Robert Weitbrecht and his telecommunication device for the deaf

Robert Weitbrecht was born deaf. He went on to become an award winning electronics scientist who invented the acoustic coupler (or modem) and a teletypewriter (or teleprinter) system allowing the deaf to communicate via a normal phone call.

A modem telephone: the telephone slots into a teletypewriter here with screen rather than printer.
A telephone modem: Image by Juan Russo from Pixabay

If you grew up in the UK in the 1970s with any interest in football, then you may think of teleprinters fondly. It was the way that you found out about the football results at the final whistle, watching for your team’s result on the final score TV programme. Reporters at football grounds across the country, typed in the results which then appeared to the nation one at a time as a teleprinter slowly typed results at the bottom of the screen. 

Teleprinters were a natural, if gradual, development from the telegraph and Morse code. Over time a different simpler binary based code was developed. Then by attaching a keyboard and creating a device to convert key presses into the binary code to be sent down the wire you code type messages instead of tap out a code. Anyone could now do it, so typists replaced Morse code specialists. The teleprinter was born. In parallel, of course, the telephone was invented allowing people to talk to each other by converting the sound of someone speaking into an electrical signal that was then converted back into sound at the other end. Then you didn’t even need to type, never mind tap, to communicate over long distances. Telephone lines took over. However, typed messages still had their uses as the football results example showed.

Another advantage of the teletypewriter/teleprinter approach over the phone, was that it could be used by deaf people. However, teleprinters originally worked over separate networks, as the phone network was built to take analogue voice data and the companies controlling them across the world generally didn’t allow others to mess with their hardware. You couldn’t replace the phone handsets with your own device that just created electrical pulses to send directly over the phone line. Phone lines were for talking over via one of their phone company’s handsets. However, phone lines were universal so if you were deaf you really needed to be able to communicate over the phone not use some special network that no one else had. But how could that work, at a time when you couldn’t replace the phone handset with a different device?

Robert Weitbrecht solved the problem after being prompted to do so by deaf orthodontist, James Marsters. He created an acoustic coupler – a device that converted between sound and electrical signals –  that could be used with a normal phone. It suppressed echoes, which improved the sound quality. Using old, discarded teletypewriters he created a usable system Slot the phone mouthpiece and ear piece into the device and the machine “talked” over the phone in an R2D2 like language of beeps to other machines like it. It turned the electrical signals from a teletypewriter into beeps that could be sent down a phone line via its mouthpiece. It also decoded beeps when received via the phone earpiece in the electrical form needed by the teleprinter. You typed at one end, and what you typed came out on the teleprinter at the other (and vice versa). Deaf and hard of hearing people could now communicate with each other over a normal phone line and normal phones! The idea of Telecommunications Device for the Deaf that worked with normal phones was born. However, they still were not strictly legal in the US so James Marsters and others lobbied Washington to allow such devices.

The idea (and legalisation) of acoustic couplers, however, then inspired others to develop similar modems for other purposes and in particular to allow computers to communicate via the telephone network using dial-up modems. You no longer needed special physical networks for computers to link to each other, they could just talk over the phone! Dial-up bulletin boards were an early application where you could dial up a computer and leave messages that others could dial up to read there via their computers…and from that idea ultimately emerged the idea of chat rooms, social networks and the myriad other ways we now do group communication by typing.

The first ever (long distance) phone call between two deaf people (Robert Weitbrecht and James Marsters) using a teletypewriter / teleprinter was:

“Are you printing now? Let’s quit for now and gloat over the success.”

Yes, let’s.

– Paul Curzon, Queen Mary University of London

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Super-plant supercapacitors

Aloe vera plant
Image by Marco from Pixabay

There are a whole range of plants that have been called superfoods for their amazing claimed health benefits because of the nutrients they contain. But plants can have other super powers too. For example, some are better at absorbing Carbon Dioxide to help with climate change, others provide medicines, or can strip our pollutants out of the air or soil. But one, Aloe Vera, is a super-plant in a new way. It can now store electricity that could be used to power portable devices – by plugging them into the plant.

Capacitors are one of the basic electronic components, like resistors and transistors, that electronic circuits are built from. They act a bit like a tiny battery, building up charge on a pair of surfaces with an insulator between so that charge cannot move directly from one to the other. Electrons build up on one plate, storing energy. When the capacitor is discharged that energy is released. They have a variety of uses including evening out power supplies. A supercapacitor is just a capacitor that can store a lot more energy so is a little like a tiny rechargeable battery, though releases the energy faster and can be charged and discharged many more times.

