Math with Roman Numerals

I’ve been reading (listening to, actually) a book on the state of scientific and technical knowledge in what has traditionally been referred to as the “Dark Ages” (roughly the 5th to 14th centuries). One of the things I became intrigued by was the use of Roman numerals for relatively complex math during this period.

Quick Review

You’re probably familiar with Roman numerals. Conceptually, they’re easy. Letters represent numbers. To get the value of a number written in Roman numerals you simply add the individual numerals (or groups of numerals). Here are the letters you use to represent values up to 3999:

I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1000

Values are read from left to right, and it is traditional to put the larger numerals first. So III is 3, XII is 12, and DCLXVI is 666 (“let the reader understand”). One complication is that you never write the same numeral more than 3 times in a row, and that a smaller numeral appearing before a larger one means it should be subtracted from the larger value. So 14 is XIV, not XIIII. 3999 is MMMCMXCIX. That being said, the Romans weren’t real picky about how these numbers were written. For simplicity, I’m going to ignore the no-more-than-3 rule from time to time in my discussion below.

I’ll use the symbol => to indicate that I’m rearranging or simplifying.

Adding and Subtracting

Addition with Roman numerals is easy: just put all the numerals in one group and rearrange them. So III + XVIII is IIIXVIII => XVIIIIII => XVVI => XXI.

Subtraction is similar. Convert larger digits into groups of smaller ones if needed for convenience, then just subtract similar numerals from each other. So XXI – III is XVVI – III, which is XVIIIIII – III, or XVIII.

Doubling and Halving

Before getting into the fun stuff (multiplication and division), it helps to think about how to double and halve a value written in Roman numerals.

Doubling is just adding the number to itself. To double a number, just write two of each numeral. Twice XXI is thus XXIXXI => XXXXII => XLII. Twice III is IIIIII => VI.

Halving is similar. Divide each numeral by 2 and append the next lesser numeral if there is a remainder. This is slightly circular, since we’re saying to divide by 2 you divide by 2. If you don’t remember what half of L is, write it out as XXXXX and take half of those X’s to get XX with a remainder of half an X, which is V. To get half of 666, follow these steps:

CCL + half of CLXVI
CCL + L + half of LXVI
CCL + L + XXV + half of XVI
CCL + L + XXV + V + half of VI
CCL + L + XXV + V + II + half of II
CCL + L + XXV + V + II + I


Multiplication is a combination of doubling, halving, and adding remainders. Note that in general A ✖️ B is the same as (A / 2) ✖️ (B ✖️ 2). That is, if we double one term and halve the other term, the result is the same. In familiar terms, 8 ✖️ 3 (24) is the same as half 8 (4) times twice 3 (6). Further note that we can do that again, so:

8 ✖️ 3 = 4 ✖️ 6 = 2 ✖️ 12 = 1 ✖️ 24 = 24

Where it gets tricky with Roman numerals is handling remainders. Consider if we reversed which term was being doubled and which halved in our example above:

8 ✖️ 3 = 16 ✖️ 1.5 = 32 ✖️ .75 ….

While the first attempt actually took us all the way to the correct answer, the second appears to be getting harder and harder. Let’s look at both with Roman numerals:



When the halving leaves a remainder, we can ignore it. I’ll mark those with an asterisk.

* drop remainder



When we have remainders, there’s one more step. We need to add the value from the “double” column to our final result to get our true result (since we technically dropped that value when we dropped the remainder):


We can apply this technique to do arbitrarily complicated multiplication:

39 ✖️ 81

* drop remainder
* drop remainder
* drop remainder
* drop remainder



Now add the value from the “double” column for each odd value in the “half” column.


I’ll leave it to the reader to verify my work.


I’m not sure if there’s an easier way to do vision, but here’s what I came up with. First, we repeatedly double the divisor (the number by which we’re dividing) until we get a value greater than the dividend (the number that we’re dividing). We’ll keep track of the multiplier for each because we’ll need those later.

Consider 365 / 12

In Roman numerals: CCCLXV / XII



exceeds dividend

We now know that XII goes into CCCLXV at least XVI times. We can make a note of that, and also subtract the doubled divisor to see what’s left:

XII goes into CCCLXV XVI times with a remainder of CLXXIII

Now check our list of doubled divisors to see what the biggest one is that goes into the remainder (CLXXIII). It looks like LXXXXVI is the largest value that is smaller than CLXXIII. Repeat the step above. Subtract LXXXXVI from our remainder and add the multiplier to our accumulated multiplier:

XII goes into CCLXV XXIV times with a remainder of LXXVII

Repeating, this time using a multipler of IV and a doubled divisor of XXXXVIII:

XII goes into CCLXV XXVIII times with a remainder of XXIX

Repeating, this time using a multiplier of II and doubled divisor XXIV:

XII goes into CCLXV XXX times with a remainder of V.

Since V is less than our original divisor, that’s as far as we can go.

Again, the proof is left to the reader.


