Lego computer science: representing numbers using position

Numbers represented with different sized common blocks

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…how do we represent numbers and how is it related to the representation Charles Babbage used in his design for a Victorian steam-powered computer?

We’ve seen there are lots of ways that human societies have represented numbers and that there are many ways we could represent numbers even just using lego. Computers store numbers using a different representation again called binary. Before we get to that though we need to understand how we represent bigger numbers ourselves and why it is so useful.

Numbers represented as colours.

Our number system was invented in India somewhere before the 4th century. It then spread, including to the west, via muslim scholars in Persia by the 9th century, so is called the Hindu-Arabic numeral system. Its most famous advocate was Muḥammad ibn Mūsā al-Khwārizmī. The word algorithm comes from the latin version of his name because of his book on algorithms for doing arithmetic with Hindu-arabic numbers.

The really clever thing about it is the core idea that a digit can have a different value depending on its position. In the number 555, for example, the digit 5 is representing the number five hundred, the number fifty and the number five. Those three numbers are added together to give the actual number being represented. Digit in the ‘ones’ column keep their value, those in the ‘tens’ column are ten times bigger, those in the ‘hundreds column a hundred times bigger than the digit, and so on. This was revolutionary differing from most previous systems where a different symbol was used for bigger number, and each symbol always meant the same thing. For example, in Roman numerals X is used to mean 10 and always means 10 wherever it occurs in a number. This kind of positional system wasn’t totally unique as the Babylonians had used a less sophisticated version and Archimedes also came up with a similar idea, those these systems didn’t get used elsewhere.

In the lego representations of numbers we have seen so far, to represent big numbers we would need ever more coloured blocks, or ever more different kinds of brick or ever bigger piles of bricks, to give a representation of those bigger numbers. It just doesn’t scale. However, this idea of position-valued numbers can be applied whatever the representation of digits used, not just with digits 0 to 9. So we can use the place number system to represent ever bigger numbers using our different versions of the way digits could be represented in lego. We only need symbols for the different digits, not for every number, of for every bigger numbers.

For example, if we have ten different colours of bricks to represent the 10 digits of our decimal system, we can build any number by just placing them in the right position, placing coloured bricks on a base piece.

The number 2301 represented in coloured blocks where black represents 0, red represents 1, blue represents 2 and where yellow represents 3

Numbers could be variable sized or fixed size. If as above we have a base plate, and so storage space, for four digits then we can’t represent larger numbers than 9999. This is what happens with the way computers store numbers. A fixed amount of space is allocated for each number in the computer’s memory, and if a number needs more digits then we get an “overflow error” as it can’t be stored. Rockets worth millions of pounds have exploded on take-off in the past because a programmer made the mistake of trying to store numbers too big for the space allocated for them. If we want bigger numbers, we need a representation (and algorithms) that extend the size of the number if we run out of space. In lego that means our algorithm for dealing with numbers would have to include extending the grey base plate by adding a new piece when needed (and removing it when no longer needed). That then would allow us to add new digits.

Unlike when we write numbers, where we write just as many digits as we need, with fixed-sized numbers like this, we need to add zeros on the end to fill the space. There is no such thing as an empty piece of storage in a computer. Something is always there! So the number 123 is actually stored as 0123 in a fixed 4-digit representation like our lego one.

The number 321 represented in coloured blocks where space is allocated for 4 digits as 0321: black represents 0, red represents 1, blue represents 2 and where yellow represents 3

Charles Babbage made use of this idea when inventing his Victorian machines for doing computation: had they been built would have been the first computers. Driven by steam power his difference engine and analytical engine were to have digits represented by wheels with the numbers 0-9 written round the edge, linked to the positions of cog-like teeth that turned them.

Wheels were to be stacked on top of each other to represent larger numbers in a vertical rather than horizontal position system. The equivalent lego version to Babbage’s would therefore not have blocks on a base plate but blocks stacked on top of each other.

The number 321 represented vertically in coloured blocks where space is allocated for 4 digits as 0321: black represents 0, red represents 1, blue represents 2 and where yellow represents 3

In Babbage’s machines different numbers were represented by their own column of wheels. He envisioned the analytical engine to have a room sized data store full of such columns of wheels.

Numbers stored as columns of wheels on the replica of Babbage’s Difference Engine at the Science Museum London. Carsten Ullrich: CC-BY-SA-2.5. From wikimedia.

So Babbage’s idea was just to use our decimal system with digits represented with wheels. Modern computers instead use binary … bit that is for next time.

This post was funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.


Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binaryand computation.

Lego Computer Science


Part 1: Lego Computer Science: pixel picture

Part 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

Part 5: Lego Computer Science: Gray code

Part 6: Lego Computer Science: Binary

Part 7: Lego Computer Science: What is computation (simple cellular automata)?

Lego computer science: representing numbers

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…what does number representation mean?

We’ve seen some different ways to represent images and how ultimately they can be represented as numbers but how about numbers themselves. We talk as though computers can store numbers as numbers but even they are represented in terms of simpler things in computers.

Lego numbers

But first what do we mean by a number and a representation of a number? If I told you to make the numbers 0 to 9 in lego (go on have a go) you may well make something like this…

But those symbols 0, 1, 2, … are just that. They are symbols representing numbers not the numbers themselves. They are arbitrary choices. Different cultures past and present use different symbols to mean the same thing. For example, the ancient Egyptian way of writing the number 1000 was a hieroglyph of a water lily. (Perhaps you can make that in lego!)

The ancient Egyptian way to write 1000 was a hieroglyph of a waterlily

What really are numbers? What is the symbol 2 standing for? It represents the abstract idea of twoness ie any collection, group or pile of two things: 2 pieces of lego, 2 ducks, 2 sprouts, … and what is twoness? … it is oneness with one more thing added to the pile. So if you want to get closer to the actual numbers then a closer representation using lego might be a single brick, two bricks, three bricks, … put together in any way you like.

Numbers represented by that number of lego bricks

Another way would to use different sizes of bricks for them. Use a lego brick with a single stud for 1, a 2-stud brick for two and so on (combining bricks where you don’t have a single piece with the right number of studs). In these versions 0 is the absence of anything just like the real zero.

Lego bricks representing numbers based on the number of studs showing.

Once we do it in bricks it is just another representation though – a symbol of the actual thing. You can actually use any symbols as long as you decide the meaning in advance, there doesn’t actually have to be any element of twoness in the symbol for two. What other ways can you think of representing numbers 0 to 9 in lego? Make them…

A more abstract set of symbols would be to use different coloured bricks – red for 1, blue for 2 and so on. Now 0 can have a direct symbol like a black brick. Now as long as it is the right colour any brick would do. Any sized red brick can still mean 1 (if we want it to). Notice we are now doing the opposite of what we did with images. Instead of representing a colour with a number, we are representing a number with a colour.

Numbers represented as colours.

Here is a different representation. A one stud brick means 1, a 2-stud brick means 2, a square 4 stud brick means 3, a rectangular 6 stud brick means 4 and so on. As long as we agreed that is what they mean it is fine. Whatever representation we choose it is just a convention that we have to then be consistent about and agree with others.

Numbers represented by increasing sized blocks

What has this to do with computing? Well if we are going to write algorithms to work with numbers, we need a way to store and so represent numbers. More fundamentally though, computation (and so at its core computer science) really is all about symbol manipulation. That is what computational devices (like computers) do. They just manipulate symbols using algorithms. We will see this more clearly when we get to creating a simple computer (a Turing Machine) out of lego (but that is for later).

We interpret the symbols in the inputs of computers and the symbols in the outputs with meanings and as a result they tell us things we wanted to know. So if we key the symbols 12+13= into a calculator or computer and it gives us back 25, what has happened is just that it has followed some rules (an algorithm for addition) that manipulated those input symbols and made it spew out the output symbols. It has no idea what they mean as it is just blindly following its rules about how to manipulate symbols. We also could have used absolutely any symbols for the numbers and operators as long as they were the ones the computer was programmed to manipulate. We are the ones that add the intelligence and give those symbols meanings of numbers and addition and the result of doing an addition.

This is why representations are important – we need to choose a representation for things that makes the symbol manipulation we intend to do easy. We already saw this with images. If we want to send a large image to someone else then a representation of images like run-length encoding that shrinks the amount of data is a good idea.

When designing computers we need to provide them with a representation of numbers so they can manipulate those numbers. We have seen that there are lots of representations we could choose for numbers and any in theory would do, but when we choose a representation of numbers for use to do computation, we want to pick one that makes the operations we are interested in doing easy. Charles Babbage for example chose to use cog-like wheels turned to particular positions to represent numbers as he had worked out how to create a mechanism to do calculation with them. But that is something for another time…


This post was funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.


Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binaryand computation.

Lego Computer Science


Part 1: Lego Computer Science: pixel picture

Part 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

Part 5: Lego Computer Science: Gray code

Part 6: Lego Computer Science: Binary

Part 7: Lego Computer Science: What is computation (simple cellular automata)?

