Free teaching aid: How does binary and 2s complement work?
Free teaching aid: How does binary and 2s complement work?

Free binary teaching tool: How does 2s complement work?

My GCSE Computer Science students need to understand how to convert positive numbers into 8 bit binary. Today, my Y10s had to teach this to the Y6s visiting for their new Y7 induction day. After the  Y6s had gone off for an early lunch, my Y10s had to discuss how they would extend what they knew about binary to allow computers to understand negative numbers as well as positive numbers.

The end result was an explanation of 2’s complement and how it works, using this free interactive binary teaching tool.

What is binary?

Binary is made up of 0s and 1s
Binary is made up of 0s and 1s

There are 26 letters in the English alphabet. Even in simplified Japanese Kanji, there are over 2,000 symbols. The digital electronics that makes computers work can only understand two possible states: electricity switched on and electricity switched off. That’s why binary is used so much in computers: a binary 1 or 0 can be processed with electronic switches and stored in electronic ‘memory’.

Even though computers only ‘understand’ 0s and 1s, they can combine multiple 0s and 1s to represent any number, image, sound or anything else.

Free teaching aid: How does binary and 2s complement work?
Free teaching aid: How does 2s complement work?

How can positive numbers be represented in binary?

In the same way that you can combine letters in the English alphabet to make any word, you can combine 0s and 1s in binary to represent any sort of data.

The easiest sort of data to represent is an unsigned integer.

Unsigned means that it’s a positive number: it has no + or – sign.

Integer means that it’s a whole number: it has no decimal point.

Each individual 0 or 1 in a binary number is called a bit. As you look from right to left at a binary number, each bit is worth twice as much as the one on its right.

Example: 01001001

The bit on the far right is known as the least significant bit: it’s only worth 1.

As you move from right to left, each bit doubles in value, until you get to the bit on the far left, which we call the most significant bit:

01001001
01001001

To convert from binary into decimal (what most people think of as ‘normal’ numbers) you have to add together all of the values of bits that are set to 1.

For our example of 01001001, you would add together 64 + 8 + 1 = 73

Experiment with this interactive tool to see how to convert from binary to decimal and vice versa.

 

How can negative numbers be represented in binary?

When asked this question, my Y10s came up with a couple of suggestions:

  • You could use twice as many bits to store each number: each bit could have an extra ‘positive or negative’ bit next to it

    This would work but would be a massive waste of memory: you’d need twice as much RAM to store negative numbers and you’d have to redesign CPUs to handle negative numbers in a different way to positive numbers. Converting between the two types would be a waste of processing power.

  • You could add an extra bit on to the end you binary number: just one ‘positive or negative’ bit at the start of each number

    This is a much better idea but you’d still be wasting memory: an 8 bit number would need 9 bits in total. A 32 bit number would need 33 bits etc… That extra bit will cause problems when you try to store it as almost all memory is designed for multiples of 8 bits: 8, 16, 32, 64, 128 bit etc…

So, the way forwards is to take the most significant bit of a binary number (the bit on the far left) and keep the same absolute value but make it negative.

8 bit 2s complement binary bit values
8 bit 2s complement binary bit values

Note that the most significant bit (MSB) is now worth -128 rather than 128.

The advantage of representing negative numbers this way is that for any positive number where the MSB was 0, the binary is the same as it would be if we’d used our good old fashioned normal way of representing binary numbers. You can also quickly tell if any number is negative by looking at the MSB: it’s negative if the MSB is 1 and it’s positive (or zero) if the MSB is 0.

This method of representing negative numbers in binary is called 2’s complement.

Experiment with this interactive tool to see how to convert from binary to decimal and vice versa. Tick the 2s complement box at the top to experiment with negative numbers.

How do you convert a 2’s complement binary number into decimal?

This is easy. You’d do exactly the same as you would for a positive number: give the least significant bit (on the right) a value of 1 and double the value of each bit as you move to the left. Add up all the values of bits that are 1s. The only difference is that the most significant bit (on the left) is negative:

Example: 10001000

10001000 in 2s complement binary
10001000 in 2s complement binary

In this example, the MSB (most significant bit, on the left) is set to 1 so we can know that the number is negative.

When you add together the values of each bit that is set to 1 you get:

-128 + 8 = -120

So the 2s complement binary number 10001000 represents the negative number -120.

The digital electronics that makes a computer work doesn’t know (or care) if 10001000 is 2s complement or not. It could just as easily represent the positive number 136 (128 + 8). The computer processes the binary data (for addition, subtraction etc…) in exactly the same way. It’s down to the software rather than the hardware to remember if the binary data is a signed (2s complement) number or an unsigned (positive) number. This can sometimes lead to some pretty catastrophic software failures when programmers aren’t aware of how computers handle negative numbers.

Experiment with this interactive tool to see how to convert from binary to decimal and vice versa. Tick the 2s complement box at the top to experiment with negative numbers.

 

Continue ReadingFree binary teaching tool: How does 2s complement work?
Free discussion tools
Free discussion tools

Free discussion tool: create live word clouds that respond as people vote

Ask: free discussion tool
Ask: free discussion tool

Get your class / audience engaged with this free quick survey maker classroom tool. This free discussion tool lets you ask a question on screen that anyone can answer by visiting a short URL or automatically created QR code. The answers that are submitted generate a dynamic word clouds or bar chart so you and your class / audience can use it as a discussion tool for debate or a quick way of gauging how much people have understood a topic.

On a Thursday lunchtime at Manor CE Academy the students who’ve been coming to Cyber Security and Digital Forensics club have been learning how to create and test web applications. The aim was to look at how you can quickly develop a web application using mashups of different APIs and external code sources before going to test the web app for security vulnerabilities and look at guarding against common exploits. This tool started as a code experiment then grew into a tool that we used a lot to promote discussion in class without always having to rely on the same students putting up their hands.

You can try out the web app here.

Anyone can create a discussion question really quickly (you don’t have to log in or create an account).

Once you’ve made a discussion question you’ve got an hour for your class / audience to respond. After that time, the survey is no longer accessible, so that people don’t post inappropriate content long after you’ve finished discussing the question in class.

If you just type in a question, you’ll get a screen where people can see their answers appear in a word cloud.

If you choose set responses (like “Fully understand”, “mostly understand”, “not sure” and “not a clue”) then instead of a word cloud you’ll see a bar chart automatically update whenever anyone chooses one of those options.

You can also set a password to allow you to moderate and remove any answers that you don’t want included in the word cloud and you can set a limit for the number of times anyone can answer each question.

The next page has some ideas for how to use this free discussion survey tool in class.

Continue ReadingFree discussion tool: create live word clouds that respond as people vote
Read more about the article micro:bit Python Simulator
Scrolling text with python on a micro:bit

micro:bit Python Simulator

I’ve been following the hype around the BBC micro:bits since they were announced but it’s been notoriously difficult to get my hands on one. My Y7s have been waiting for their class set to arrive since they were promised in September and although I know there are plenty of schools out there who’ve received them, I’d pretty much given up hope this academic year.

Then, my lovely science technician told me she’d got some spare “computer gadget things” that came bundled in a in a kit that she wasn’t going to be needing and she wondered if some of my students would be interested.

Oh yes.

So, for the last week, I’ve set all sorts of different aged / ability students loose with the “computer gadget things” to see if they live up to the hype.

And they do.

Continue Readingmicro:bit Python Simulator