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Chapter 1 - PPT - Mano & Kime - 3rd EdSlide 1 Lecture 1 Number Systems and Base ConversionLecture 1 Number Systems and Base Conversion
Slide 2 OutlineOutline Number System Representation Converting Binary to Decimal Converting Decimal to Binary Octal and Hexadecimal Conversion
Slide 3 Number Systems RepresentationNumber Systems Representation Positive radix, positional number systems A number with radix r is represented by a string of digits: An - 1An - 2 … A1A0 . A- 1 A- 2 … A- m + 1 A- m in which 0 £ Ai < r and . is the radix point. The string of digits represents the power series: ( ) ( ) (Number)r = å å + j = - m j j i i = 0 i r A r A (Integer Portion) + (Fraction Portion) i = n - 1 j = - 1
Slide 4 Number Systems ExamplesNumber Systems Examples
Slide 5 Special Powers of 2Special Powers of 2 210 (1024) is Kilo, denoted "K" 220 (1,048,576) is Mega, denoted "M" 230 (1,073, 741,824)is Giga, denoted "G"
Slide 6 Positive Powers of 2Useful for Base Conversion Exponent Value Exponent Value 0 1 11 2,048 1 2 12 4,096 2 4 13 8,192 3 8 14 16,384 4 16 15 32,768 5 32 16 65,536 6 64 17 131,072 7 128 18 262,144 19 524,288 20 1,048,576 21 2,097,152 8 256 9 512 10 1024 Positive Powers of 2
Slide 7 Converting Binary to DecimalTo convert to decimal, use decimal arithmetic to form S (digit × respective power of 2). Example:Convert 110102 to N10: Converting Binary to Decimal
Powers of 2: 43210 110102 => 1 X 24 = 16 + 1 X 23 = 8 + 0 X 22 = 0 + 1 X 21 = 2 + 0 X 20 = 0 2610
Slide 8 Converting Decimal to BinaryMethod 1 Subtract the largest power of 2 (see slide 6) that gives a positive remainder and record the power. Repeat, subtracting from the prior remainder and recording the power, until the remainder is zero. Place 1 s in the positions in the binary result corresponding to the powers recorded; in all other positions place 0 s. Example: Convert 62510 to N2 Converting Decimal to Binary
625 512 = 113 => 9 113 64 = 49 => 6 49 32 = 17 => 5 17 16 = 1 => 4 1 1 = 0 => 0 Placing 1 s in the result for the positions recorded and 0 s elsewhere, 9 8 7 6 5 4 3 2 1 0 1 0 0 1 1 1 0 0 0 1
Slide 9 Converting Decimal to BinaryMethod 1 Subtract the largest power of 2 (see slide 14) that gives a positive remainder and record the power. Repeat, subtracting from the prior remainder and recording the power, until the remainder is zero. Place 1 s in the positions in the binary result corresponding to the powers recorded; in all other positions place 0 s. Example: Convert 62510 to N2 Converting Decimal to Binary
625 512 = 113 => 9 113 64 = 49 => 6 49 32 = 17 => 5 17 16 = 1 => 4 1 1 = 0 => 0 Placing 1 s in the result for the positions recorded and 0 s elsewhere, 9 8 7 6 5 4 3 2 1 0 1 0 0 1 1 1 0 0 0 1
Slide 10 Commonly Occurring BasesCommonly Occurring Bases Name Radix Digits Binary 2 0,1 Octal 8 0,1,2,3,4,5,6,7 Decimal 10 0,1,2,3,4,5,6,7,8,9 Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F The six letters (in addition to the 10 integers) in hexadecimal represent: 10, 11, 12, 13, 14, 15
Answer: The six letters A, B, C, D, E, and F represent the digits for values 10, 11, 12, 13, 14, 15 (given in decimal), respectively, in hexadecimal. Alternatively, a, b, c, d, e, f are used.
