# Binary Representation. Number Systems. Base 10, Base 2, Base 16. Positional Notation. Conversion of Any Base to Decimal.

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1 Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must be able to handle more than just values for real world problems 1, 0, 56 True, False, Maybe Hot, Cold, LukeWarm, Cool On, Off, Leaky V Number Systems To talk about binary data, we must first talk about number systems The decimal number system (base 10) you should be familiar with! A digit in base 10 ranges from 0 to 9. A digit in base 2 ranges from 0 to 1 (binary number system). A digit in base 2 is also called a bit. A digit in base R can range from 0 to R-1 A digit in Base 16 can range from 0 to 16-1 (0,1,2,3,4,5,5,6,7,8,9,A,B,C,D,E,F). Use letters A-F to represent values 10 to 15. Base 16 is also called Hexadecimal or just Hex. V Positional Notation Value of number is determined by multiplying each digit by a weight and then summing. The weight of each digit is a POWER of the BASE and is determined by position = 9 x x x x x 10-2 = = % = 1x x x x x x2-2 = = \$ A2F = 10x x x16 0 = 10 x x x 1 = = 2607 V Base 10, Base 2, Base 16 The textbook uses subscripts to represent different bases (ie. A2F 16, , ) I will use special symbols to represent the different bases. The default base will be decimal, no special symbol for base 10. The \$ will be used for base 16 ( \$A2F) Will also use h at end of number (A2Fh) The % will be used for base 2 (% ) If ALL numbers on a page are the same base (ie, all in base 16 or base 2 or whatever) then no symbols will be used and a statement will be present that will state the base (ie, all numbers on this page are in base 16). V Common Powers Conversion of Any Base to Decimal 2-3 = = = = = = = = = = = = = = = = = 1 = = 16 = = 256 = = 4096 = = 1024 = 1 K 2 20 = = 1 M (1 Megabits) = 1024 K = 2 10 x = = 1 G (1 Gigabits) V Converting from ANY base to decimal is done by multiplying each digit by its weight and summing. Binary to Decimal % = 1x x x x x x2-2 = = Hex to Decimal A2Fh = 10x x x16 0 = 10 x x x 1 = = 2607 V

2 Conversion of Decimal Integer To ANY Base Divide Number N by base R until quotient is 0. Remainder at EACH step is a digit in base R, from Least Significant digit to Most significant digit. Convert 53 to binary 53/2 = 26, rem = 1 Least Significant Digit 26/2 = 13, rem = 0 13/2 = 6, rem = 1 6 /2 = 3, rem = 0 3/2 = 1, rem = 1 1/2 = 0, rem = 1 Most Significant Digit 53 = % = 1x x x x x x2 0 = = 53 V Least Significant Digit Most Significant Digit 53 = % Most Significant Digit (has weight of 2 5 or 32). For base 2, also called Most Significant Bit (MSB). Always LEFTMOST digit. Least Significant Digit (has weight of 2 0 or 1). For base 2, also called Least Significant Bit (LSB). Always RIGHTMOST digit. V More Conversions Convert 53 to Hex 53/16 = 3, rem = 5 3 /16 = 0, rem = 3 53 = 35h = 3 x x 16 0 = = 53 V Hex (base 16) to Binary Conversion Each Hex digit represents 4 bits. To convert a Hex number to Binary, simply convert each Hex digit to its four bit value. Hex Digits to binary: \$ 0 = % 0000 \$ 1 = % 0001 \$2 = % 0010 \$3 = % 0011 \$4 = % 0100 \$5 = % 0101 \$6 = % 0110 \$7 = % 0111 \$8 = % 1000 Hex Digits to binary (cont): \$ 9 = % 1001 \$ A = % 1010 \$ B = % 1011 \$ C = % 1100 \$ D = % 1101 \$ E = % 1110 \$ F = % 1111 V Hex to Binary, Binary to Hex A2Fh = % h = % Binary to Hex is just the opposite, create groups of 4 bits starting with least significant bits. If last group does not have 4 bits, then pad with zeros for unsigned numbers. % = % = 51h Padded with a zero V A Trick! If faced with a large binary number that has to be converted to decimal, I first convert the binary number to HEX, then convert the HEX to decimal. Less work! % = % = D F 3 = 13 x x x16 0 = 13 x x x 1 = = 3571 Of course, you can also use the binary, hex conversion feature on your calculator. Too bad calculators won t be allowed on the first test, though... V

