CHAPTER ONE numbering systems ask many people what the most commonly used number system is, and they would probably reply after a bit of thought, of course, the decimal system, but actually, many numbers, systems and counting systems are used without the user thinking much about it. For example, clocks encompasses use the ancient Babylonian number system based on 60 rather than the decimal system based on 10. Why? Because 60 is easier to divide into equal segments. It can be evenly divided by 12345 612 1520 and 30, etc. This is much better for applications such as time or degrees of angle.

The base 10, which can only be divided into parts by one, two and five. Many counting systems are ancient in origin are still used today because they're useful for particular purposes. Using the decimal system, it's easier to count up to 10 fingers for example, using just the fingers of two hands. In Northern Britain, farmers used an ancient Celtic counting system based on 20 also called a score to count their animals, and it's still used even today in the second half of the 20th century. Another special number system is the binary system used by digital electronic devices because digital circuits work on an electrical on or off a two state system, a number system Based on two that is the binary counting system is much easier for electronic devices to use. However, binary is not an actual choice for human counting or calculations.

This course explains how the numbering systems used in electronics work, and how computers and calculators use different forms of binary to carry out their calculations. Let's compare two of the most common numbering systems the ones that we're going to be using mostly in this course. It's the decimal or base 10 and the binary or base two. Now these terms base 10, and base two will become more clear as we go through the next few slides. For now, it's just a way of naming the decimal system. They call it base 10 and the binary system To the decimal system has 10 values 012345678 and nine, zero to 910 values.

If larger values than nine are needed extra columns are added to the left of the number. Each column value is 10 times the value to the call of the column to the right. For example, the decimal value 12 is written one two, where you have 110 plus two ones. And as I said before, the base 10 figure is becoming clear now, when we see how we have to treat the columns to the left of the one before it by increasing it by a value of times 10 The binary has only two values zero and one. If larger values than one are needed, extra columns are added to the left. Each column value is now two times or twice the value of the column, the column to the right.

Hence, the base two. Description. For example, the decimal value, three is written one, one in binary, one, two, plus one, one. So a binary number is a number that includes only ones and zeros. The number could be of any length. The following are examples of binary numbers, and they're not equated to each other in any way, shape, or form.

I just wanted to demonstrate how very All links and formations can be made up using binary numbers, ones and zeros. As I said before, another name for a binary for a binary system is base two, and it's pronounced base two. The numbers that we are used to seeing are called decimal numbers. decimal numbers consist of digits from zero through nine. The following are examples of decimal numbers and there's an infinite number of them but I just chose these few to demonstrate and another name for decimal numbers, our base 10 numbers, pronounced base 10. And even though they look exactly the same, the value of the Bible Number 101 is different from the value of the decimal number 101.

The value of the binary number 101 is equal to the decimal number five. The value of the decimal number 101 is equal to 101. Every binary number has a corresponding decimal value and vice versa. There's a one to one relationship for each number. In the numbering systems have the binary number and the decimal numbers. Here are some examples.

One is the same in decimal as it is in binary. So a zero we haven't got it listed here, but those are the only two numbers that are are the same. One, zero in binary number is the decimal that has a decimal equivalent of 211 as a decimal equivalent of three, and 1010111 has the decimal equivalent of 87. And here are some more examples. And putting them in tabular form sometimes is very useful because rather than try to do the, the mathematical calculation for the equivalent just look it up in table so here we have the numbers from zero to seven, represented by the binary number zero to triple one. And if you go on from eight to 15, you see that the decimal equivalents just to the right, go from 1000 to 1111.

And these are the numbers from 16 to 23. You'll notice That after 15, you need an extra binary number to the left in order to continue on counting and you'll find this as in a decimal number. As you go up in quantity, or value in quantity, you're going to have to add digits to the left becoming more and more significant. Here's another example of equivalency. We have 25. In in the decimal system, equated to 11001 in the binary system, and you'll notice we've written in and this is a standard notation of 10 in the bottom right hand corner as a subscript to the 25 denoting it that it is to the base Can it's a decimal number to the base can, and the two written to the subscript on the number to the right is denoting and added as a binary number to the base two.

This is a standard way of writing numbers sometimes not always. But when you want to keep track of two systems at the same time, it helps to write these identifiers in in the form of a subscript. Another way of considering a number is in terms of its base and its weight, or position from right to left. Each column is given a weight based on its position, the least significant digit being given the weight of zero. The next significant digit being given the weight of one the next or most In this case, significant digit being given the weight of two. The numbers then can be calculated in terms of the sum of each of the single digits multiplied by its base raise to the power of its Wait, I'm gonna say that again.