Various teams around the world have explored the use of aloe vera in supercapacitors. A team of researchers, led by Yang Zhao from Beijing Institute of Technology, has succeeded in creating a supercapacitor made completely from materials extracted from the plant (apart from one gold wire). The parts were made by heating a part of the leaf of the plant, and by freezing its juice. The advantage of this is that the supercapacitor is biodegradable unlike traditional ones made from oil-based synthetic materials. It also makes them biocompatible in that they can be inserted into aloe vera and similar plants without doing them harm and potentially make use of electricity generated by the plant. Her team has inserted these tiny capacitors inside other plants including cacti and aloe vera plants to show this idea works in principle.

So plants can be superheroes and aloe vera more than most: it looks nice on your window cill, you can make soap from it, it supposedly has medicinal value, it is being used in research to remove pollutants from the air and soon it could provide you with electricity too. So next time you are lost in a cactus filled wilderness make sure you have aloe vera capacitors with you so you can charge your gadgets while waiting to be rescued.

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The wrong trousers? Not any more!

A metal figure sitting on the floor head down
Image by kalhh from Pixabay

Inspired by the Wallace & Gromit film ‘The Wrong Trousers’, Johnathan Rossiter of the University of Bristol builds robotic trousers. We could all need them as we get older.

Think of a robot and you probably think of something metal: something solid and hard. But a new generation of robot researchers are exploring soft robotics: robots made of materials that are squishy. When it comes to wearable robots, being soft is obviously a plus. That is the idea behind Jonathan’s work. He is building trousers to help people stand and walk.

Being unable to get out of an armchair without help can be devastating to a person’s life. There are many conditions like arthritis and multiple sclerosis, never mind just plain old age, that make standing up difficult. It gets to us all eventually and having difficulty moving around makes life hard and can lead to isolation and loneliness. The less you move about, the harder it gets to do, because your muscles get weaker, so it becomes a vicious circle. Soft robotic trousers may be able to break the cycle.

We are used to the idea of walking sticks, frames, wheelchairs and mobility scooters to help people get around. Robotic clothes may be next. Early versions of Jonathan’s trousers include tubes like a string of sausages that when pumped full of air become more solid, shortening as they bulge
out, so straightening the leg. Experiments have shown that inflating trousers fitted with them, can make a robot wearing them stand. The problem is that you need to carry gas canisters around, and put up with the psshhht! sound whenever you stand!

The team have more futuristic (and quieter) ideas though. They are working on designs
based on ‘electroactive polymers’. These are fabrics that change when electricity
is applied. One group that can be made into trousers, a bit like lycra tights, silently shrink with an electric current: exactly what you need for robotic trousers. To make it work you need a computer control system that shrinks and expands them in the right places at the right time to move the leg
wearing them. You also need to be able to store enough energy in a light enough way that the trousers can be used without frequent recharging.

It’s still early days, but one day they hope to build a working system that really can help older people stand. Jonathan promises he will eventually build the right trousers.

– Paul Curzon, Queen Mary University of London (from the archive)

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The rise of the robots [PORTAL]


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Music-making mates for Mortimer

Drumming Robot after Mortimer
Image by CS4FN

Robots are cool. Fact. But can they keep you interested for more than a short time? Over months? Years even? Louis McCallum of Queen Mary University of London tells us about his research using Mortimer a drumming robot.

Roboticists (thats what we’re called) have found it hard to keep humans engaged with robots once the novelty wears off. They’re either too simple and boring, or promise too much and disappoint. So, at Queen Mary University of London we’ve built a robot called Mortimer that can not only play the drums, but also listen to humans play the piano and jam along. He can talk (a bit) and smile too. We hope people will build long term relationships with him through the power of music.

Robots have been part of our lives for a long time, but we rarely see them. They’ve been building our cars and assembling circuit boards in factories, not dealing with humans directly. Designing robots to have social interactions is a completely different challenge that involves engineering and artificial intelligence, but also psychology and cognitive science. Should a robot be polite? How long and accurate should a robot’s memory be? What type of voice should it have and how near should it get to you?