4 34h - 27h Dh 4-7 = D; with borrow of 1 from next column 3-1 (borrow) - 2 = 0. answer = \$ 0D. Hex Subtraction Decimal check. 34h = 3 x = 52 27h = 2 x = = 13 0Dh = 13!! Hex subtraction again Why is 4h 7h = \$D with a borrow of 1? The borrow has a weight equal to the BASE (in this case 16). BORROW +4h 7h = = 20-7 = 13 = Dh. Dh is the result of the subtraction with the borrow. Exactly the same thing happens in decimal. 3-8 = 5 with borrow of 1 borrow = = 13-8 = 5. V V Fixed Precision With paper and pencil, I can write a number with as many digits as I want: 1,027,80,032,034,532,002,391,030,300,209,399,302,992,092,920 A microprocessor or computing system usually uses FIXED PRECISION for integers; they limit the numbers to a fixed number of bits: \$ AF DEFA bit number, 16 hex digits \$ 9DEFA bit number, 8 hex digits \$ A bit number, 4 hex digits \$ 31 8 bit number, 2 hex digits High end microprocessors use 64 or 32 bit precision; low end microprocessors use 16 or 8 bit precision. V Unsigned Overflow In this class I will use 8 bit precision most of the time, 16 bit occassionally. Overflow occurs when I add or subtract two numbers, and the correct result is a number that is outside of the range of allowable numbers for that precision. I can have both unsigned and signed overflow (more on signed numbers later) 8 bits -- unsigned integers 0 to or 0 to bits -- unsigned integers 0 to or 0 to V Unsigned Overflow Example Assume 8 bit precision; ie. I can t store any more than 8 bits for each number. Lets add = 256. The number 256 is OUTSIDE the range of 0 to 255! What happens during the addition? 255 = \$ FF /= means Not Equal + 1 = \$ /= \$00 \$F + 1 = 0, carry out \$F + 1 (carry) + 0 = 0, carry out Carry out of MSB falls off end, No place to put it!!! Final answer is WRONG because could not store carry out. V Unsigned Overflow A carry out of the Most Significant Digit (MSD) or Most Significant Bit (MSB) is an OVERFLOW indicator for addition of UNSIGNED numbers. The correct result has overflowed the number range for that precision, and thus the result is incorrect. If we could STORE the carry out of the MSD, then the answer would be correct. But we are assuming it is discarded because of fixed precision, so the bits we have left are the incorrect answer. V

6 Hex to Signed Decimal (cont) STEP 2 (positive sign): If the sign is POSITIVE, then just convert the hex value to decimal. \$64 is a positive number, decimal value is 6 x = 100. Final answer is \$64 as an 8 bit twos complement integer = +100 Hex to Signed Decimal (cont) STEP 2 (negative sign): If the sign is Negative, then need to compute the magnitude of the number. We will use the trick that - (-N) = + N i.e. Take the negative of a negative number will give you the positive number. In this case the number will be the magnitude. For 2s complement representation, complement and add one. \$F0 = % => % = % = \$10 = 16 V V Hex to Signed Decimal (cont) STEP 3 : Just combine the sign and magnitude to get the result. \$F0 as 8 bit twos complement number is -16 \$64 as an 8 bit twos complement integer = +100 Signed Decimal to Hex conversion 2 s complement Step 1: Ignore the sign, convert the magnitude of the number to binary. 34 = 2 x = \$ 22 = % = 1 x = \$ 14 = % Step 2 (positive decimal number): If the decimal number was positive, then you are finished! +34 as an 8 bit 2s complement number is \$ 22 = % V V Signed Decimal to Hex conversion (cont) Step 2 (negative decimal number): Need to do more if decimal number was negative. To get the final representation, we will use the trick that: -(+N) = -N i.e., if you take the negative of a positive number, get Negative number. For 2s complement, complement and add one. 20 = % => % = % = \$EC Signed Decimal to Hex conversion (cont) Final results: +34 as an 8 bit 2s complement number is \$ 22 = % as an 8 bit 2s complement number is \$ EC = % V V

8 Codes for Characters Also need to represent Characters as digital data. The ASCII code (American Standard Code for Information Interchange) is a 7-bit code for Character data. Typically 8 bits are actually used with the 8th bit being zero or used for error detection (parity checking). 8 bits = 1 Byte. (see Table 2.5, pg 47, Uffenbeck). ASCII American Standard Code for Information Interchange A = % = \$41 & = % = \$26 7 bits can only represent 2 7 different values (128). This enough to represent the Latin alphabet (A-Z, a-z, 0-9, punctuation marks, some symbols like \$), but what about other symbols or other languages? V V UNICODE UNICODE is a 16-bit code for representing alphanumeric data. With 16 bits, can represent 2 16 or different symbols. 16 bits = 2 Bytes per character. \$ A A-Z \$ A a-z Some other alphabet/symbol ranges \$3400-3d2d Korean Hangul Symbols \$ F Hiranga, Katakana, Bopomofo, Hangul \$4E00-9FFF Han (Chinese, Japenese, Korean) UNICODE used by Web browsers, Java, most software these Codes for Decimal Digits There are even codes for representing decimal digits. These codes use 4-bits for EACH decimal digits; it is NOT the same as converting from decimal to binary. BCD Code 0 = % = % = % = % = % = % = % = % = % = % 1001 In BCD code, each decimal digit simply represented by its binary equivalent. 96 = % = \$ 96 (BCD code) Advantage: easy to convert Disadvantage: takes more bits to store a number: 255 = % = \$ FF (binary code) 255 = % = \$ 255 (BCD code) takes only 8 bits in binary, takes 12 bits in BCD. days. V V Gray Code 0 = % = % = % = % = % = % = % = % = % = % 1000 Gray Code for decimal Digits A Gray code changes by only 1 bit for adjacent values. This is also called a thumbwheel code because a thumbwheel for choosing a decimal digit can only change to an adjacent value (4 to 5 to 6, etc) with each click of the thumbwheel. This allows the binary output of the thumbwheel to only change one bit at a time; this can help reduce circuit complexity and also reduce signal noise. What do you need to Know? Convert hex, binary integers to Decimal Convert decimal integers to hex, binary Convert hex to binary, binary to Hex N binary digits can represent 2 N values, unsigned integers 0 to 2 N -1. Addition, subtraction of binary, hex numbers Detecting unsigned overflow Converting a decimal number to Twos Complement Converting a hex number in 2s complement to decimal V V

9 What do you need to know? (cont) Number ranges for 2s complement Overflow in 2s complement Sign extension in 2s complement ASCII, UNICODE are binary codes for character data BCD code is alternate code for representing decimal digits Gray codes can also represent decimal digits; adjacent values in Gray codes change only by one bit. V

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