I’m not going to dwell on fractions except to say that the only fractions the Romans seemed to use were twelfths and halves. Each twelfth was a dot following the number. Since we coincidentally divided by 12 in the previous example, our remainder of 5 would be represented as 5 dots following XXX. I believe they were written in some kind of pattern, perhaps like we use on dice or playing cards.

Halves were represented using the letter S. So 7/12 would be “S..” or maybe “S:”.


While this is fun, and while our Middle Ages ancestors could do amazing things using these simple techniques, it’s pretty clear why the hard math was being done using base-10 and place value representation.

It’s Time to Move Christmas to January

Today’s post comes to you from my perch high atop Mt. Crumpit.

I live in Iowa, in the heart of what is colloquially known as “the midwest” – even though it is not really “west” of anything but is more “mid mid” longitudinally. Latitudinally, we are north of most of you but still in the southern north half of the country.

It snows here. A lot, some years. We usually get our first snow on or about Thanksgiving. It may melt before Christmas, but about half the time our Christmases are white. It gets worse before it gets better; we’ll get about two feet of snow over the course of an average winter.

I live in a little town in Iowa. Marion has about 30,000 residents. My office downtown (we call it “uptown” here because we think that makes us sound sophisticated) is in a 150-year-old building across the street from Marion Square Park, which features a pavilion with the roof and footprint of the old railroad station that used to sit about a block east of its current location. About three quarters of the buildings I can see from my window are as old or older than the one I’m in. About a half dozen of them house antique shops.

The week after Thanksgiving, Santa Claus arrives at the park by fire truck and takes up residence in a little outhouse in the park. Kids line up around the block to see him. For the rest of the Christmas shopping season, Christmas carols are played over loudspeakers in the park. If we open our office windows (which we wouldn’t dare do), we can hear the music from the park across the street.

In these parts, looking forward to Christmas is what makes the crappy, cold weather bearable. The thought of Christmas, family, carols, cookies, and presents gives us the strength to get through the first month of winter weather.

The problem is, Christmas comes too soon. Sure, we’re happy basking in the knowledge that the miserable weather is what keeps the rest of you out of our quaint little town the rest of the year. But this twisted schadenfreude is not quite enough to make it worthwhile. Many of us actually leave town and go spend the winter in warmer climes. The rest of us tough it out by virtue of our mid-mid-south-north pluck.

I think it’s time we do something about this. I think it’s time we move Christmas to January. I think there’s a good case for doing this.

First, we need something to look forward to through more of winter. Moving Christmas to January 25 gives us another four weeks to revel in the anticipation and forget about the weather.

Second, Black Friday to Christmas Eve just isn’t long enough to get everything done that needs to be done. Retailers would love having another month of Christmas shopping.

Third, there’s no way Jesus was born in December anyway, let alone December 25th. We all know that a bunch of nascent Catholics abducted the Roman Saturnalia festival back in the fourth century and attached a bunch of churchy language to it. Jesus’ feelings won’t be hurt if we move his birthday celebration. We already moved it once when we decided to celebrate it on December 25.

Fourth, we all know that Martin Luther King, Jr. Day is just a thinly disguised way for white people to assuage their guilt over slavery and something about lunch counters and bus seats. It’s kind of insulting to African Americans if you think about it, to think that we can make up for thousands of years of oppression with a Monday holiday. I, for one, am not comfortable insulting African Americans. If we move Christmas to January 25, we could move MLK Day to December 25 and make it the launch day of Kwanza, lending some credibility to a somewhat questionable, made-up holiday. Lending mutual credibility to two somewhat questionable, made-up holidays now that I think of it.

All-in-all, this is a great idea whose time has come. It will let us separate Jesus from the false gods of the Greeks and Romans; will let us give more attention to what will now be the unquestionably valid holidays of Kwanza and MLK Day (further removing us innocent white people from our evil, slave-owning ancestors); and will give us something to look forward to while shoveling driveways at 6AM so we won’t be late for work.

Little known Christmas fact: Did you know the woman who did the voice of Cindy Lou Who also did the voice of Bullwinkle’s side-kick Rocket J. “Rocky” Squirrel? Knowing this makes you smarter than your friends, but will ruin How the Grinch Stole Christmas for you, as you hear what sounds like a young flying squirrel asking, “Santie Claus, why?”


“Dad, When Did the Internet Start?”

Dillon and I were talking this morning about people who write checks and keep a running balance in the back of their checkbooks. I got thinking back and figured out I probably stopped keeping a paper check register in 1987 and stopped keeping an electronic one in the early 90’s. Nowadays, my bank keeps track of that for me and I can access it from my phone.

That led to the question, “When did the world wide of web begin?” And that question took me back…

I think my first experiences with any kind of online computing was during the BBS days of the 1980’s. I was a member of the “Hawkeye BBS” run by Ben Blackstock, a local attorney. For $15/year you could dial into Ben’s PC and access the various discussion lists and files that were kept there.

In about 1987 I started paying bills online with CheckFree. There was no Web and no dial-up access to the internet for most people at that time. Your computer called CheckFree directly and send payment requests. CheckFree wrote a physical check against your account and mailed it to the vendor for you. Or they would do an EFT transaction and write the check against their own account.

