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Numbering Systems and Arithmetic operations on Hex, binary, and octal
Chapter 0 Numbering Systems and Arithmetic operations on Hex, binary, and octal By: Bryar M. Shareef
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Common Number Systems System Base Symbols Used by humans?
Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 Octal 8 0, 1, … 7 Hexa- decimal 16 0, 1, … 9, A, B, … F
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Quantities/Counting (1 of 3)
Decimal Binary Octal Hexa- decimal 1 2 10 3 11 4 100 5 101 6 110 7 111 p. 33
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Quantities/Counting (2 of 3)
Decimal Binary Octal Hexa- decimal 8 1000 10 9 1001 11 1010 12 A 1011 13 B 1100 14 C 1101 15 D 1110 16 E 1111 17 F
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Quantities/Counting (3 of 3)
Decimal Binary Octal Hexa- decimal 16 10000 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 10100 24 14 10101 25 15 10110 26 10111 27 Etc.
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Conversion Among Bases
The possibilities: Decimal Octal Binary Hexadecimal pp
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Quick Example 2510 = = 318 = 1916 Base
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Decimal to Decimal (just for fun)
Octal Binary Hexadecimal Next slide…
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Weight 12510 => 5 x 100 = x 101 = x 102 = Base
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Binary to Decimal Decimal Octal Binary Hexadecimal
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Binary to Decimal Technique
Multiply each bit by 2n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results
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Example Bit “0” => 1 x 20 = x 21 = x 22 = x 23 = x 24 = x 25 = 32 4310
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Octal to Decimal Decimal Octal Binary Hexadecimal
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Octal to Decimal Technique
Multiply each bit by 8n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results
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Example 7248 => 4 x 80 = x 81 = x 82 =
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Hexadecimal to Decimal
Octal Binary Hexadecimal
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Hexadecimal to Decimal
Technique Multiply each bit by 16n, where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results
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Example ABC16 => C x 160 = 12 x 1 = B x 161 = 11 x 16 = A x 162 = 10 x 256 = 2560 274810
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Decimal to Binary Decimal Octal Binary Hexadecimal
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Decimal to Binary Technique Divide by two, keep track of the remainder
First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc.
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Example 12510 = ?2 12510 =
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Octal to Binary Decimal Octal Binary Hexadecimal
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Octal to Binary Technique
Convert each octal digit to a 3-bit equivalent binary representation
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Example 7058 = ?2 7058 =
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Hexadecimal to Binary Decimal Octal Binary Hexadecimal
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Hexadecimal to Binary Technique
Convert each hexadecimal digit to a 4-bit equivalent binary representation
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Example 10AF16 = ?2 A F 10AF16 =
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Decimal to Octal Decimal Octal Binary Hexadecimal
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Decimal to Octal Technique Divide by 8 Keep track of the remainder
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Example = ?8 8 19 2 8 2 3 8 0 2 = 23228
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Decimal to Hexadecimal
Octal Binary Hexadecimal
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Decimal to Hexadecimal
Technique Divide by 16 Keep track of the remainder
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Example = ?16 77 2 16 = D 0 4 = 4D216
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Binary to Octal Decimal Octal Binary Hexadecimal
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Binary to Octal Technique Group bits in threes, starting on right
Convert to octal digits
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Example = ?8 = 13278
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Binary to Hexadecimal Decimal Octal Binary Hexadecimal
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Binary to Hexadecimal Technique Group bits in fours, starting on right
Convert to hexadecimal digits
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Example = ?16 B B = 2BB16
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Octal to Hexadecimal Decimal Octal Binary Hexadecimal
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Octal to Hexadecimal Technique Use binary as an intermediary
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Example 10768 = ?16 E 10768 = 23E16
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Hexadecimal to Octal Decimal Octal Binary Hexadecimal
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Hexadecimal to Octal Technique Use binary as an intermediary
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Example 1F0C16 = ?8 1 F C 1F0C16 =
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Exercise – Convert ... Decimal Binary Octal Hexa- decimal 33 1110101
703 1AF Don’t use a calculator! Skip answer Answer
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Exercise – Convert … Decimal Binary Octal Hexa- decimal 33 100001 41
Answer Decimal Binary Octal Hexa- decimal 33 100001 41 21 117 165 75 451 703 1C3 431 657 1AF
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What is the two’s complement?
