Tag: representation

Why the Romans were pants at maths

Paul Curzon, Queen Mary University of London

The Romans were great at counting and addition but they were absolutely pants at multiplication. It wasn’t because they were stupid. It was because they hadn’t invented a good way to represent numbers, and that meant they needed really convoluted algorithms.

The Roman system is based on an earlier really simple way of writing numbers. You just put a line for each thing you’ve counted. Its probably the way shepherds kept count of sheep, drawing a line for each sheep. Those lines turned into the Roman letter I. To add 1 to a number you just add another I. You count: I, II, III, and so on and it makes counting easy.

Roman number blocks image by Clker-Free-Vector-Images from Pixabay

This system is called unary – whereas binary involves counting with two symbols, 1 and 0, in unary you only have one symbol to count with. Addition in unary is easy too at least for small numbers. Take the first number and add on the end all the Is for the second and you’ve got the answer number. This is exactly the way we all start doing addition on our fingers.To add 2+3, hold up 2 fingers (II) then hold up another three fingers (III) and you have the answer (IIIII).

This is fine for small numbers but it gets a bit tedious as the numbers increase (and you run out of fingers!) Comparing numbers is easy in principle – do you have the same number of Is, but hard in practice for large numbers. We can’t keep all those Is in our head so a large number is hard to think about. To get round this the Romans invented new letters to stand for groups of Is. This is what we do when we tally numbers making a crossbar for every fifth number we count. It helps us keep track of larger numbers. The Romans invented a whole bunch of symbols to help: so for example in the Roman numeral system, V stands for 5 (IIIII), X stands for 10, L for 50, C for 100, D for 500 and M for 1000. They had invented a new way to represent numbers.

Image by Katie Rose from Pixabay

This makes it much easier to write and compare larger numbers. Now when counting and you get up to 5 you just replace all those Is with a V and then carry on adding Is: VI, VII, VIII, VIIII. Then you get to VIIIII (10) so replace it all with an X, starting again adding a new lot of Is: XI, XII, XIII, XIIII, XV, and so on. Counting large numbers is now a bit more involved – the algorithm involves more than just adding an I on the end, but it is much more convenient. The addition algorithm has now become more complicated, though it is still fairly simple too. Take any two numbers to add like VII and VIII and string them together: VIIVIII. Now group together the same letters: VVIIIII. Anywhere you have enough to replace symbols with the next character do so. VV can be replaced by X and IIIII can be replaced by V to give XV in the above. Keep making replacements until you can make no more. Put the symbols in order from largest to smallest symbol and you have your answer.

Now the Romans were obviously a bit lazy as bored with writing even four Is in a row they sometimes introduced a new set of abbreviations, so that IIII became IV and VIIII became IX. Putting a smaller symbol (like I) before a larger one (like X) instead of after meant subtract it to get the number. so IX means “one less than 10” or 9. Counting just got a tiny bit more complicated to get the advantage of writing fewer symbols. Addition now needs a more convoluted algorithm though. There are several ways to do it. The easiest is actually just to change the numbers to add to the simpler form (so IV goes back to IIII). You them do the addition that way, and convert back at the end. Addition just got that little bit harder, and all because of a change in representation.

Worse, doing any more complicated maths is even harder still using the Roman number representation. See if you can work out how to multiply Roman numbers. The Roman number system doesn’t help at all. The only really easy way is to just repeatedly add ( so III x VI is VI + VI + VI). That just isn’t practical for large numbers. Try it on XXIII x LXV1. There are other possible ways including one that is actually based on the binary multiplication algorithms computers use – multiplying and dividing repeatedly by 2. See if you can work out how to do it. Whatever way you do it, its clear that the number system the Romans chose made maths hard for them to do!

A good representation makes maths easy. A bad one makes it much harder to do

Image by Michael Kauer from Pixabay

Luckily, Indian and Arabian scholars understood that the representation they used mattered. They invented, and spread, the Hindu-Arabic numbers and decimal system we use today. What is special about it is that rather than introducing new symbols for bigger and bigger numbers, the position of a symbol is used instead. As we go from nine to ten we go back to the start of our symbols, from 9 back to 0, but stick a 1 in a new 10s column to count how many 10s we have. Counting is still pretty easy but suddenly not only is the algorithm for addition straightforward but we can come up with fairly simple algorithms for multiplication and division too. They are the algorithms you learn at school – though as with any algorithm making sure you follow the steps exactly and don’t miss steps is hard for a human (unlike for a computer). That is why we tend to find learning maths hard at first and it gets easier the more we practice.

