Example 1: Denary = 1. Answer: Binary = (1 * 1) = 1. Example 2: Denary = 3. Answer: Binary = (1 * 1) + (2 * 1) = 3

Transcription

1 1.1.1 Binary systems In mathematics and digital electronics, a binary number is a number expressed in the binary numeral system, or base-2 numeral system, which represents numeric values using two different symbols: typically 0 (zero) and 1 (one). More specifically, the usual base-2 system is a positional notation with a radix of 2. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computer-based devices such as mobile phones. Each digit is referred to as a bit. Calculating binary numbers Likewise, when working with binary numbers, the position is important in order for you to be able to calculate the correct value. The value 1 in binary represents the value one, the value 0 represents zero. For base-two binary numbers, you need to multiply each digit on the left by a progressive factor of 2. As with denary numbers, the calculations always work from right to left. Example 1: Denary = Answer: Binary = (1 * 1) = 1 Example 2: Denary = Answer: Binary = (1 * 1) + (2 * 1) = 3

2 Example 3: Denary = Answer: Binary = (1 * 0) + (2 * 1) + ( 4 * 1) = 6 Example 4: Denary = Answer: Binary = (1 * 0) + (2 * 0) + (4 * 1) + (8 * 1) = 12 Example 5: Denary = Answer: Binary = (1 * 1) + (2 * 0) + (4 * 1) + (8 * 0) + (16 * 0) + (32 * 1) = 37 Example 6: Denary = Answer: Binary = (1 * 1) + (2 * 1) + (4 * 0) + (8 * 0) + (16 * 1) + (32 * 1) + (64 * 1) = 115

3 Example 7: Denary = Answer: Binary = (the maximum 8 digit binary number) (1 * 1) + (2 * 1) + (4 * 1) + (8 * 1) + (16 * 1) + (32 * 1) + (64 * 1) = 255 if the digit is a 0 then just don't add that value to the total. The number below has a 0 in the 32 position and the binary number in decimal is: x x64 + 0x32 + 1x16 + 1x8 + 1x4 + 1x2 + 1x1 = 223 decimal Let's try a smaller binar y number: What is in decimal? Answer: x16 + 0x8 + 1x4 + 0x2 + 1x1 = 21 decimal

4 Bit A 'bit' (short for Binary Digit) is the smallest unit of data that can be stored by a computer. Each 'bit' is represented as a binary number, either 1 (tr ue) or 0 (false). Byte A 'byte' contains 8 bits, so for example, it could be stored as A single keyboard character that you type, such as the letter A or the letter T takes up one byte of storage. Nibble This is not a very commonly used term compared to bit and byte. It is the term given to a group of four bits. Therefore two nibbles make a byte. The reason it is not very common is because most microprocessors use groups of 8 bits and higher i.e. they use one or more bytes to process data. Not many devices make use of a nibble. Kilobyte A kilobyte is a unit of storage capacity. It can be written as kb or kbyte. A kilobyte is generally thought of as 1,000 bytes. However, to be completely correct, it is actually 1,024 bytes which is 2 to the power of 10. This would be equivalent to 1024 characters on the screen.

5 Images usually take up more storage space than text characters. The images below will help you put storage sizes into context. Storage size = 7 kilobytes Storage size = 33 kilobytes So a simple, limited colour image as seen on the left will take up much less storage than the detailed (but cute) image on the right. Megabyte A megabyte is a unit of storage capacity. It can be written as MB or mbyte. A Megabyte is generally thought of as one million bytes or 1,000 kilobytes. However, to be completely correct, it is actually 1,048,576 bytes or 1,024 kilobytes. PowerPoint presentations on the other hand can be quite large. For example, a ten slide presentation with an image on each slide and a theme applied to the slide backgrounds could be 2 or 3 megabytes in size. A typical MP3 song can be anywhere between 3 to 5 megabytes in size. A CD can store up to 650 megabytes of data.. Gigabyte A Gigabyte is equivalent to 1024 Megabytes although you will commonly see it referred to as 1,000 Megabytes (which isn't correct). It can be written as GB or gbyte, but not Gb (used for gigabit). 1 Gigabyte could hold the contents of about 10 yards of books on a book shelf. A DVD can store a Hollywood blockbuster film which would be around 4-8 gigabytes in size. Hard disks are measured in gigabytes, with a typical hard disk size being around 160 gigabytes or more.