A two's-complement system is a system in which negative numbers are represented by the two's complement of the absolute value
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Example Suppose we're working with 8 bit and suppose we want to find how -28 would be expressed in two's complement notation. First we write out 28 in binary form. Then we invert the digits. 0 becomes 1, 1 becomes 0. Then we add 1. That is how one would write -28 in 8 bit binary.
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Common Powers (1 of 2) Base 10 Power Preface Symbol pico p nano n
10-12 pico p 10-9 nano n 10-6 micro 10-3 milli m 103 kilo k 106 mega M 109 giga G 1012 tera T Value .001 1000
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Common Powers (2 of 2) Base 2 What is the value of “k”, “M”, and “G”?
Preface Symbol 210 kilo k 220 mega M 230 Giga G Value 1024 What is the value of “k”, “M”, and “G”? In computing, particularly w.r.t. memory, the base-2 interpretation generally applies
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Example In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties / 230 =
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Review – multiplying powers
For common bases, add powers ab ac = ab+c 26 210 = 216 = 65,536 or… 26 210 = 64 210 = 64k
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Binary Addition (1 of 2) Two 1-bit values A B A + B 1 10 “two”
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Binary Addition (2 of 2) Two n-bit values Add individual bits
Propagate carries E.g., 1 1
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Multiplication (1 of 3) Decimal (just for fun)
35 x
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Multiplication (2 of 3) Binary, two 1-bit values A B A B 1
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Multiplication (3 of 3) Binary, two n-bit values
As with decimal values E.g., x
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Fractions Decimal to decimal (just for fun)
3.14 => 4 x 10-2 = x 10-1 = x 100 =
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Fractions Binary to decimal
=> 1 x 2-4 = x 2-3 = x 2-2 = x 2-1 = x 20 = x 21 =
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Fractions Decimal to binary 3.14579 11.001001...
x x x x x x etc. Decimal to binary
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Exercise – Convert ... Decimal Binary Octal Hexa- decimal 29.8
3.07 C.82 Don’t use a calculator! Skip answer Answer
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Exercise – Convert … Decimal Binary Octal Hexa- decimal 29.8
Answer Decimal Binary Octal Hexa- decimal 29.8 … 35.63… 1D.CC… 5.8125 5.64 5.D 3.07 3.1C 14.404 C.82
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Arithmetic Operations on Hex, Binary, Octal
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Decimal Addition 111 3758 + 4657 8415 3 7 5 8 What is going on?
(carry) (subtract the base)
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Binary Addition Rules. 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1
1 + 1 = 2 = 102 = 0 with 1 to carry = 3 = 112 = 1 with 1 to carry
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Binary Addition Verification 5510 + 2810 8310 = = 8310
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Binary Addition ex Verification = 1 0 0 1 1 1 + 0 1 0 1 1 0 + ___
___ ___________ =
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Octal Addition 1 1 6 4 3 78 + 2 5 1 08 9 9 - 8 8 (subtract Base (8))
9 9 (subtract Base (8))
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Octal Addition ex (subtract Base (8))
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Hexadecimal Addition 1 1 7 C 3 916 + 3 7 F 216 20 18 11
1 1 7 C F 216 (subtract Base (16)) B B16
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Hexadecimal Addition 8 A D 416 D 616 (subtract Base (16)) 16
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Decimal Subtraction How it was done? ( add the base 10 when borrowing) 10 10
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Binary Subtraction Verification 8310 5510 = = 5510
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Binary Subtraction ex Verification = 1 0 0 1 1 1 - 0 1 0 1 1 0 - ___
___ ___________ =
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Octal Subtraction
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Octal Subtraction ex
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Hexadecimal Subtraction
B 16 7 C F
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Hexadecimal Subtraction
16
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Thanks
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