In fact Romans needing to do serious maths probably used a variation of an abacus representing numbers with stones. They would do a calculation on the abacus and then convert the answer back into the Roman number system. And guess what. The Roman Abacus uses columns to represent larger numbers in a very similar way to the Hindu-Arabic system. The Romans understood that representation matters too.

Image by Hans from Pixabay

Sometimes things are hard to do just because we make them hard! The secret of coming up with good algorithms is often to come up with a good representation first. In programming too, if you come up with a good way to represent data, a good data structure, you can often then make it much easier to write an efficient program.

This article was first published on the original CS4FN website.

EPSRC supports this blog through research grant EP/W033615/1.

Letters from the Victorian Smog: Braille: binary, bits & bytes

We take for granted that computers use binary: to represent numbers, letters, or more complicated things like music and pictures…any kind of information. That was something Ada Lovelace realised very early on. Binary wasn’t invented for computers though. Its first modern use as a way to represent letters was actually invented in the first half of the 19th century. It is still used today: Braille.

Reading Braille - a hand on a page of Braille
Image by Myriams-Fotos from Pixabay

Braille is named after its inventor, Louis Braille. He was born 6 years before Ada though they probably never met as he lived in France. He was blinded as a child in an accident and invented the first version of Braille when he was only 15 in 1824 as a way for blind people to read. What he came up with was a representation for letters that a blind person could read by touch.

Choosing a representation for the job is one of the most important parts of computational thinking. It really just means deciding how information is going to be recorded. Binary gives ways of representing any kind of information that is easy for computers to process. The idea is just that you create codes to represent things made up of only two different characters: 1 and 0. For example, you might decide that the binary for the letter ‘p’ was: 01110000. For the letter ‘c’ on the other hand you might use the code, 01100011. The capital letters, ‘P’ and ‘C’ would have completely different codes again. This is a good representation for computers to use as the 1’s and 0’s can themselves be represented by high and low voltages in electrical circuits, or switches being on or off.

He was inspired by an earlier ‘Night Writing’ system developed by Charles Barbier to allow French soldiers in the 1800s to read military messages without using a lamp (which gave away their position, putting them at risk).

The first representation Louis Braille chose wasn’t great though. It had dots, dashes and blanks – a three symbol code rather than the two of binary. It was hard to tell the difference between the dots and dashes by touch, so in 1837 he changed the representation – switching to a code of dots and blanks.

He had invented the first modern
form of writing based on binary.

Braille works in the same way as modern binary representations for letters. It uses collections of raised dots (1s) and no dots (0s) to represent them. Each gives a bit of information in computer science terms. To make the bits easier to touch they’re grouped into pairs. To represent all the letters of the alphabet (and more) you just need 3 pairs as that gives 64 distinct patterns. Modern Braille actually has an extra row of dots giving 256 dot/no dot combinations in the 8 positions so that many other special characters can be represented. Representing characters using 8 bits in this way is exactly the equivalent of the computer byte.

Modern computers use a standardised code, called Unicode. It gives an agreed code for referring to the characters in pretty well every language ever invented including Klingon! There is also a Unicode representation for Braille using a different code to Braille itself. It is used to allow letters to be displayed as Braille on computers! Because all computers using Unicode agree on the representations of all the different alphabets, characters and symbols they use, they can more easily work together. Agreeing the code means that it is easy to move data from one program to another.

The 1830s were an exciting time to be a computer scientist! This was around the time Charles Babbage met Ada Lovelace and they started to work together on the analytical engine. The ideas that formed the foundation of computer science must have been in the air, or at least in the Victorian smog.

– Paul Curzon and Jo Brodie, Queen Mary University of London

More on…

This post was first published on CS4FN and also appears on page 7 of Issue 20 of the CS4FN magazine. You can download a free PDF copy of the magazine here as well as all of our previous magazines and booklets, at our free downloads site.

A different way of representing letters is Morse Code which is a series of audible short and long sounds with spaces that was used to communicate messages very rapidly via telegraphy.

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