Binary 4 – Floating Point Binary Fractions 1

This is the fourth in a series of videos about binary. I'm going to talk about how we can represent real numbers using floating point binary notation. Most modern computers use floating point binary to represent real numbers. Why? Because floating point binary makes particularly efficient use of a computer register when it comes to representing either extremely large values, extremely small values, or values that require a high degree of precision. Floating point binary is similar to standard scientific notation in base 10. It's worth getting to know this first. Here are some special values in scientific notation. You might recognise some of these if you study science. This is the speed of light in a vacuum, measured in metres per second. This is Avogadro's number. It's also called a mole. It's the number of particles you'll have if you measure out the molecular weight of a substance in grams. Suffice to say, it's a very large value. This is known as the elementary charge. It's the electrical charge of an electron, measured in coulombs. This number is tiny. And this is the age of the universe in seconds. At least it was not so long ago. Don't worry if you're not a scientist and you're not familiar with these values. What really matters to us is that these are either very large or very small. Let's take a look at what this notation means. Here's the mole again. And again, don't worry if you're not a scientist. All you need to appreciate is that this is a very big number. This part of the notation is called the mantissa. This part of the notation is known as the exponent. To convert this notation into something more familiar, you have to float the decimal point 23 places to the right, and then fill with zeros. You can see that this is a very large value indeed, with so many digits that it occupies a lot of space when it's written down in full. If you're able to use more digits in the mantissa, you can express this value with greater precision. Floating the point 23 places to the right, just like last time, gives us a more accurate value. But look at all those zeros. It's still an approximation, just a better one than before. So as you can see, the number of digits you're allowed to use in the mantissa governs the precision of the value written in this way. Similarly, the number of digits available for the exponent governs the range. With only two digits for the exponent, you wouldn't be able to float the point any more than 99 places to the right. You'll see why this is important in computer science soon. Here's the charge on an electron again. This time it's an example of a very small number. Notice that the exponent is a negative number. This means that you have to float the decimal point 19 places to the left to get a more familiar form. Clearly this is a very small fraction indeed. So how does this apply to the representation of real numbers in a digital computer? This is a 16-bit register inside a central processing unit. In this theoretical processor, by design, 10 bits on the left have been allocated to the mantissa and 6 bits on the right have been allocated to the exponent. The mantissa is stored in 2's complement and so is the exponent. An imaginary binary point is located immediately to the right of the sine bit. The sine bit is the leftmost bit in the register. Historically, floating point formats have varied from machine to machine. For example, some put the exponent on the left and the mantissa on the right and some use sine and magnitude instead of 2's complement. It wasn't until 1985 that the Institute of Electrical and Electronics Engineers agreed on the standard format used in most of today's computers. I'll say more about the IEEE 754 standard in another video but the format I'm going to describe in detail now illustrates the fundamental principles rather well. Furthermore, the format I'm going to describe now is the one you're expected to know about if you sit an A-level Computer Science exam in the United Kingdom. Looking at this register you can see that the sine bit of the mantissa is a 0 which, in 2's complement means that the mantissa is a positive number. In fact, it means that this 16-bit register contains a positive number. If the sine bit of the mantissa was a 1 you'd be looking at a negative number here. You can also see that the exponent is positive because the leftmost bit of the exponent is 0. A positive exponent indicates that the binary point should be floated to the right. If the leftmost bit of the exponent was a 1 instead you'd be looking at a negative exponent indicating that the binary point should be floated to the left. To convert this number into base 10 First, work out the value of the exponent in denary. In this case, it's positive 3. So we have something times 2 to the power of 3. Remember where the binary point is. Float it 3 places to the right. So here's the original number in pure binary. There's no need to concern yourself with the trailing 0's on the right-hand side. They're insignificant. Put this into a conversion gadget and you'll arrive at 6.5 in base 10. When you do this final conversion remember that this is a positive number so the leftmost 1 has a positive place value in this case 4. The original register contents therefore represent the denary value 6.5. Let's see another example. Again, we have a 16-bit register with 10 bits for the mantissa and 6 for the exponent. At a glance, you can see it's a positive mantissa and a positive exponent because they both start with 0. First, work out the value of the exponent in denary. It's positive 2. So we have something times 2 to the power of 2. Remember where the binary point is and float it 2 places to the right. So here's the original number in pure binary without the trailing 0's. Plug it into a conversion gadget remembering that this is a positive number so the leftmost 1 must be a positive place value. And you'll arrive at 2.625 in base 10. And here's one more example. At a glance, you can see it's a positive value because the mantissa is positive but this time it's a negative exponent. Work out the value of the exponent in this case, minus 2. So we have the original mantissa times 2 to the power of minus 2. Float the point 2 places to the left. So here's the value in pure binary. Plug this into a gadget and this time we have 0.125. Here are some conversions you can try yourself. Pause the video now if you want to give them a go and I'll show you the solutions in just a moment. And here are the solutions. Here's a couple more examples for you to try. Notice that these values are represented with 4 bits for the mantissa and 4 bits for the exponent. Solutions in a moment. Pause the video if you want to give them a try. And here are the solutions. Let's see an example of a negative number. You can see it's negative at a glance because the first bit of the mantissa, the sign bit is a 1 this time. Using the same method as before convert the exponent into denary. So we have something times 2 to the power of 3. Remember where the binary point is and float it 3 places to the right. Now, when you convert from pure binary remember that the leftmost 1 has a negative place value. So this time we have minus 8 plus 4 plus 2 plus 0.5 which gives us minus 1.5. And one more example. This time the mantissa is negative which means that the value in the register is negative and the exponent is negative as well. So the binary point will float to the left. Convert the exponent into denary. We have minus 2 so we have something times 2 to the power of minus 2. Remember where the point is and float it 2 places to the left. When you convert the pure binary back into denary remember that the leftmost 1 has a negative place value. So we have minus 0.25 plus 0.125 which gives us minus 0.125. Here are some negative number conversions you can try for yourself. Pause the video now if you want to give them a go and I'll show you the solutions in just a moment. And here are the solutions. In the next video of this series I'm going to talk about the relationship between range and precision.