The number then can be calculated in terms of the sum of each of the single digits, multiplied by its base raised to the power of its weight. And you can see here that the least significant digit is a five, and that's going to be multiplied by the base value 10 raise to the weight which is zero and 10 to the zero is one. So it's five times one and a value of five. The second digit is two. And that's again multiplied by the base value 10 raised to the value of its position from the right, it's in our two positions to the, to the left, giving it a value that not a zero but have one. So it is 10 to the 110 to the one is 10.

So two times 10 to the one is equal to 20. And the third and most significant digit is one. And it's multiply again by the base raised to the power of two, because of its position being in the two position 012. It's one times 10 squared 10 squared is 100. So one times 100 is 100. You add those figures up and they come to 125.

Now this may seem cumbersome way of handling, but it is As a way of dealing with conversions between systems anyway, it's it's helpful to note that a number can be calculated by its position from right to left, and which is given its weight value, and by its base value. This time, let's consider a buying the binary number 1101. In terms of its base and weight, remember, each column is given a weight based on this position. And this time the base value because we're dealing with a binary number is two. So the least significant digit being given the weight of zero, the next significant digit is given one, the third significant digit is given a weight of two, and the fourth is given a weight of three. Now this may seem incongruous because you're dealing with a zero at the beginning and a zero is the first count.

And the second one is one, the third one is two, etc, etc. The number then can then be calculated in terms of the sum of each of its signal, single digits, so you take the least significant digit, it's one times its base value, the base being two in this case, because the digital number and the weight is zero, so two to the zero is one. So one times one is one. The next significant digit is given the weight of one so you the next significant digit is zero. And you're multiplying it by the base value two raised to the power of one and two to the to the power of one is two, but zero times two is still zero. The next significant digit being given is the weight of two.

So you have the digit one times two to the second power, or two squared, two times two is four. So you take one times four is equal to four. The next a significant digit is one, and it's multiplied by the base value two, given by it's raised to the power of its weight value, which is three. So two times two times two is eight, or two cubed is eight, times one is eight. And if you sum that number up, you become Come up with third team. And if you bring back our old table, you can see that 13 is indeed 1101.

So what we have here is a way of converting binary numbers to decimal numbers by just going through this process. So we have just discovered a way of converting binary numbers to a decimal number. Using the following technique. You multiply each bit by two to the nth power, where n is the weight of the bit. The weight is the position of the bits starting from the right and it's given a weight of zero, and the next digits one the next digit to the next section three. Once you had all these numbers multiplied by their weighted values, then you just add up all the term, all the numbers that you come up with, and the total is the decimal equivalent.

So let's just work through another example to see how it works system. We'll start off with the, the binary number 101011. And it of course is a binary number, so as to the base two, so we have a subscript two in the bottom right hand side of that terminology. So given our methodology, we will take the first number which is one and multiply it by the base value to the race, the power of the weight and the first one, the weight is zero. So two to the zero is one one times one is one. The next one is a source.

The next digit is a one and We're going to multiply that again by the base value raised to the weight this time, the weight is one. So it's two to the first power, which is just the number itself two, so one times two is two. The next number is a zero, so regardless of what we multiply it by, it's going to come out as zero, but we're going to multiply it by two square, but it's still going to come out to zero because we multiply the number by zero. The next number is one, and it's going to be multiplied again by the base value raised to the power of the weight and it's third is the fourth digit in so taking into account that the first one is zero, the fourth digit in gives a weight of three. So it's two cube, which is eight. One times eight is eight.

Again, the next value is going to be raised to the fourth power, but because it we were starting with zero We're multiplying it by zero, we end up with a zero. So the most significant digit is a one. And that's going to be multiplied by two raised to the fifth power, and five raised to the fifth power is 32. So the number one times two to the fifth power is 32. The decimal equivalent then is the sum of all these numbers that we've just calculated, which works out to 43 and 43 is to the base 10. That's what we've got it as a subscript there.

So we now have a method of calculating a decimal equivalent of a binary number. So how about let's go in the other direction we have, we're going to start with a decimal number and this time we want to come up with a binary number. So we need A methodology of converting a decimal number to a binary number. So in converting a decimal number to a binary number we use or we can use the following technique. Take the number that is the decimal number and you divide it by two you get a quotient plus a remainder. And because you're dividing by two, the remainder will be either one or zero.