It turns out that making a robot interact like a human is tricky, even the slightest errors make people feel weird. Just getting a robot to speak naturally and understand what we’re saying is far from easy. And if we could, would we get bored of them asking the same questions every day? Would we believe their concern if they asked how we were feeling?

Would we believe their concern
if they asked how we were feeling?

Music is emotionally engaging but in a way that doesn’t seem fake or forced. It also changes constantly as we learn new skills and try new ideas. Although there have been many examples of family bands, duetting couples, and band members who were definitely not friends, we think there are lots of similarities between our relationships with people we play music with and ‘voluntary non-kin social relationships’ (as robotocists call them – ‘friendships’ to most people!). In fact, we have found that people get the same confidence boosting reassurance and guidance from friends as they do from people they play music with.

So, even if we are engaged with a machine, is it enough? People might spend lots of time playing with a guitar or drum machine but is this a social relationship? We tested whether people would treat Mortimer differently if it was presented as a robot you could socially interact with or simply as a clever music machine. We found people played for longer uninterrupted and stopped the robot whilst it was playing less often if they thought you could socially interact with it. They also spent more time looking at the robot when not playing and less time looking at the piano when playing. We think this shows they were not only engaged with playing music together but also treating him in a social manner, rather than just as a machine. In fact, just because he had a face, people talked to Mortimer even though they’d been told he couldn’t hear or understand them!

So, if you want to start a relationship with a creative robot, perhaps you should learn to play an instrument!

– Louis McCallum, Queen Mary University of London (from the archive)

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Watch the video Louis made with the Royal Institution about Mortimer

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Philippa Gardner bringing law and order to a wild west

Verified Trustworthy Software

Image by CS4FN

The computing world is a wild west, with bugs in software the norm, and malicious people and hostile countries making use of them to attack people, companies and other nations. We can do better. Just as in the original wild west, advances have happened faster than law and order can keep up. Rather than catch cyber criminals we need to remove the possibility. In software the complexity of our computers and the programs they run has increased faster than ways have been developed and put in place to ensure they can be trusted. It is important that we can answer precisely questions such as “What does this code do?” and “Does it actually do what is intended?”, but can also assure ourselves of what code definitely does NOT do: it doesn’t include trapdoors for criminals to subvert, for example. Philippa Gardner has dedicated her working life to rectifying this by providing ways to verify software, so mathematically prove such trust-based properties hold of it.

Programs are incredibly complicated. Traditionally, software has been checked using testing. You run it on lots of input scenarios and check it does the right thing in those cases. If it does you assume it works in all the cases you didn’t have time to check. That is not good enough if you want code to really be trustworthy. It is impossible to check all possibilities, so testing alone is just not good enough. The only way to do it properly is to also use engineering methods based on mathematics. This is the case, not just for application programs, but also for the software systems they run within, and that includes programming languages themselves. If you can’t trust the programming language then you can’t trust any programs written in that language. Building on decades of work by both her own team and others, Philippa has helped provide tools and techniques that mean complex industrial software and the programming languages they are written in can now be verified mathematically to be correct. Helping secure the web is one area she is making a massive contribution via the W3C WebAssembly (Wasm) initiative. She is helping ensure that programs of the future that run over the web are trustworthy. 

Programs written in programming languages are compiled (translated) into low level code (ie binary 1s and 0s) that can actually be run on a computer. Each kind of computer has its own binary instructions. Rather than write a compiler for every different machine, compilers often now use common intermediary languages. The idea is you have what is called a virtual machine – an imaginary one that does not really exist in hardware. You compile your code to run on the imaginary machine. A compiler is written for each language to compile it into the common low level language for that virtual machine. Then a separate, much simpler, translator can be written to convert that code into code for a particular real machine. That two step process is much easier than writing compilers for all combinations of languages and machines. It is also a good approach to make programs more trustworthy, as you can separately verify the separate, simpler parts. If programs compile to the virtual machine, then to be sure they cannot do harm (like overwrite areas of memory they shouldn’t be able to write to) you also only have to be sure that programs running on the virtual machine programs cannot , in general, do such harm.

The aim of Wasm is to make this all a reality for web programming, where visiting a web page may run a program you can’t trust. Wasm is a language with linked virtual machine that programming language compilers can be compiled into that itself will be trustworthy even when run over the web. It is based on a published formal specification of how the programming language and the virtual machine should behave.