6 Terabyte Commonly written as TB. A terabyte is equivalent to 1024 Gigabytes although you will commonly see it referred to as 1,000 Gigabytes (which isn't correct) Hexadecimal numbers This is name given to numbers using the base-16 notation. Allowed Hexadecimal digits are: Decimal Hexadecimal A B C D E F The decimal numbers 10 to 15 are represented with the letters A to F. There are 16 values, hence why it is a base-16 number system. Hexadecimal is widely used in computing because it is a much shorter way of representing a byte of data. If we were to r epresent a byte of data in binar y, it would require 8 digits, e.g However, that same byte of data could be represented in hexadecimal in just two digits e.g. FF - much more compact and user friendly than a binary number. The table above only shows hexadecimal numbers up to denary 15. The largest byte value is 255 and you can use hexadecimal numbers to repr esent up to this value. We can extend the table above to show the next set of hexadecimal numbers: Decimal Hexadecimal A 1B 1C 1D 1E 1F Once we have gone past the first 16 numbers, a 1 is added in front of the next 16 numbers (similar to denary when we go from 0-9 and then the next 10 numbers have a 1 in front of them) The next set of hexadecimal numbers would have a 2 in front of them: Decimal Hexadecimal A 2B 2C 2D 2E 2F See if you can complete the table for the next set: Decimal Hexadecimal

7 Hexa to decimal HEX A B C D E F DEC Here is a HEX number: 1E5DF To convert this to a DEC, we need to define the base for our power function. Since HEX is based on 16 different digits [0-9A-F], our base is 16. To convert from HEX to DEC, follow these steps: We know that F = 15 in DEC so we use this formula (15*16 0 ) = 15 We know that D = 13 in DEC so we use this formula (13*16 1 ) = 208 We know that 5 = 5 in DEC so we use this formula (5*16 2 ) = 1280 We know that E = 14 in DEC so we use this formula (14*16 3 ) = We know that 1 = 1 in DEC so we use this formula (1*16 4 ) = Now we add all of the numbers together to get the DEC number for HEX number 1E5DF: = Hexa to binary Example #1 Convert (4E) 1 6 (4) = (0100) (E) = (1110) to binary: S o (4E) 16 = ( ) 2 Example #2 Convert (4A01) (4) = (0100) (A) = (1010) 16 2 (0) = (0000) (1) = (0001) to binary: S o (4A01) = ( ) 16 2

8 0 Uses of hexadecimal in computing Here are two important aspects to the beauty of using Hexadecimal with computers: First, it can represent 16-bit words in only four Hex digits, or 8- bit bytes in just two; thus, by using a numeration with more symbols, it is both easier to work with (saving paper and screen space) and makes it possible to understand. This is why programs such as DEBUG, use only Hexadecimal to display the actual Binary bytes of a Memory Dump rather than a huge number of ones and zeros! The second aspect is closely related: Whenever it is necessary to convert the Hex representation back into the actual Binary bits, the process is simple enough to be done in your own mind. For example, FAD7 hex is (F=1111, A=1010, D=1101, 7=0111) in Binary. A common use of hexadecimal numbers is to describe colors on web pages. Each of the three primary colors (i.e., red, green and blue) is represented by two hexadecimal digits to create 255 possible values, thus resulting in more than 16 million possible colors. For example, the HTML (hypertext markup language) code telling a browser to render the background color of a web page as red is <body bgcolor="#ff0000"> and that telling it to render the page as white is <body bgcolor="#ffffff">. MAC addresses consist of hex values, and a Media Access Control (MAC) address looks like this: Af-14-b3-c The first oddity with MAC addresses is that we've got numbers and letters in the address. Since every single networking device should have a unique MAC address, we have to use hex rather than the usual decimals we're all familiar with. Using hex to express an address allows us to have many more possible combinations than we'd have if we just used decimals.

