Number Systems Number Representation Every number like a

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Number Systems

Number Systems

Number Representation Every number like ‘a’ can be represented as

Number Representation Every number like ‘a’ can be represented as

Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10

Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexadecimal 16 0, 1, … 9, A, B, … F No No

Quantities/Counting Hexa. Decimal Binary Octal decimal 0 0 Hexa. Decimal Binary Octal decimal 10

Quantities/Counting Hexa. Decimal Binary Octal decimal 0 0 Hexa. Decimal Binary Octal decimal 10 1010 12 A 1 2 1 10 1 2 11 12 1011 1100 13 14 B C 3 4 5 6 7 8 11 100 101 110 111 1000 3 4 5 6 7 10 3 4 5 6 7 8 13 14 15 16 17 18 1101 1110 1111 10000 10001 10010 15 16 17 20 21 22 D E F 10 11 12 9 1001 11 9 19 10011 23 13

Conversion Among Bases The possibilities: Decimal Octal Binary Hexadecimal

Conversion Among Bases The possibilities: Decimal Octal Binary Hexadecimal

The others to Decimal

The others to Decimal

Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal Next slide…

Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal Next slide…

Weight 12510 => 5 x 100 2 x 101 1 x 102 Base =

Weight 12510 => 5 x 100 2 x 101 1 x 102 Base = 5 = 20 = 100 125

Binary to Decimal Octal Binary Hexadecimal

Binary to Decimal Octal Binary Hexadecimal

Binary to Decimal Technique ◦ Multiply each bit by 2 n, where n is

Binary to Decimal Technique ◦ Multiply each bit by 2 n, 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

Example Bit “ 0” 1010112 => 1 1 0 1 x x x 20

Example Bit “ 0” 1010112 => 1 1 0 1 x x x 20 21 22 23 24 25 = = = 1 2 0 8 0 32 4310

Octal to Decimal Octal Binary Hexadecimal

Octal to Decimal Octal Binary Hexadecimal

Octal to Decimal Technique ◦ Multiply each bit by 8 n, where n is

Octal to Decimal Technique ◦ Multiply each bit by 8 n, 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

Example 7248 => 4 x 80 = 2 x 81 = 7 x 82

Example 7248 => 4 x 80 = 2 x 81 = 7 x 82 = 4 16 448 46810

Hexadecimal to Decimal Octal Binary Hexadecimal

Hexadecimal to Decimal Octal Binary Hexadecimal

Hexadecimal to Decimal Technique ◦ Multiply each bit by 16 n, where n is

Hexadecimal to Decimal Technique ◦ Multiply each bit by 16 n, 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

Example ABC 16 => C x 160 = 12 x 1 = 12 B

Example ABC 16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810

Decimal to Binary Decimal Octal Binary Hexadecimal

Decimal to Binary Decimal Octal Binary Hexadecimal

Decimal to Binary Technique ◦ ◦ Divide by two, keep track of the remainder

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.

Example 12510 = ? 2 2 125 2 62 2 31 2 15 7

Example 12510 = ? 2 2 125 2 62 2 31 2 15 7 2 3 2 1 2 0 1 1 1 12510 = 11111012

Octal to Binary Decimal Octal Binary Hexadecimal

Octal to Binary Decimal Octal Binary Hexadecimal

Octal to Binary Technique ◦ Convert each octal digit to a 3 -bit equivalent

Octal to Binary Technique ◦ Convert each octal digit to a 3 -bit equivalent binary representation

Example 7058 = ? 2 7 0 5 111 000 101 7058 = 1110001012

Example 7058 = ? 2 7 0 5 111 000 101 7058 = 1110001012

Hexadecimal to Binary Decimal Octal Binary Hexadecimal

Hexadecimal to Binary Decimal Octal Binary Hexadecimal

Hexadecimal to Binary Technique ◦ Convert each hexadecimal digit to a 4 -bit equivalent

Hexadecimal to Binary Technique ◦ Convert each hexadecimal digit to a 4 -bit equivalent binary representation

Example 10 AF 16 = ? 2 1 0 A F 0001 0000 1010

Example 10 AF 16 = ? 2 1 0 A F 0001 0000 1010 1111 10 AF 16 = 000101011112

Decimal to Octal Decimal Octal Binary Hexadecimal

Decimal to Octal Decimal Octal Binary Hexadecimal

Decimal to Octal Technique ◦ Divide by 8 ◦ Keep track of the remainder

Decimal to Octal Technique ◦ Divide by 8 ◦ Keep track of the remainder

Example 123410 = ? 8 8 8 1234 154 19 2 0 2 2

Example 123410 = ? 8 8 8 1234 154 19 2 0 2 2 3 2 123410 = 23228

Decimal to Hexadecimal Decimal Octal Binary Hexadecimal

Decimal to Hexadecimal Decimal Octal Binary Hexadecimal

Decimal to Hexadecimal Technique ◦ Divide by 16 ◦ Keep track of the remainder

Decimal to Hexadecimal Technique ◦ Divide by 16 ◦ Keep track of the remainder

Example 123410 = ? 16 16 1234 77 4 0 2 13 = D

Example 123410 = ? 16 16 1234 77 4 0 2 13 = D 4 123410 = 4 D 216

Binary to Octal Decimal Octal Binary Hexadecimal

Binary to Octal Decimal Octal Binary Hexadecimal

Binary to Octal Technique ◦ Group bits in threes, starting on right ◦ Convert

Binary to Octal Technique ◦ Group bits in threes, starting on right ◦ Convert to octal digits