As Philippa has pointed out, while some companies have good processes for ensuring their software is good enough, these are often kept secret.  But given we all rely on such software we need much better assurances. Processes and tools need to be inspectable by anyone. That has been one of the areas she has focussed on. Working on Wasm is a way she has been doing that. Much of her work over 30 years or so has been around the development and use of logics that can be used to mathematically verify that concurrent programs are correct. Bringing that experience to Wasm has allowed her to work on the formal specification conducting proofs of properties of Wasm that show it is trustworthy in various way, correcting definitions in the specification when problems are found. Her approach is now being adopted as the way to do such checking.

Her work with Wasm continues but she has already made massive steps to helping ensure that the programs we use are safe and can be trusted. As a result, she was recently awarded the BCS Lovelace medal for her efforts.

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Soft squidgy robots

A smiling octopus
Image by OpenClipart-Vectors from Pixabay

Think of a robot and you probably think of something hard, metal, solid. Bang into one and it would hurt! But researchers are inventing soft robots, ones that are either completely squidgy or have squidgy skins.

Researchers often copy animals for new ideas for robots and lots of animals are soft. Some have no bones in them at all nor even hard shells to keep them safe: think slugs and octopuses. And the first soft robot that was “fully autonomous”, meaning it could move completely on its own, was called Octopod. Shaped like an Octopus, its body was made of silicone gel. It swam through the water by blowing gas into hollow tubes in its arms like a balloon, to straighten them, before letting the gas out again. 

Soft, squidgy animals are very successful in nature. They can squeeze into tiny spaces for safety or to chase prey, for example. Soft squidgy machines may be useful for similar reasons. There are plenty of good reasons for making robots soft, including

  • they are less dangerous around people, 
  • they can squeeze into small spaces,
  • they can be made of material that biodegrades so better for the planet, and
  • they can be better at gently gripping fragile things.

Soft robots might be good around people for example in caring roles. Squeezing into small spaces could be very useful in disaster areas, looking for people who are trapped. Tiny ones might move around inside an ill person’s body to find out what is wrong or help make them better.

Soft robotics is an important current research area with lots of potential. The future of robotics may well be squidgy.

Paul Curzon, Queen Mary University of London

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An Wang’s magnetic memory

A golden metal torus
Image by Hans Etholen from Pixabay

An Wang was one of the great pioneers of the early days of computing. Just as the invention of the transistor led to massive advances in circuit design and ultimately computer chips, Wang’s invention of magnetic core memory provided the parallel advance needed in memory technology.

Born in Shanghai, An went to university at Harvard in the US, studying for a PhD in electrical engineering. On completing his PhD he applied for a research job there and was set the task of designing a new, better form of memory to be used with computers. It was generally believed that the way forward was to use magnetism to store bits, but no one had worked out a way to do it. It was possible to store data by for example magnetising rings of metal. This could be done by running wires through the rings. Passing the current in one direction set a 1, and in the other a 0 based on the direction of the magnetic field created.

If all you needed was to write data it could be done. However, computers, needed to be able to repeatedly read memory too, accessing and using the data stored, possibly many times. And the trouble was, all the ways that had been thought up to use magnets were such that as soon as you tried to read the information stored in the memory, that data was destroyed in the process of reading it. You could only read stored data once and then it was gone!

An was stumped by the problem just like everyone else, then while walking and pondering the problem, he suddenly had a solution. Thinking laterally, he realised it did not matter if the data was destroyed at all. You had just read it so knew what it was when you destroyed it. You could therefore write it straight back again, immediately. No harm done!

Magnetic-core memory was born and dominated all computer memory for the next two decades, helping drive the computer revolution into the 1970s. An took out a patent for his idea. It was drafted to be very wide, covering any kind of magnetic memory. That meant even though others improved on his design, it meant he owned the idea of all magnetic based memory that followed as it all used his basic idea.

On leaving Harvard he set up his own computer company, Wang Laboratories. It was initially a struggle to make it a success. However, he sold the core-memory patent to IBM and used the money to give his company the boost that it needed to become a success. As a result he became a billionaire, the 5th richest person in the US at one point.

Paul Curzon, Queen Mary University of London

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Aaron and the art of art

Aaron is a successful American painter. Aaron’s delicate and colourful compositions on canvas sell well in the American art market, and have been exhibited worldwide, in London’s Tate Modern gallery and the San Francisco Museum of Modern Art for example. Oh and by the way, Aaron is a robot!