Lego computer science: compression algorithms

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…

A giraffe as a pixel image.
Colour look-up table
Black 0
Blue 1
Yellow 2
Green 3
Brown 4

We saw in the last post how images are stored as pixels – the equivalent of square or round lego blocks of different colours laid out in a grid like a mosaic. By giving each colour a number and drawing out a gird of numbers we give ourself a map to recreate the picture from. Turning that grid of numbers into a list (and knowing the size of the rectangle that is the image) we can store the image as a file of numbers, and send it to someone else to recreate.

Of course, we didn’t really need that grid of numbers at all as it is the list we really need. A different (possibly quicker) way to create the list of numbers is work through the picture a brick at a time, row by row and find a brick of the same colour. Then make a long line of those bricks matching the ones in the lego image, keeping them in the same order as in the image. That long line of bricks is a different representation of the image as a list instead of as a grid. As long as we keep the bricks in order we can regenerate the image. By writing down the number of the colour of each brick we can turn the list of bricks into another representation – the list of numbers. Again the original lego image can be recreated from the numbers.

The image as a list of bricks and numbers
Colour look-up table: Black 0: Blue 1: Yellow 2: Green 3: Brown 4

The trouble with this is for any decent size image it is a long list of numbers – made very obvious by the very long line of lego bricks now covering your living room floor. There is an easy thing to do to make them take less space. Often you will see that there is a run of the same coloured lego bricks in the line. So when putting them out, stack adjacent bricks of the same colour together in a pile, only starting a new pile if the bricks change colour. If eventually we get to more bricks of the original colour, they start their own new pile. This allows the line of bricks to take up far less space on the floor. (We have essentially compressed our image – made it take less storage space, at least here less floor space).

Now when we create the list of numbers (so we can share the image, or pack all the lego away but still be able to recreate the image), we count how many bricks are in each pile. We can then write out a list to represent the numbers something like 7 blue, 1 green, … Of course we can replace the colours by numbers that represent them too using our key that gives a number to each colour (as above).

If we are using 1 to mean blue and the line of bricks starts with a pile of seven black bricks then write down a pair of numbers 7 1 to mean “a pile of seven blue bricks”. If this is followed by 1 green bricks with 3 being used for green then we next write down 1 3, to mean a pile of 1 green bricks and so on. As long as there are lots of runs of bricks (pixels) of the same colour then this will use far less numbers to store than the original:

7 1 1 3 6 1 2 3 1 1 1 2 3 1 2 3 2 2 3 1 2 3 …

We have compressed our image file and it will now be much quicker to send to a friend. The picture can still be rebuilt though as we have not lost any information at all in doing this (it is called a lossless data compression algorithm). The actual algorithm we have been following is called run-length encoding.

Of course, for some images, it may take more not less numbers if the picture changes colour nearly every brick (as in the middle of our giraffe picture). However, as long as there are large patches of similar colours then it will do better.

There are always tweaks you can do to algorithms that may improve the algorithm in some circumstances. For example in the above we jumped back to the start of the row when we got to the end. An alternative would be to snake down the image, working along the adjacent rows in opposite directions. That could improve run-length encoding for some images because patches of colour are likely the same as the row below, so this may allow us to continue some runs. Perhaps you can come up with other ways to make a better image compression algorithm

Run-length encoding is a very simple compression algorithm but it shows how the same information can be stored using a different representation in a way that takes up less space (so can be shared more quickly) – and that is what compression is all about. Other more complex compression algorithms use this algorithm as one element of the full algorithm.

Activities

Make this picture in lego (or colouring in on squared paper or in a spreadsheet if you don’t have the lego). Then convert it to a representation consisting of a line of piles of bricks and then create the compressed numbered list.

An image of a camel to compress: Colour look-up table: Black 0: Blue 1: Yellow 2: Green 3: Brown 4

Make your own lego images, encode and compress them and send the list of numbers to a friend to recreate.


Find more about Lego Art at lego.com.

Find more pixel puzzles (no lego needed, just coloured pens or spreadsheets) at https://teachinglondoncomputing.org/pixel-puzzles/


This post was funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.

Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binaryand computation.

Lego Computer Science


Part 1: Lego Computer Science: pixel picture

Part 2: Lego Computer Science: compression algorithms

Part 3: Lego Computer Science: representing numbers

Part 4: Lego Computer Science: representing numbers using position

Part 5: Lego Computer Science: Gray code

Part 6: Lego Computer Science: Binary

Part 7: Lego Computer Science: What is computation (simple cellular automata)?

