Introduction to Numbering Systems and Binary The Decimal

Introduction to Numbering Systems and Binary

The Decimal Number System (base-10) The numbering system we use is called decimal. It consists of 10 numbers: 0123456789. It is used by most civilizations today. In the past, some civilizations used other numbering systems. For example, the Babylonians used the sexagesimal system, based on the number 60. The Babylonians divided the circle into 360 degrees (a multiple of 60). The Mayas used the vigesimal system, based on the number 20.

The Unary Numbering System (base-1) A base-1 numbering system was used by some people in ancient times. In this system, you simply put a mark (or a 1) for each item you are counting. For example, bones have been found with 29 and 30 notches, possibly corresponding to the cycle of the moon: Placing a mark for every item you are counting would be quite cumbersome if you need to count larger quantities. A better way is to have special symbols to represent larger numbers. For example, Roman numerals use V for 5, X for 10, L for 50, etc. The ancient Egyptians also used symbols to represent larger quantities. The number below is 2, 382, 358.

The Binary Numbering System (base-2) A bit (short for binary digit) is the smallest unit of data in a computer. A bit has a single binary value, either 0 or 1. It is implemented with an electronic switch that is either on or off. If the switch is on, it is equal to 1. If the switch is off, it is equal to 0. You may notice the symbol on power switches. This symbol composed of the 1 and 0 (on and off). When we have a lot of these on/off electronic switches together we come up with a numbering system that only has two digits – 1 and 0. This numbering system is called binary. On Off On On This table shows six switches, or bits. 1 0 1 1 It stores the binary number 101011.

Bytes A bit can store either a 0 or 1. If we group eight bits together we form a byte. To convert between decimal and binary, you can use the Windows calculator in programmer mode or use an online binary calculator. Convert these bytes from binary to decimal: (a) 11110000 Answer: 240 (b) 1010 Answer: 170 The smallest number a byte can be is: 0000 = 0 (decimal) The largest number a byte can be is: 1111 = 255 (decimal) Therefore, a byte can store a decimal number from 0 to 255.

Binary Prefixes Unit Size What it Can Store Bit 0 or 1 One bit of information can store answers to yes/no questions: "Do you have a driver’s license? " Byte 8 bits A byte can store a decimal number from 0 to 255. Using ASCII, a byte can store one text character. Kilobyte (KB) 1, 000 bytes 2 KB could store one page of a document. 50 KB can store a very low resolution photograph. Megabyte (MB) 1, 000 bytes 1 MB can store a novel. 5 MB can store the complete works of Shakespeare or 30 seconds of TV-quality video. 640 MB can store a music CD. Gigabyte (GB) 1, 000, 000 bytes 4 GB can store a DVD movie. 16 GB can store a Blu-ray or HD movie. Terabyte (TB) 1, 000, 000 bytes 10 TB can store the printed section of the Library of Congress. Petabyte (PB) 1, 000, 000 bytes 250 PB can store everything ever printed on Earth. Exabyte (EB) 1, 000, 000 bytes The capacity of large data centers such as Google or Facebook are in the exabytes. Zettabyte (ZB) 1, 000, 000, 000 bytes In 2012, it was estimated that the total amount of global data is 2. 7 ZB. Yottabyte (YB) 1, 000, 000, 000 bytes

Doing Calculations with Bytes Example 1: You have some video files that are 2 GB each. How many of these files will fit on a 1 TB external hard drive? Solution: 1 TB 1, 000, 000 1, 000 ------- = ------------ = 500 video files 2 GB 2, 000, 000 2 Example 2: How 8 MB photos will fit on a phone with 24 GB free space? Solution: 24 GB 24, 000, 000 24, 000 ----- = ----------- = 3000 photos 8 MB 8, 000 8

Storing Text using Binary Since computers only store bytes of data (a number between 0 and 255), in order to store text we need a coding system to translate numbers into text. One that has been used for decades is called ASCII. Each character is stored using a byte.

Unicode – Since the ASCII coding system only has the characters for Latin derived languages, Unicode is a newer coding system used to store the characters for almost all writing systems (see table on right). Each Unicode character requires 2 bytes of storage giving a total of 65, 536 different characters. Here is a complete Unicode chart: http: //www. tamasoft. co. jp/en/g eneral-info/unicode. html Most Writing Systems Used Today Arabic Gurmukhi (Punjabi) Myanmar (Burmese) Armenian Han (Kanji, Hanja, Hanzi) N'Ko Bengali Hangul (Korean) Oriya Braille Hebrew Osmanya Canadian Aboriginal Syllabics Hiragana and Katakana (Japanese) Silot Nagri Cherokee International Phonetic Alphabet (IPA) Syriac Coptic Khmer (Cambodian) Tamil Cyrillic Kannada Telugu Devanagari Lao Thai Ethiopic Latin Tibetan Georgian Limbu Tifinagh Greek Malayalam Yi Gujarati Mongolian Zhuyin (Bopomofo) Extinct Writing Systems for Academic Purposes Cuneiform Ogham Runic Deseret Old Italic (Etruscan) Shavian Linear B Old Persian Ugaritic Kharoshti Phoenician

Summary: Storing Text on a Computer

Counting in Binary (for programming majors) Counting in binary can be tricky to understand for newcomers. One method that might help you count in binary would be to use the decimal system, but only keep the numbers that contain 0's and 1's. The table above shows these numbers in bold. Binary Decimal 0 0 1 1 10 2 11 3 100 4 The table on the right shows how to count to 12 in binary. 101 5 110 6 Now you can understand a classic programmer joke: 111 7 1000 8 1001 9 1010 10 1011 11 1100 12 There’s 10 types of people. Those who understand binary and those who don’t.

Converting Binary to Decimal Manually (for programming majors) The easiest way to convert a binary number to decimal is to use the Windows calculator. To do it manually follow these steps: (1) write the binary number (e. g. 10101110) (2) on the rightmost number, put a 1 on top of it. On the next number to the left, put a 2 on top of it. Keep doubling the numbers until you have a number on top of each binary digit. (3) add up all the numbers that have a 1 below them. Example: Convert binary 10101110 to decimal 128 1 64 0 32 1 16 0 8 1 4 1 128 + 32 + 8 + 4 + 2 = 174 2 1 1 0