.g. this site) as “The first really popular arithmetic in English”. The title is short for The grou[n]d of artes: teachyng the worke and practise of arithmetike, moch necessary for all states of men. After a more easyer [et] exacter sorte, then any lyke hath hytherto ben set forth: with dyuers newe additions. Here’s a woodcut from the 1543 version

and here’s the title page from the 1658 edition, with its fancy modern spelling:

(Heh heh — I just ran that page through Adobe’s Optical Character Recognition, and it pretty much didn’t recognize a thing.)

I couldn’t track down any copies of what exactly was in the book, but this article about medicine in English texts quotes quite a bit from Recorde’s book. For example:

In 1552, Record amplified his arithmetical text with, among other material, a “second part touchyng fractions”. This new section included additional chapters on various rules of proportion, including “The Rule of Alligation”, so-called “for that by it there are divers parcels of sundry pieces, and sundry quantities alligate, bounde, or myxed togyther”.

The rule of alligation, we are told, “hath great use in composition of medicines, and also in myxtures of mettalles, and some use it hath in myxtures of wynes, but I wyshe it were lesse used therein than it is now a daies”. Despite Record’s regret about the adulteration of wine, the first problem the Master uses to exemplify his discussion of alligation in fact deals with mixing wines; the second involves a merchant mixing spices; and the remainder involve the mixing of metals. None of the examples involve medicine, although the merchant’s spices are once called “drugges”.

A third part was added in 1582 by John Mellis. Here’s a page (from Newcastle University) showing one of the pictures in the margin (possibly from that year, or possibly a later edition; I couldn’t tell for certain).

Robert Recorde is actually better known for a later book: The Whetstone of Witte, which was published in 1557 and so is another candidate for An Original Thanksgiving Math Text.  It was called because, according to the poem on the front, “Here, if you list your wittes to whette, Moche sharpenesse therby shall you gette,” and appears (using Google books) to have been well-known enough to be referenced by Shakespeare in Act 1 Scene 2 of As You Like It (written around 1600) when Celia says, “…for always the dulness of the fool is the whetstone of the wits. How now, Witte! whither wander you?”

The Whetstone is best known for its introduction of the equal sign =, which Recorde explains below:

Nowbeit, for easy alteration of equations.  I will pro(?provide?) a few examples, because the extraction of their roots, may the more aptly be wrought.  And to avoid the tedious repetition of these words “is equal to”, I will set as I do often in work use a pair of parallels or Gemowe lines of one length, thus: ======, because no two things can be more equal….

So there you have it: math that some of the first Thanksgiving folk could possibly have studied.

### Perfection

November 26, 2008

In the episode “The Boneless Bride in the River” in Season 2 of the TV show Bones, the body of a young woman is found in a river, and it’s discovered that she was likely brought over to the US on a fiancée visa. About 14 ½ minutes into the episode, two of the main characters (Special Agent Seeley Booth and Forensic Anthropologist Dr. Temperance “Bones” Brennan) have this conversation:

Booth: Homeland Security says the fiancée visa was expedited by a lawyer on retainer into a smaller bride agency here in town called “The Perfect Wife”.
Brennan: Oh that sounds archaic.
Booth:
No, you know, in therapy I learned that superlatives like perfect are meaningless.
Brennan:
Not in science. A perfect number is a number whose divisors add up to itself, as in one plus two plus three equals six.
Booth:
Well, in therapy I learned that definitive statements are by their very nature, wrong.
Brennan:
Isn’t the statement “definitive statements are by their very nature wrong”, definitive, and thus wrong?

Speaking of wrong, Brennan was a little bit wrong in her definition: a perfect number is one whose proper divisors add up to itself. But still, neat math in a neat show is always worth a mention. And perfect numbers are pretty neat, because like so much in number theory they’re simple but there are still open problems about them.

A bit of history [where “a bit” apparently means “a lot because I don’t know how to edit today”]: perfect numbers were studied by Pythagoras, which makes the concept at least 2500 years old. Euclid also talked about perfect numbers a few hundred years later in Book IX of The Elements. In particular, Thomas Heath’s translation of Proposition 36 states:

If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect.

As a translation of this translation, this is saying that if 1+2+22+…+2k is prime, then that sum times the last number in the sum (2k) must be perfect. For example, 1+2=3 is prime so 3·2=6 is perfect. Likewise, 1+2+4=7 is prime, and 7·4=28 is perfect. [As a note, the formula is is sometime written out algebraically with 2k+1-1 used instead of 1+2+22+…+2k; in addition, when it’s written that way it’s sometimes reindexed so that k is used in place of k+1, and the statement becomes “If 2k-1 is prime, then (2k-1)·2k-1 is perfect.”]

A few hundred years after Euclid, Nicomachus wrote some more information about perfect numbers. According to the St. Andrew’s web site, he made
five claims:

(1) The nth perfect number has n digits.
(2) All perfect numbers are even.
(3) All perfect numbers end in 6 and 8 alternately.
(4) Euclid’s algorithm [described above] to generate perfect numbers will give all perfect numbers
(5) There are infinitely many perfect numbers.

Nicomachus’s word was law as far as perfect numbers were concerned, and his claims, while unproven, were believed for decades centuries a really really long time. Of course, there were only four perfect numbers known at that time (6, 28, 496, and 8128), so they really didn’t have much to go on. In reality, the 5th and 6th perfect numbers (33,550,336 and 8,589,869,056 respectively) disprove claims (1) and the “alternately” portion of (3), but it took a while for someone to discover those.

(Which leads to a little tangent: who did discover those numbers? Ismail ibn Ibrahim ibn Fallus (1194-1239) made a list of ten numbers he thought were perfect. Three of them weren’t, but the other seven were. Sadly, a lot of other mathematicians had no idea about this list: mathematicians in Western Europe had to wait another 350 years for those numbers to enter their collective psyche, during which time a few other mathematicians found the 5th and 6th perfect numbers and were equally ignored.)

But even after it was known that Nicomachus’s claims weren’t themselves perfect (ba DUM!), mathematicians continued to study the numbers. In the 1600s Pierre Fermat tried to find patterns, and ended up discovering his Little Theorem as a consequence. Marin Mersenne also spent some time on it, and in fact his exploration of when 2k-1 is prime, as a part of that theorem of Euclid’s mentioned above, led to the notion of Mersenne primes (primes of the form 2k-1 where k itself is prime).

In the 1700s Leonhard Euler entered the fray. He couldn’t prove that Euclid’s formula generated all perfect numbers, but he did show that it generated all even perfect numbers. And a bunch of other mathematicians spent a lot of time trying to show that numbers were or were not perfect (which was related to showing that specific numbers of the form 2k-1 were or were not prime), a challenging task in the pre-computer days.

Not that we’re doing much better now. As of this moment, we still only know of 46 perfect numbers, and they’re pretty big. We do know a few cool things about perfect numbers in general:

• Even perfect numbers end in 6 or 8.
• Even perfect numbers are triangular numbers (e.g. 6=1+2+3 and 28=1+2+3+4+5+6+7) where the ending digit is one less than a power of 2.
• The reciprocals of the divisors of perfect numbers all add up to 2:
$\frac{1}{1} + \frac{1}{2}+\frac{1}{3} + \frac{1}{6}=2$
$\frac{1}{1} + \frac{1}{2}+\frac{1}{4} + \frac{1}{7}+\frac{1}{14} + \frac{1}{28}=2$

But there’s a lot we still don’t know:

• We don’t know if all perfect numbers are even.
• We don’t know if there are a finite or infinite number of perfect numbers.

In other words, perfection continues to eludes us.