Keep track of the one or zero remainder. This remainder becomes the least significant bit of your binary number. Now, you take the quotient and you divided by two again and you will get a Second quotient, a smaller number because it's we're getting smaller as we keep dividing, but the second quotient plus a remainder and that remainder will be there be one or zero. Again, because we're dividing by two, track it, it will become the next significant bits of our binary number. So you keep dividing the quotients by two, and the number keeps getting smaller all the time because you're dividing by two all the time. You have to track the remainders.

There are going to be one or zero, and you keep dividing by two until and sorry, the as the remainders are generated, they become the next significant bit of the binary number. And you keep on dividing the quotient until the quotient is zero. At that point, your conversion is finished. And your binary number is the ones and zeros of the remainders placed in order as how you found them. The first one being the least significant and the rest of them being more significant as you go up. So the technique is a little bit confusing, confusing if you've never used it before, and you're going to look at just what the technique is without going through an example.

Once we go through an example, of course, you'll, you'll see, you'll see and remember exactly how to do this conversion. It's very simple. So we want to convert, in this example the number 125, which is a decimal number to the base 10. We want to convert it to a binary number or a base two number. So using or following our technique, we will divide the number by two and we're given a quotient of 62 plus a remainder of one, you track the remainder. And that becomes our least significant digit of our binary number that we're looking for.

Now we divide the quotient by two again, and we get 31. This time there is no leftover, no remainder, you put a zero, but you got to track the zero because that becomes the next significant digit of our binary number. You then divide 31 by two and you get 15 with a remainder of one, divide 15 by two and you get seven with a remainder of one divide seven by two and you get three with a remainder of one divide three by two, you get one with a remainder of one, and you divide one by two, you get zero with a remainder of one. Because our quotient is now zero. We're finished the conversion process. It's pretty simple.

And we can now rewrite the number And this time, we just have to keep track of which is our least significant digit is the first remainder that we came across. And everything follows in order from right to left. So you got one is the first number zero was the second number, and all the rest were ones. So I converted number 125 is actually 111101. And of course, there's another technique that you can use for converting in either direction. And that is you can go on the internet and you can use your computer you can also download apps so it will do the conversion in a blink of an eye.

However, if your battery fails and your cell phone or your computer fails, you still have your manual way of converting one to the other. And as I promised that, we would look at the decimal to binary conversion technique again, you divide the number by two, you get a quotient and the remainder, the remainder will be a one or a zero. This remainder becomes our least significant bit of our binary number. You divide the first quotient by two and you get a second quotient plus a remainder one or zero, you track the one or zero. And this remainder is now the next significant bit. You keep dividing the quotients by to tracking the remainders.

And they they then become the next significant bits of our convert converted number and you stopped doing the dividing when the quotient becomes zero. Just before finishing off this chapter, I want to talk briefly or just introduce you to some binary nomenclature, what you're going to run into out there and it becomes a little bit confusing from time to time because These terms weren't thought of ahead of time, they've kind of evolved with the system. So it helps to know a little bit about what their roots are. This the smallest bit, if you want to have a binary number a binary nomenclature is the bit. It's either a one or a zero. And the term bit I guess, is derived from the two words binary digit.

The next size that you might be referred to, and it's not often referred to as a nibble, it's a combination of four bits of information. You heard the term a bite. A bite is a combination of eight bits. A word is made up of 16 bits and a double word is made up of 32 bits. A kilobyte Or a kilobyte is 1024 bytes of information. It's a closest binary digit number to 1000.

And that is one zero to four bytes of information. a megabyte is 1024 kilobytes of information. And a gigabyte is 1024 megabytes of information. And this table expresses some of the common terminologies that are used in the decimal system are the base 10 Pico, and often you'll heard things called Pico fer ads when you're referring to electronic devices or Pico anything. Pico is a very small quantity, it's 10 to the minus 12. So It's actually 1 million millionth a very small number, the next largest number two, that is a nano and it's 10 to the minus nine micro is 10 to the minus six or one 1 million.

And a milli a milli is one 1,000th 10 to the minus three, going in the other direction, keto is 1000 or 10 to the third power, mega is a million. It's made up of 10 to the six or a million and a Giga is 10 to the ninth, and a terra is 10 to the 12th. And you can see the short form symbols there as well. This ends chapter one