Yes, Aaron is a robot, controlled by artificial intelligence, and part of a lifelong experiment undertaken by the late Harold Cohen to create a creative machine. Aaron never paints the same picture twice; it doesn’t simply recall pictures from some big database. Instead Aaron has been programmed to work autonomously. That is, once it starts there is no further human intervention, Aaron just draws and paints following the rules for art that it has been taught.

Perfecting the art of painting

Aaron’s computer program has grown and developed over the years, and like other famous painters, has passed though a number of artistic periods. Back in the early 1970s all Aaron could do was draw simple shapes, albeit shapes that looked hand drawn – not the sorts of precise geometric shapes that normal computer graphics produced. No, Aaron was going to be a creative artist. In the late 1970s Aaron learned something about artistic perspective, namely that objects in the foreground are larger than objects in a picture’s background. In the late 80s Aaron could start to draw human figures, knowing how the various shapes of the human body were joined together, and then learning how to change these shapes as a body moved in three dimensions. Now Aaron knows how to add colour to its drawings, to get those clever compositions of shades just spot on and to produce bold, unique pictures, painted with brush on canvas by its robotic arm.

It’s what you know that counts

When creating a new painting Aaron draws on two types of knowledge. First Aaron knows about things in the real world: the shapes that make up the human body, or a simple tree. This so called declarative (declared) knowledge is encoded in rules in Aaron’s programming. It’s a little like human memory: you know something about how the different shapes in the world work. This information is stored somewhere in your brain. The second type of knowledge Aaron uses is called procedural knowledge. Procedural knowledge allows you to move (process) from a start to an end through a chain of connected steps. Aaron, for example, knows how to proceed through painting areas of a scene to get the colour balance correct and in particular, getting the tone or brightness of the colour right. That is often more artistically important than the actual colours themselves. Inside Aaron’s computer program these two types of knowledge, declarative and procedural, are continuously interacting with each other in complex ways. Perhaps this blending of the two types of knowledge is the root of artistic creativity?

Creating Creativity

Though a successful artist, and capable of producing pleasing and creative pictures, Aaron’s computer program still has many limitations. Though the pictures look impressive, that’s not enough. To really understand creativity we need to examine the process by which they have been made. We have an ‘artist’ that we can take to pieces and examine in detail. Studying what Aaron can do, given we know exactly what’s been programmed into it, allows us to examine human creativity. What about it is different from the way humans paint, for example? What would we need to add to Aaron to make its process of painting more similar to human creativity?

Not quite human

Unlike a human artist Aaron cannot go back and correct what it does. Studies of great artist’s paintings often show that under the top layer of paint there are many other parts of the picture that have been painted out, or initial sketches that have been redrawn as the artist progresses through the work, perfecting it as they go. Aaron always starts in the foreground of the picture and moves toward painting the background later, whereas human artists can chop and change which part of a picture to work on to get it just right. Perhaps in the future, with human help Aaron or robots like him will develop new human-like painting skills and produce even better paintings. Until then the art world will need to content itself with Aaron’s early period work.

the CS4FN team (updated from the archive)

Some of Aaron’s (and Harold COhen’s) work is on display at the Tate modern until June 2025 as part of the Electric Dreams exhibition.

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Sue Sentance: Teaching the world to program

A figure sprinting: silhouette binary
Image by Gerd Altmann from Pixabay (edited)

How do you learn to program? How do you best teach programming. When the English school curriculum changed, requiring even primary school students to learn programming, suddenly this became an important question for school teachers who had previously not had to even think about it. Teaching is about much more than knowing the subject, or even knowing how to teach. It also needs knowledge and skill in how to teach each specific subject. Many teachers had to learn these new skills often with no background at all. Luckily, Sue Sentance came to the rescue with PRIMM, a simple framework for how to teach programming suitable for schools. She was awarded the BCS Lovelace prize for the work.

If you were a novice wanting to develop maker skills, whether building electronics, making lego, doing origami or knitting, you might start by following instructions created by someone else for a creation of theirs. Many assumed that was a sensible way to teach programming too, but it isn’t. This approach, sometimes called “copy code” where a teacher provides a program, and students type it in, is a very poor way to learn to program. But if you can’t do the obvious, what do you do?

Sue came up with PRIMM as a way to help teachers. It stands for Predict, Run, Investigate, Modify and Make, giving a series of steps a programming lesson should follow.