Slide 11 Numbers in Different BasesDecimal (Base 10) Binary (Base 2) Octal (Base 8) Hexa decimal (Base 16) 00 00000 00 00 01 00001 01 01 02 00010 02 02 03 00011 03 03 04 00100 04 04 05 00101 05 05 06 00110 06 06 07 00111 07 07 08 01000 10 08 09 01001 11 09 10 01010 12 0A 11 0101 1 13 0B 12 01100 14 0C 13 01101 15 0D 14 01110 16 0E 15 01111 17 0F 16 10000 20 10 Good idea to memorize! Numbers in Different Bases
Slide 12 Conversion Between BasesConversion Between Bases Method 2 To convert from one base to another: 1) Convert the Integer Part 2) Convert the Fraction Part 3) Join the two results with a radix point
Slide 13 Conversion DetailsConversion Details To Convert the Integral Part: Repeatedly divide the number by the new radix and save the remainders. The digits for the new radix are the remainders in reverse order of their computation. If the new radix is > 10, then convert all remainders > 10 to digits A, B, … To Convert the Fractional Part: Repeatedly multiply the fraction by the new radix and save the integer digits that result. The digits for the new radix are the integer digits in order of their computation. If the new radix is > 10, then convert all integers > 10 to digits A, B, …
Slide 14 Example: Convert 46.687510 To Base 2Example: Convert 46.687510 To Base 2 Convert 46 to Base 2 46/2 = 23 rem 0 23/2 = 11 rem 1 11/2 = 5 rem 1 5/2 = 2 rem 1 2/2 = 1 rem 0 1/2 = 0 rem 1 Reading remainders in reverse: 1011102 Convert 0.6875 to Base 2: 0.6875 x 2 = 1.3750 int = 1 0.3750 x 2 = 0.7500 int = 0 0.7500 x 2 = 1.5000 int = 1 0.5000 x 2 = 1.0000 int = 1 0 Reading int portion in forward direction: 0.10112 Join the results together with the radix point: 101110.10112
Answer 1: Converting 46 as integral part: Answer 2: Converting 0.6875 as fractional part: 46/2 = 23 rem = 0 0.6875 * 2 = 1.3750 int = 1 23/2 = 11 rem = 1 0.3750 * 2 = 0.7500 int = 0 11/2 = 5 remainder = 1 0.7500 * 2 = 1.5000 int = 1 5/2 = 2 remainder = 1 0.5000 * 2 = 1.0000 int = 1 2/2 = 1 remainder = 0 0.0000 1/2 = 0 remainder = 1 Reading off in the forward direction: 0.10112 Reading off in the reverse direction: 1011102 Answer 3: Combining Integral and Fractional Parts: 101110. 10112
Slide 15 Additional Issue - Fractional PartAdditional Issue - Fractional Part Note that in this conversion, the fractional part became 0 as a result of the repeated multiplications. In general, it may take many bits to get this to happen or it may never happen. Example: Convert 0.6510 to N2 0.65 = 0.1010011001001 … The fractional part begins repeating every 4 steps yielding repeating 1001 forever! Solution: Specify number of bits to right of radix point and round or truncate to this number.
Point out here that this is why banks don t like to use binary representations of numbers: 65 cents in your account should be 65 cents, not something close to it. Later we will learn about BCD, an alternative representation that avoids this problem.
Slide 16 Checking the ConversionChecking the Conversion To convert back, sum the digits times their respective powers of r. From the prior conversion of 46.687510 1011102 = 1·32 + 0·16 +1·8 +1·4 + 1·2 +0·1 = 32 + 8 + 4 + 2 = 46 0.10112 = 1/2 + 1/8 + 1/16 = 0.5000 + 0.1250 + 0.0625 = 0.6875
Slide 17 Octal (Hexadecimal) to Binary and BackOctal (Hexadecimal) to Binary and Back Octal (Hexadecimal) to Binary: Restate the octal (hexadecimal) as three (four) binary digits starting at the radix point and going both ways. Binary to Octal (Hexadecimal): Group the binary digits into three (four) bit groups starting at the radix point and going both ways, padding with zeros as needed in the fractional part. Convert each group of three bits to an octal (hexadecimal) digit.
Slide 18 Octal to Hexadecimal via BinaryOctal to Hexadecimal via Binary Convert octal to binary. Use groups of four bits and convert as above to hexadecimal digits. Example: Octal to Binary to Hexadecimal 6 3 5 . 1 7 7 8 110|011|101 . 001|111|1112 Regroup: 1|1001|1101 . 0011|1111|1(000)2 Convert: 1 9 D . 3 F 816
Answer 1: 6 3 5 . 1 7 7 8 110|011|101 . 001|111|111 2 Regroup: 1|1001|1101 . 0011|1111|1(000)2 Convert: 1 9 D . 3 F 816 Answer 2: Marking off in groups of three (four) bits corresponds to dividing or multiplying by 23 = 8 (24 = 16) in the binary system.
Slide 19 SummarySummary Number System Representation Converting Binary to Decimal Converting Decimal to Binary Octal and Hexadecimal Conversion
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