Introduction to number systems and binary | Pre-Algebra | Khan Academy

From the beginning of mankind, man counted things and looking for ways to get an overview and view the things he counted. For example, imagine that you are a person in the early days and you try to keep track of the last day, the last time it rained. You can say, great, it didn\’t rain today, so it\’s been 1 day. ” We now use the word “one”, but then they did not have to use the word. The next day is coming. And more. And the next day it rained. Now that his friend is coming, and he asks, “How long has it not rained?” And you would answer “It\’s been so long!” And your friend would say, “OK, it pretty much describes it.” And after a while, they thought it would be useful to name them. And so they called them “one, two, three, four, five, six, seven.” Of course, each language in the world named them a little differently. We are sure there were languages which had different names for them. You quickly realize this way is a relatively bulky inefficient way of representing numbers. First, it takes a long time to write them down, they also take up a lot of space, and if someone wants to read them later, they have to sit and count for a long time. Counting 7 is already difficult, and imagine if you had 27. Or 1000. Maybe it would take the whole page and, moreover, it would be easy to make a mistake. To solve this problem, humanity invented a number system It’s something we take for granted. You can say, “That\’s how we always counted.” But I hope that during this video you will begin to appreciate the beauty of the number system and you will realize that our system is not the only one in the world. Most of us are used to the decimal number system. It is a system with a basis of 10. And why 10? Probably because I have 10 fingers. At least most of us have. So it’s natural to think in groups of 10 or have 10 symbols. Whatever you want to count, you can use your fingers or even symbols. And because we need ten symbols, we invented zero, one, two, three, four, five, six, seven, eight and nine. These ten digits represent ten symbols, which we use in the decimal system. Let’s repeat how we use them to describe the number 231. What does that actually mean? It’s nice that every number has its place. This place on the right are the units. This is literally number one on the unit site. We have dozens in the middle. Specifically, we have 3 tens for this issue. And this two is in place of hundreds. So this represents 2 hundred. Now we add the numbers down and we really get 231. It’s actually 200 plus 30 plus 1. In the decimal system, it works so that every time I move a place to the left, it is a place with ten times the value of the one on the right. So these are units. And if I multiply one by ten, I get to tens. If we want to go to the next place, we multiply by ten and we are in the hundreds. If you control powers, you know that 1 is the same as 10 to 0. 10 is again the same as 10 to 1. So I write it to the decimal place. And finally 100 is the same as 10 on 2. This is how we could go on indefinitely. This is the strength of the decimal system. Maybe you thought the base might not be 10. What if we .. ..own actually this can be seen as a system with basis 1, we have only one symbol here. What if we had some other basis, something more complex, like two. You may be pleased that such a system exists and is called binary. The previous one is called a decimal, this one is a two. This system is the basis for computer technology. Computers communicate through this system. In the binary system you have 2 symbols, zero and one. Why is this important for computers? Because all the hardware in modern computers, transistors, logic circuits they can be either on or off. When we use a calculator, we count in decimal, but she still uses the binary system inside. But how can we express numbers using the binary system? We can derive a similar hierarchy as in the previous case. Just instead of being a multiple of ten places apart, these will be multiples of two. So let’s make room for numbers. On the far right will be 2 to 0, which is, like 10 to 0, one. So this place can be the same name of the unit. Then we can move one place to the left, it will be 2 on 1. We can call it a binary place, I’ll write it here. It used to be a place of dozens. And we can continue, this place will be 2 on 2, or four place. And we can continue. This place corresponds … Try stopping the video and creating it yourself. This corresponds to 2 on 3, ie octal. Notice that we always move to the left by multiples of two. As in the previous case, we moved by multiples of ten. So the system is the same, just instead of tens, there are two everywhere. We can continue. We will try to rewrite this number using the binary system. This place will be a place for .. ..I will change my color .. this place is 2 to 4, so we can call it a hexadecimal place. And one more to the left will be 2 to 5, which we will call the thirty-double place. The next one will be 2 to 6, which will be sixty-four places. It tells us how many zeros and ones we have, but we will show it all. And one more side will be 2 to 7, which we can call a one hundred and twenty-eight place. We could go on, but this should be enough for us, to write this number. In the next videos I will show you how to do it. Now you have to believe me that this number can be written as 11100111 in the binary system. What does that actually mean? This means that I have 128 once, plus 64 plus 32 once plus 0 times 16 plus 0 times 8 plus 1 times 4 plus 1 times 2 plus 1 times 1. Let’s see if they are really the same numbers. So we have 128 plus 64 plus 32 .. … we don’t have 16 or 8, there is 0 in place of sixteen and eights, so I’m not adding them .. ..plus 4 plus 2 plus 1 Now let’s add this. I would also like to point out that we are now using the decimal system we know. And when we add it up, you’ll see that this is just the otherwise written number 231. It’s just another representation. The only reason we transferred it was that we were used to this notation. This is how we are used to performing mathematical operations. Perhaps you find it interesting. It completely opened my mind, such a force of the decimal system. In the following videos, we will explore other number systems. In addition to the most used decimal and binary, there are also, for example, hexadecimal, where instead of two or ten symbols we have sixteen. We will show all this together with the way of transfer between them.