What is a byte? What is it? How would you use it? How does it apply to computers?

Binary, Decimal, Hexadecimal,& Octal Goofy video Numbers Systems Binary, Decimal, Hexadecimal,& Octal Goofy video

Common Bases Base 2 1010 0001 Base 8 735 Base 10 129 Base 16 BC94E

Positional Notation So how can we express the numbers in different bases? Positional notation is writing out each symbol and the calculation that represents that position. For example: BASE TEN is our number system. Think of the number 56,423 broken down in base 10 positional notation. The base 10 number 56,423 becomes: (5 x 104) + (6 x 103) + (4 x 102) + (2 x 101) + (3 x 100) 10000 1000 100 10 1 104 103 102 101

Base 2 - BINARY IF… BASE 10 multiplies a number 0-9 by 10position and then adds the numbers together Then… Base 2 multiplies a number _______ by _____position and then adds the numbers together BASE 2 number system is called the binary number system. Computers use binary numbers to communicate because a 0 or 1 is simply and on or off. One binary digit is referred to as bit. A byte contains 8 bits.

Base 2 - BINARY BASE 2 multiplies a number from 0-1 by 2position and then adds the numbers together So let’s look at a typical base 2 number. For example 0010 0111 becomes:   0 x 27 + 0 x 26 + 1 x 25 + 0 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20 = 0 + 0 + 1 x 32 + 0 + 0 + 1 x 4 + 1 x 2 + 1 x 1= 32 + 4 + 2 + 1 =39 128 64 32 16 8 4 2 1 27 26 25 24 23 22 21 20

Base 2 - BINARY 128 64 32 16 8 4 2 1 27 26 25 24 23 22 21 20   GO to the netschool page and click on the BINARY game. Practice your binary numbers. See how high you can score…

Base 8 - Octal The BASE 8 or OCTAL NUMBER SYSTEM is based on 8 numbers 0-7. Let’s look at the octal number 362 converting to decimal form. 0 x 83 + 3 x 82 + 6 x 81 + 2 x 80 = 0 x 512 + 3 x 64 + 6 x 8 + 2 x 1 = 0 + 192+ 48 + 2 = 242 4096 512 64 8 1 84 83 82 81 80

Base 16 - Hexadecimal The HEXADECIMAL NUMBER SYSTEM is based on 16 numbers 0-9 and A-F. Notice that the letters A - F are used in hex to represent the digits 10 – 15. BASE 16 multiplies a number 0-9 & A-F by 16position and then adds the numbers together. D3C216 = D x 163 + 3 x 162 + C x 161 + 2 x 160 = 53248 + 768 + 192 + 2 = 54210 65536 4096 256 16 1 164 163 162 161 160

CONVERSIONS: You can convert numbers between decimal and binary with two possible techniques: the ladder technique or positional notation technique. Start with a decimal number. For example 39 using the ladder technique: 39/2 = 19 remainder 1 19/2 = 9 remainder 1 9/2 = 4 remainder 1 4/2 = 2 remainder 0 2/2 = 1 remainder 0 1/2 = 0 remainder 1 You continue to divide the result until you get 0. The remainder numbers are what represent the binary numbers. So 39 in binary form = 100111. The first remainder you get becomes the first bit.

CONVERSIONS: The other technique is called the positional notation technique. It looks at the base 2 values and pulls out the largest value from the decimal number and then finds the other combination of numbers going from largest to smallest. For example 39 using the positional notation technique: First you look for the largest base 2 position less than 39. For 39 the largest number is 32 or 25. Then the largest number less than 39-32 = 7 which is 4 or 22. Then look for the largest number less than 7-4 = 3 which is 2 or 21. And finally the largest number less than 3-2 = 1 which is 1 or 20. 39=25+22+21+20 or 00100111 128 64 32 16 8 4 2 1 27 26 25 24 23 22 21 20

Data Representation Now computers store, present, and help us modify several different types of data: Numbers Text Images and graphics Audio Video All of this data is stored in the computer as binary digits. 

Analog and Digital Data Signal Analog signal is a continuous signal which represents physical measurements. Digital signals are discrete time signals generated by digital modulation. Applications Thermometer PCs, PDAs Bandwidth Analog signal processing can be done in real time and consumes less bandwidth. There is no guarantee that digital signal processing can be done in real time and consumes more bandwidth to carry out the same information. Memory Stored in the form of wave signal Stored in the form of binary bit Power Analog instrument draws large power Digital instrument draws only negligible power Cost Low cost and portable Cost is high and not easily portable While Analog is cheaper it has more errors and size limitations.

Numeric Representation - Integers Remember bit - binary digit, a 0 or 1 byte - a series of 8 bits, 10010101 The largest positive integer that can be represented with one byte is 11111111 (25510). So we have to use multiple bytes to store larger integers. What about negative integers? Most computers use two’s compliment. (see bwagner’s presentation of two’s compliment) Addition of different bases needed