Chapter 2. Binary and Number Representation
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Computers are machines that do stuff with information. They let you view, listen, create, and edit information in documents, images, videos, sound, spreadsheets and databases.
They let you compute and calculate with numerical information; they let you send and receive information over networks. To make representing integers in binary trading easier to build and keep them reliable, everything is represented using just two values. You may have seen these two values represented as 0 and 1, but on a computer they are represented by anything that can be in two states.
For example, in memory a low or high voltage is used to store each 0 or 1. On a magnetic disk it's stored with magnetism whether a tiny spot on the disk is magnetised north or south. The idea that everything stored and transmitted in our digital world is stored using just two values might seem somewhat fantastic, but here's an exercise that will give you a little experience using just black and white cards to represent numbers.
In the following interactive, click on the last card on the right to reveal that it has one dot on it. Now click on the previous card, which should have two dots on it. Before clicking on the next one, how many dots do you predict it will have?
Carry on clicking on each card moving left, trying to guess how many dots each has. Representing integers in binary trading the interactive online at http: The challenge for you now is to find a way to have exactly 22 dots showing the answer is in the spoiler below. Now try making up other numbers of dots, such as 11, 29 and Is there any number that can't be represented? To test representing integers in binary trading, try counting up from 0.
You may have noticed that each card shows twice as many dots as the one to its right. This is an important pattern in data representation on computers.
The number 22 requires the cards to be "white, black, white, white, black", 11 representing integers in binary trading "black, white, black, white, white", 29 is "white, white, white, black, white", and 19 is "white, black, black, black, white". You should have found that any number from 0 to 31 can be represented with 5 cards. Each of the numbers could be communicated using just two words: For example, 22 dots is "white, representing integers in binary trading, white, white, black".
Or you could decode "black, black, white, white, representing integers in binary trading to the number 7. This is the basis of data representation - anything that can have two different states can represent anything on a digital device. For example, a piece of computer memory could have the following voltages:. They are just using physical mechanisms such as high and low voltage, north or south polarity, and light or dark materials.
The use of the two digits 0 and 1 is so common that some of the best known computer jargon is used for them. Since there are only two digits, the system is called binary. The short word for a "binary digit" is made by taking the first two letters and the last letter a bit is just a digit that can have two values.
Every file you save, every picture representing integers in binary trading make, every download, every digital recording, every web page is just a whole lot of bits.
These binary digits are what make digital technology digital! And the nature of these digits unlock a powerful world of storing and sharing a wealth of information and entertainment.
Computer scientists don't spend a lot of time reading bits themselves, but knowing how they are stored is really important because it affects the amount of space that data will use, the amount of time it takes to send the data to a friend as data that takes more space takes longer to send!
Understanding what the representing integers in binary trading are doing enables you to work out how much space will be required to get high-quality colour, hard-to-crack secret codes, a unique ID for every device in the world, or text that uses more characters than the usual English representing integers in binary trading.
This chapter is about some of the different methods that computers use to code different kinds of information in patterns of these bits, and how this affects the cost and quality of what we do on the computer, or even if something is feasible at all. To begin with, we'll look at Braille.
Braille is not actually a way that computers represent data, but is a great introduction to the topic. More than years ago a year-old French boy invented a system for representing text using combinations of flat and raised dots representing integers in binary trading paper so that they could be read by touch.
The system became very popular with people who had visual impairment as it provided a relatively fast and reliable way to "read" text without seeing it. Louis Braille's system is an early example of a "binary" representation of data there are only two symbols raised and flatand yet combinations of them can be used to represent reference books and works of literature. Each character in braille is represented with a cell of 6 dots.
Each dot can either be raised or not raised. Different numbers and letters can be made by using different patterns of raised and not raised dots. Let's work out how many different patterns can be made using the 6 dots in a Braille character. If braille used only 2 dots, there would be 4 patterns.