Example 10110101112 = ? 8 1 010 111 1 3 2 7 10110101112 =

Example 10110101112 = ? 8 1 010 111 1 3 2 7 10110101112 = 13278

Binary to Hexadecimal Decimal Octal Binary Hexadecimal

Binary to Hexadecimal Decimal Octal Binary Hexadecimal

Binary to Hexadecimal Technique ◦ Group bits in fours, starting on right ◦ Convert

Binary to Hexadecimal Technique ◦ Group bits in fours, starting on right ◦ Convert to hexadecimal digits

Example 1010112 = ? 16 10 1011 2 B B 1010112 = 2 BB

Example 1010112 = ? 16 10 1011 2 B B 1010112 = 2 BB 16

Octal to Hexadecimal Decimal Octal Binary Hexadecimal

Octal to Hexadecimal Decimal Octal Binary Hexadecimal

Octal to Hexadecimal Technique ◦ Use binary as an intermediary

Octal to Hexadecimal Technique ◦ Use binary as an intermediary

Example 10768 = ? 16 1 0 7 6 001 000 111 110 2

Example 10768 = ? 16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23 E 16

Hexadecimal to Octal Decimal Octal Binary Hexadecimal

Hexadecimal to Octal Decimal Octal Binary Hexadecimal

Hexadecimal to Octal Technique ◦ Use binary as an intermediary

Hexadecimal to Octal Technique ◦ Use binary as an intermediary

Example 1 F 0 C 16 = ? 8 1 0001 1 F 0

Example 1 F 0 C 16 = ? 8 1 0001 1 F 0 1111 7 C 0000 4 1100 1 4 1 F 0 C 16 = 174148

Exercise – Convert. . . Decimal 33 Binary Octal Hexadecimal 1110101 703 1 AF

Exercise – Convert. . . Decimal 33 Binary Octal Hexadecimal 1110101 703 1 AF Don’t use a calculator! Skip answer Answer

Exercise – Convert … Answer Decimal 33 117 Binary 100001 1110101 Octal 41 165

Exercise – Convert … Answer Decimal 33 117 Binary 100001 1110101 Octal 41 165 451 431 111000011 110101111 703 657 Hexadecimal 21 75 1 C 3 1 AF

Common Powers (1 of 2) Base 10 Power Preface Symbol Value 10 -12 pico

Common Powers (1 of 2) Base 10 Power Preface Symbol Value 10 -12 pico p . 0000001 10 -9 nano n . 00001 10 -6 micro . 000001 10 -3 milli m . 001 103 kilo k 1000 106 mega M 1000000 109 giga G 100000 1012 tera T 1000000

Common Powers (2 of 2) Base 2 Power Preface Symbol Value 210 kilo k

Common Powers (2 of 2) Base 2 Power Preface Symbol Value 210 kilo k 1024 220 mega M 1048576 230 Giga G 1073741824 • What is the value of “k”, “M”, and “G”? • In computing, particularly w. r. t. memory, the base-2 interpretation generally applies

Example In the lab… 1. Double click on My Computer 2. Right click on

Example In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties / 230 =

Exercise – Free Space Determine the “free space” on all drives on your personal

Exercise – Free Space Determine the “free space” on all drives on your personal computer! Free space Drive A: C: D: E: etc. Bytes GB

Binary Addition (1 of 2) Two 1 -bit values A 0 0 1 1

Binary Addition (1 of 2) Two 1 -bit values A 0 0 1 1 B 0 1 A+B 0 1 1 10 “two”

Binary Addition (2 of 2) Two n-bit values ◦ Add individual bits ◦ Propagate

Binary Addition (2 of 2) Two n-bit values ◦ Add individual bits ◦ Propagate carries ◦ E. g. , 1 1 10101 + 11001 101110 21 + 25 46

Multiplication (1 of 3) Decimal (just for fun) 35 x 105 175 000 35

Multiplication (1 of 3) Decimal (just for fun) 35 x 105 175 000 35 3675

Multiplication (2 of 3) Binary, two 1 -bit values A 0 0 1 1

Multiplication (2 of 3) Binary, two 1 -bit values A 0 0 1 1 B 0 1 A B 0 0 0 1

Multiplication (3 of 3) Binary, two n-bit values ◦ As with decimal values ◦

Multiplication (3 of 3) Binary, two n-bit values ◦ As with decimal values ◦ E. g. , 1110 x 1011 1110 0000 1110 10011010 Exercise- Search about binary division?