9 1.1.2 Data storage Understanding different formats to store sound, pictures, video, text and numbers 1 Plain text (.txt). In this case, the individual characters in the document (letters, punctuation, newlines etc.) are each encoded into bytes using the ASCII encoding, and stored in a simple sequence. This format only stores the text itself, with no information about formatting, fonts, page size, or anything like that. It is portable across all Computer systems and can be read and modified by a huge range of software applications. The details of the format are freely available and standardized. If the storage media are damaged, any undamaged sections can be recovered without problems. 2 Word document (.doc). In this case the text plus formatting, page size and so on is stored in a complex encoding. 3 PDF (Portable Document Format,.pdf). In this case, the text plus formatting, page size and similar information are stored in a moderately complex encoding. While the details of this encoding are freely available, the format is owned by Adobe and can be changed by them at any time, for any reason. The document can be viewed and printed on all major platforms, using free software provided by Adobe (or others). PDF documents cannot be readily edited. 4 HTML (Hypertext Markup Language,.html). The text, plus simple formatting, is stored in a simple encoding.this format is freely available and controlled by a public-interest standards body. The document can be viewed in any web browser. It can be edited in a text editor by someone who knows HTML, or in any number of rich text editors, word processors, HTML editors and so on. 5 An image format such as JPEG (.jpg) or TIFF. While the image can be viewed on any modern platform using a wide range of software, editing the content of the document (the sequence of characters, words etc.) will be extremely cumbersome. This format preserves appearance but loses all structure. However this may be an option for documents that only exist on paper, perhaps with a plan to use optical character recognition (OCR) to migrate to a text-based format in the future. Binary Representation of Bitmap Images All bitmap images are stored as an array of pixels. A monochrome bitmap will store a 1 for a black pixel and 0 for a white pixel (or vice-versa depending on the encoding protocol).

10 This image could be represented by the following 35 binary digits (5 bytes): It would also be necessary to store the dimensions of the image. If the image were in colour, using a colour palette of 256 possible shades, each pixel would need to translate to a value between 0 and 256 (8 bits). Thus we would need 35 bytes to store the image. It is common for colours to be recorded by quantity or Red, Green and Blue (RGB) and this is stored using 3 bytes per pixel so we would need 105 bytes to store the image. Bitmap images can be encoded in order to reduce the file size. Examples of encoded bitmaps in clued JPG, PNG and GIF file types. Binary Representation of Sound In order to digital record sound, a microphone is used, a device that converts the analogue sound into a digital form by altering the output signal from the device. In order to store this digitally, the voltage is sampled at frequent intervals (typically times per second or 48kHz) and stored as a binary code (typically 16 or 32 bits per sample). This equates to slightly over 1.5 million bits per second, or 88 MB per minute. An 80 minute album stored in such a manner would require 6.9 GB of storage. There are three ways of altering this: Sample Rate : By reducing the sample rate (e.g. to 22 khz), you reduce the amount of data you need to store. This has quite serious effects on the quality of the audio. Contact the teacher:

11 Sample Depth : Much as with colour, you can reduce the precision of each datum, using only 8 bits per sample instead of 32, for example. Compression : Ideally you would sample at the highest practical rate you can and these use an encoding algorithm to reduce the output file size (this is why Audacity projects are much bigger in size than the files output). This can be Lossless compression (e.g. FLAC [and potentially MP3/AAC/Ogg Vorbis files]) or Lossy compression (e.g. most MP3, AAC and Ogg Vorbis files). WAV and AIFF files are used for storing uncompressed audio.