And with 3 dots there would be 8 patterns. You may have noticed that there are twice as many patterns with 3 dots as there are with 2 dots. It turns out that every time you add an extra dot, that gives twice as many patterns, so with 4 dots there are 16 patterns, 5 dots has 32 patterns, and 6 dots has 64 patterns. Can you come up with an explanation as to why this doubling of the number of patterns representing integers in binary trading The reason that the number of patterns doubles with each extra dot is that with, say, 3 dots you have 8 patterns, so with 4 dots you can use all the 3-dot representing integers in binary trading with the 4th dot flat, and all of them representing integers in binary trading it raised.
This gives 16 4-dot patterns. And then, you can do the same with one more dot to bring it up to 5 dots. This process can be repeated infinitely. So, Braille, with its 6 dots, can make 64 patterns. That's enough for all the letters of the alphabet, and other symbols too, such as digits and punctuation. The reason representing integers in binary trading looking at Representing integers in binary trading in this chapter is because it is a representation using bits. That is, it contains 2 different values raised and not raised and contains sequences of these to represent different patterns.
The letter m, for example, could be written aswhere "1" means raised dot, and "0" means not raised dot assuming we're reading from left to right and then down. This is the same as how we sometimes use 1's and 0's to show how a computer is representing data.
Braille also illustrates why binary representation is so popular. It would be possible to have three kinds of dot: A skilled braille reader could distinguish them, and with three values per dot, you would only need 4 dots to represent 64 patterns.
The trouble is that you would need more accurate devices to create the dots, and people would need to be more accurate at sensing them.
If a page was squashed, even very slightly, it could leave the information unreadable. Digital representing integers in binary trading almost always use two values binary for similar reasons: Using ten digits like we do in our every day decimal counting system would obviously be too challenging. Why are digital systems so hung up on only using two digits? After all, you could do all the same things with a 10 digit system? As it happens, people have tried to build decimal-based computers, but it's just too hard.
Recording representing integers in binary trading digit between 0 and 9 involves having accurate equipment for reading voltage levels, magnetisation or reflections, and it's a lot easier just to check if it's mainly one way or the other. Watch the video online at https: In this section, we will look at how computers represent numbers.
To begin with, we'll revise how the base number system that we use representing integers in binary trading day works, and then look at binary, which is base After that, we'll look at some other charactertistics of numbers that computers must deal with, such as negative numbers and numbers with decimal points.
The number system that humans normally use is in base 10 also known as decimal. It's worth revising quickly, because binary numbers use the same ideas as decimal numbers, just with fewer digits!
In decimal, the value of each digit in a number depends on its place in the number. Each representing integers in binary trading value in a number is worth 10 times more than the place value to its right, i. Also, there are 10 different digits 0,1,2,3,4,5,6,7,8,9 that can be at each of those place values. If you were only able to use one digit to represent a number, then the largest number would be 9.
After that, you need a second digit, which goes to the left, giving you the next ten numbers 10, 11, It's because we have 10 digits that each one is worth 10 times as much as the one to its right. For example, if you want to write the number in expanded form you might have written it as:. Remember that any number to the power of 0 is 1. All this probably sounds really obvious, but it is worth thinking about consciously, because binary numbers have the same properties.
As discussed earlier, computers can only store information using bits, which only have 2 possible states. This means that they cannot represent base 10 numbers using digits 0 to 9, the way we write down numbers in decimal. Instead, they must represent numbers using just 2 digits -- 0 and 1.
Binary works in a very similar way to Decimal, even though it might not initially seem that way. Because there are only 2 digits, this means that each digit is 2 times the value of the one immediately to the right.
The base 10 decimal system is sometimes called denary, which is more consistent with the the name binary for the base 2 system. The word "denary" also refers to the Roman denarius coin, which was worth ten asses an "as" was a copper or bronze coin. The interactive below illustrates how this binary number system represents numbers. Have a play around with it to see what patterns you can see. Representing integers in binary trading ensure you are understanding correctly how to use the interactive, verify that when you enter the binary number it shows that the decimal representation is 45, that when you enter it shows that the decimal representation is 32, and when you enter it shows the decimal representation is What is the largest number you can make with the interactive?
What is the smallest? Are there any numbers with more than one representation?