Fractions Decimal to decimal (just for fun) 3. 14 => 4 x 10 -2

Fractions Decimal to decimal (just for fun) 3. 14 => 4 x 10 -2 = 0. 04 1 x 10 -1 = 0. 1 3 x 100 = 3 3. 14

Fractions Binary to decimal 10. 1011 => 1 1 0 1 x x x

Fractions Binary to decimal 10. 1011 => 1 1 0 1 x x x 2 -4 2 -3 2 -2 2 -1 20 21 = = = 0. 0625 0. 125 0. 0 0. 5 0. 0 2. 6875

Fractions Decimal to binary 3. 14579 11. 001001. . 14579 x 2 0. 29158

Fractions Decimal to binary 3. 14579 11. 001001. . 14579 x 2 0. 29158 x 2 0. 58316 x 2 1. 16632 x 2 0. 33264 x 2 0. 66528 x 2 1. 33056 etc.

Exercise – Convert. . . Decimal 29. 8 Binary Octal Hexadecimal 101. 1101 3.

Exercise – Convert. . . Decimal 29. 8 Binary Octal Hexadecimal 101. 1101 3. 07 C. 82 Don’t use a calculator! Skip answer Answer

Exercise – Convert … Answer Decimal 29. 8 5. 8125 3. 109375 12. 5078125

Exercise – Convert … Answer Decimal 29. 8 5. 8125 3. 109375 12. 5078125 Binary Octal 11101. 110011… 35. 63… 101. 1101 5. 64 11. 000111 1100. 10000010 3. 07 14. 404 Hexadecimal 1 D. CC… 5. D 3. 1 C C. 82

Signed Binary number Sign-magnitude 1’s Complement 2’s Complement In all the three methods, the

Signed Binary number Sign-magnitude 1’s Complement 2’s Complement In all the three methods, the left digit for the positive numbers is zero and for the negative numbers is one!. . Sign bit n - 1

Sign-magnitude The left bit is the sign Positive numbers Negative numbers 0000=+0 1000=-0 0001=+1

Sign-magnitude The left bit is the sign Positive numbers Negative numbers 0000=+0 1000=-0 0001=+1 1001=-1 0010=+2 1010=-2 0011=+3 1011=-3 0100=+4 1100=-4 0101=+5 1101=-5 0110=+6 1110=-6 0111=+7 1111=-7

Sign-magnitude(cont. ) For a number with n bits: ◦ the biggest number is ◦

Sign-magnitude(cont. ) For a number with n bits: ◦ the biggest number is ◦ the smallest one is We have two values for zero: +0 and -0 Adding two numbers is not easy! 0101+1011? ?

1’s Complement (r-1)’s complement of a (with n digits) is: Positive numbers Negative numbers

1’s Complement (r-1)’s complement of a (with n digits) is: Positive numbers Negative numbers 0000=+0 1111=-0 0001=+1 1110=-1 0010=+2 1101=-2 0011=+3 1100=-3 0100=+4 1011=-4 0101=+5 1010=-5 0110=+6 1001=-6 0111=+7 1000=-7

1’s Complement to Decimal 1 0 1 1 = +20 + 21 - 23

1’s Complement to Decimal 1 0 1 1 = +20 + 21 - 23 + 1= - 4

1’s Complement(cont. ) We have two values for zero: +0 and -0 Summation of

1’s Complement(cont. ) We have two values for zero: +0 and -0 Summation of two numbers: ◦ if a carrier value is generated, add it with the result Exe: What is the biggest and smallest numbers which can be represented by this method?

2’s Complement (r)’s complement of a (with n digits) is: Positive numbers Negative numbers

2’s Complement (r)’s complement of a (with n digits) is: Positive numbers Negative numbers 0000=0 1111=-1 0001=+1 1110=-2 0010=+2 1101=-3 0011=+3 1100=-4 0100=+4 1011=-5 0101=+5 1010=-6 0110=+6 1001=-7 0111=+7 1000=-8

2’s Complement to Decimal 1001011 = +20 + 21 + 23 – 26 =

2’s Complement to Decimal 1001011 = +20 + 21 + 23 – 26 = - 53

2’s Complement(cont. ) We have one value for zero

2’s Complement(cont. ) We have one value for zero

Multiplication (Example 1) Sign extension

Multiplication (Example 1) Sign extension

Multiplication(Example 2)

Multiplication(Example 2)

Exercise- Search about: How compute the 1’s complement multiplication/summation? How compute 1’s and 2’s

Exercise- Search about: How compute the 1’s complement multiplication/summation? How compute 1’s and 2’s complement division?