After I started working at Parsons Technology in 1988, Bob Parsons had me start using Quicken as a way to keep an eye on the competition. Quicken integrated with CheckFree, and MoneyCounts did too, eventually. Eventually Quicken had their own bill payment option and I think I used that for a while.

About that same time, I signed up for CompuServe. CompuServe was another dial-up service that was not unlike the BBS systems from ten years before. It was text-based — you got a menu of a dozen choices of things to do, entered a number to select an item, then you got another menu. All of this in the form of scrolling text — no graphics.

Parsons started doing tech support on CompuServe long before other companies, and we did beta testing there as well using a private forum. CompuServe had its own email service. When they eventually hooked up with the internet, my CompuServe email address may have been my first. As I recall it was Easy to remember.

Sometime in the early 90’s a friend of mine at church started going on and on about the cool things he was doing on the internet. He gave me a phone number to call and told me what to ask for to get a “PTP” account that would let me dial in and have access to the internet. I don’t recall if I was using a Web browser at that point or if it was all just FTP, USENET, Archie, Gopher, and other early protocols. I downloaded instructions to build a nuclear bomb, of course.

In about that same time period, America Online (AOL) came along. For you youngsters, AOL was like the Web in a box. You dialed into AOL and they served up graphical pages not unlike the Web. No Web addresses, though. Instead it was AOL “screen names” and “keywords”. So I was CRAIGR (screen name) and Parsons Technology was PARSONS (keyword). Even today you’ll sometimes see companies say to “enter the internet keyword ‘company name'” to find them on the Web. They’re still living in the AOL of the 1990’s.

Around 1994 or so, Microsoft started MSN, which was their answer to AOL and CompuServe. But the writing was on the wall and the World Wide Web was destined to be the online destination. Both AOL and CompuServe offered connections to the Web, and MSN kind of disappeared and Internet Explorer came along. It shipped with Windows 95. I tend to date most people’s awareness of the internet and the Web to Windows 95, which shipped in August 1995.

In the summer of 1996 I registered and signed up with a company called SimpleNet for Web hosting. I created You can see a very early version of that site from December 1996 here. SimpleNet was eventually purchased by Yahoo, but not before I had a chance to visit them while on a business trip to California. The entire company was in a 3-bedroom condo with CAT5 cable running from room to room. It was pretty cool. They gave me a coffee mug and said I was the only customer who had ever visited them.

Catholics, Protestants, Denominations, and Christianity

Catholics Expressing their Unity

A friend of mine recently commented that one of his Catholic relatives refused to listen to the gospel, saying, “All you Protestants are always fighting with each other and starting a new denomination. How can you claim that any one of them is correct?” While this sounds like a good argument, it’s based on a false understanding of church history.

In the years following Jesus’ death, local churches were formed in the cities to which the gospel message spread. With the exception of their deference to the Apostles in the very early years, these churches had no common leadership, hierarchy, or organizational structure. Each was independent, and each considered its sister churches in other cities to be a part of the larger “body of Christ” on Earth.

Over 200 years after the death of the last of the Apostles, a Roman emperor with Christian and pagan roots brought together hundreds of church leaders from throughout the Roman Empire and began a process that would result in the formation of the Roman Catholic Church.

By adopting a central leadership and placing the opinions of its bishops over the authority of God’s Word, Catholicism separated itself from orthodox Christianity. Catholicism was the first successful “denomination” that split from Christianity in those early years.

In the early 1500’s, a group of Catholics grew dissatisfied with the rituals and doctrines of their church. They split from the Catholic Church, which labeled them “Protestants” because of their protest.

Protestants, argued by my friend’s relative to just be a bunch of disagreeable folks who can’t figure out what they believe, are disgruntled Catholics, not disgruntled Christians. When Protestants (disgruntled Catholics) split from each other, they become yet another group of disgruntled Catholics.

Throughout history — before and during the rise of Catholicism — there have been churches that held to the fundamental doctrines of Jesus and the Apostles. They may have varied on some points, but they retained their independence from hierarchy, their congregational polity, their reliance on the Bible as their sole authority on matters of faith and practice, their commitment to evangelism, and their belief in salvation by faith apart from baptism or other “sacraments”. These churches were severely persecuted by the nascent Catholic Church and continue to be opposed by the Catholic Church and its Protestant brothers and sisters.

So despite my friend’s relative’s claims, it is the Catholic Church that split from Truth, and it is the Catholic Church that is fraught with schisms that manifest themselves as Protestant denominations. Meanwhile Christ’s true church continues undeterred; persecuted but prevailing; united under the umbrella of fundamental doctrines that are unchanged from the first century. It can do this because Christianity isn’t a local church, it isn’t a denomination, it isn’t a bishop, and it isn’t a hierarchy. It is a personal relationship with God through the finished work of Jesus Christ, entirely separate from any organization or ritual. The true church is the universal collection of such people. It is undivided and indivisible, as opposed to Catholicism, which was founded in division and whose history — often incorrectly identified as the “history of Christianity” — is marked by and moves forward through division.