Lego computer science: pixel pictures

by Paul Curzon, Queen Mary University of London

It is now after Christmas. You are stuffed full of turkey, and the floor is covered with lego. It must be time to get back to having some computer science fun, but could the lego help? As we will see you can explore digital images, cryptography, steganography, data compression, models of computing, machine learning and more with lego (and all without getting an expensive robot set which is the more obvious way to learn computer science with lego though you do need lots of lego). Actually you could also do it all with other things that were in your stocking like a bead necklace making set and probably with all that chocolate, too.

First we are going to look at understanding digital images using lego (or beads or …)

Raster images

Digital images come in two types: raster (or bitmap) images and vector images. They are different kinds of image representation. Lego is good for experimenting with the former through pixel puzzles. The idea is to make mosaic-like pictures out of a grid of small coloured lego. Lego have recently introduced a whole line of sets called Lego Art should you want to buy rather amazing versions of this idea, and you can buy an “Art Project” set that gives you all the bits you need to make your own raster images. You can (in theory at least) make it from bits and pieces of normal lego too. You do need quite a lot though.

Raster images are the basic kind of digital image as used by digital cameras. A digital image is split into a regular grid of small squares, called pixels. Each pixel is a different colour.

To do it yourself with normal lego you need, for starters, to collect lots of the small circle or square pieces of different colours. You then need a base to put them on. Either use a flat plate piece if you have one or make a square base of lego pieces that is 16 by 16. Then, filling the base completely with coloured pieces to make a mosaic-like picture. That is all a digital image really is at heart. Each piece of lego is a pixel. Computer images just have very tiny pieces, so tiny that they all merge together.

Here is one of our designs of a ladybird.

A pixel image of a ladybird

The more small squares you have to make the picture, the higher the resolution of the image With only 16 x 16 pixels we have a low resolution image. If you only have enough lego for an 8×8 picture then you have lower resolution images. If you are lucky enough to have a vast supply of lego then you will be able to make higher resolution, so more accurate looking images.

Lego-by-numbers

Computers do not actually store colours (or lego for that matter). Everything is just numbers. So the image is stored in the computer as a grid of numbers. It is only when the image is displayed it is converted to actual colours. How does that work. Well you first of all need a key that maps colours to numbers: 0 for black, 1 for red and so on. The number of colours you have is called the colour depth – the more numbers and linked colours in your key, the higher the colour depth. So the more different coloured lego pieces you were able to collect the larger your colour depth can be. Then you write the numbers out on squared paper with each number corresponding to the colour at that point in your picture. Below is a version for our ladybird…

The number version of our ladybird picture

Now if you know this is a 16×16 picture then you can write it out (so store it) as just a list of numbers, listed one row after another instead: [5,5,4,4,…5,5,0,4,…4,4,7,2] rather than bothering with squared paper. To be really clear you could even make the first two numbers the size of the grid: [16,16,5,5,4,4,…5,5,0,4,…4,4,7,2]

That along with the key is enough to recreate the picture which has to be either agreed in advance or sent as part of the list of numbers.

You can store that list of numbers and then rebuild the picture anytime you wish. That is all computers are doing when they store images where the file storing the numbers is called an image file.

A computer display (or camera display or digital tv for that matter) is just doing the equivalent of building a lego picture from the list of numbers every time it displays an image, or changes an old one for something new. Computers are very fast at doing this and the speed they do so is called the frame rate – how many new pictures or frames they can show every second. If a computer has a frame rate of 50 frames per second, then it as though it can do the equivalent of make a new lego image from scratch 50 times every second! Of course it is a bit easier for a computer as it is just sending instructions to a display to change the colour shown in each pixels position rather than actually putting coloured lego bricks in place.

Sharing Images

Better still you can give that list of numbers to a friend and they will be able to rebuild the picture from their own lego (assuming they have enough lego of the right colours of course). Having shared your list of numbers, you have just done the equivalent of sending an image over the internet from one computer to another. That is all that is happening when images are shared, one computer sends the list of numbers to another computer, allowing it to recreate a copy of the original. You of course still have your original, so have not given up any lego.

So lego can help you understand simple raster computer images, but there is lots more you can learn about computer science with simple lego bricks as we will see…


Find more about Lego Art at lego.com.

Find more pixel puzzles (no lego needed, just coloured pens or spreadsheets) at https://teachinglondoncomputing.org/pixel-puzzles/


This post was funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.

Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binaryand computation.