The teacher still provides programs, but instead of typing the code in line by line the students first read it and try to predict what it does. This follows the way people learn to write – they first read (lots!)

Having made a prediction, the students run the program. (They don’t type it in at all, as there is little point in doing that, but are given the file ready to run), They now act like a scientist and see if their prediction is correct. Perhaps they predict the program prints 

Hello World

all on one line. By running the program they find out if they were right or not. If they were then it confirms their understanding. If it doesn’t then this suggests there was something more to understand. If the program instead printed

Hello
World

over two lines, then there is something to work out about what makes a program move to another line. The class discuss the results and compare their predictions with the results. Can they explain why it behaved the way it did?

Next they investigate the program in more depth. The teacher can set a variety of exercises to do this. One very powerful way is stepping through program fragments line by line (doing what in industry is called a code walkthrough and is also called dry running or tracing the code). 

Based on the deeper understanding gained by this they then attempt to modify the original program to do something very slightly different – for example, to print 

Hello, Paul. 
How are you?

This is more exprementation to check and expand their understanding. By making deliberate changes with specific results in mind, they can now purposefully make sure they really do understand a programming construct. As before, if the program does something different to expected then the reason can be explored and that is used to correct what they thought.

If they have fully understood the code then this should by now be fairly easy.

Finally they make a program of their own. Based on the understanding gained they create a new specific  program that uses the new constructs (like how to print a message, get input or make decisions) that they now understand. This program should solve a different problem. For example if they just played with a program containing an if statement, they might now write a simple quiz program, or simulates a vending machine where items cost different amounts. .

Part of the reason that PRIMM has been successful is that it is not only a good way to learnt to program but it gives a clear structure to lessons that can be repeated with each construct to be covered and so makes a natural framework for planning lessons around.

Sue originally developed PRIMM with local schools she was working with in mind, but it works so well, solving a specific problem teachers had everywhere, that it is now used worldwide in countries introducing programming in schools.

Women do not figure greatly in the early history of science and maths just because societal restrictions, prejudices and stereotypes meant few were given the chance. Those who were like Maria Cunitz, showed their contributions could be amazing. It just took the right education, opportunities, and a lot of dedication. That applies to modern computer science too, and as the modern computer scientist, Karen Spärck Jones, responsible for the algorithm behind search engines said: “Computing is too important to be left to men.”

– Paul Curzon, Queen Mary University of London

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Ancient Egyptian Numerals

How data is represented is an important part of computer science. There are lots of ways numbers can be represented. Choosing a good representation can make things easier or harder to do. The Ancient Egyptians had a simple way using hieroglyphs (symbols). It is similar to Roman Numerals but simpler. 

They represented numbers 1 to 9 with a hieroglyph with that number of straight lines. They arranged them into patterns (a bit like we do dots on a dice). The patterns make them easier to recognise. They used an upside down U shape for 10, two of these for 20, and so on. Their symbol for 10 also meant a “cattle hobble”. They then had a new symbols for each power of 10 up to a million. So 100 is the hieroglyph for a coil of rope.

Egyptian Numbers hundreds.jpg
Image by CS4FN

The hieroglyph for the number 1000 was a water lily.

The Ancient Egyptian hieroglyph for a waterlily that also means 1000 with 1000 written next to it
The ancient Egyptian way to write 1000 was a hieroglyph of a waterlily. Image by CS4FN

The hieroglyph for a million, which also rather sensible meant ‘many’, was just the hieroglyph of the god Hey who was the personification of eternity.

To make a number you just combined the hieroglyph for the ones, tens, hundreds and so on.

The Ancient Egyptian number system makes it very easy to write numbers and to add and subtract numbers. Big numbers are fairly compact, though take up more space than our decimals. It is easy to convert a tally representation into this system too. More complicated things like multiplication are harder to do. Computers use binary representation because they make all the main operations easy to do using logic. Ultimately it is all about algorithms. The Egyptians had easy to follow algorithms for addition and subtraction to go with their number representation. We have devised algorithms that allow computers to do all the calculations they do as quickly as possible using a binary representation

Paul Curzon, Queen Mary University of London

To do…

Try doing some sums as an Ancient Egyptian would – without converting to our numbers. What is the algorithm for adding Egyptian numbers? Do multiplication using a repeated addition algorithm – to do 3 x 4 you 4 to zero 3 times.

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An earlier version of this article first appeared on Teaching London Computing.

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