FLIPPED CLASS...Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal...
Transcript of FLIPPED CLASS...Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal...
FLIPPED CLASS
F#3- Number System
Department of CSE, Coimbatore 1CTPS 2018
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
Hexa-
decimal16
0, 1, … 9,
A, B, … FNo No
2Department of CSE, Coimbatore
The decimal system (base 10)
• The word decimal is derived from the Latin root decem (ten). In this
system the base b = 10 and we use ten symbols.
• The symbols in this system are often referred to as decimal digits or
just digits.
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
3Department of CSE, Coimbatore
Integers
• Integers are generally represented in the form as …
Eg.
• The place values for the integer number +24 is …
4Department of CSE, Coimbatore
Real Numbers
• Real Numbers are generally represented in the form as …
Eg.
• The place values for the real number +24.13 is …
5Department of CSE, Coimbatore
The binary system (base 2)
• The word binary is derived from the Latin root bini(or two by two).
• In this system the base b = 2 and we use only two symbols.
• The symbols in this system are often referred to as binary digits or bits(binary digit).
6
S = {0, 1}
Department of CSE, Coimbatore
Decimal to Binary Conversion
• The number (156)10 in decimal is the same as (10011100)2 in binary.
• Let us see how the conversion is done!
Department of CSE, Coimbatore 7
Decimal to Binary Conversion Cntd..
• If the decimal number to be converted is (156.6875)10 , let us see how the
fractional part can be converted to binary!
Department of CSE, Coimbatore 8
• If the decimal number to be converted is (156.6875)10 , let us see how the fractional part
can be converted to binary!
• The binary of (156)10 is (10011100)2 and the binary of (0.6875)10 is (0.1011)2
So the complete answer is (156.6875)10 = (10011100.1011)2
Department of CSE, Coimbatore 9
• The number (11001)2 in binary is the same as 25 in decimal and is represented as…
• The equivalent decimal number is N = 16 + 8 + 0 + 0 + 1 = 25.
Department of CSE, Coimbatore 10
• The number (101.11)2 in binary is equal to the number 5.75 in decimal .
Department of CSE, Coimbatore 11
✓Can you perform the indicated conversion?
Department of CSE, Coimbatore
Decimal to Binary
a) 1310 = ?
b) 2210 = ?
c) 4310 = ?
d) 15810 = ?
Binary to Decimal
a) 0110 2 = ?
b) 11010 2 = ?
c) 0110101 2 = ?
d) 11010011 2 = ?
12
Decimal → Binary ………..Answers
a) 1310 = ?
b) 2210 = ?
c) 4310 = ?
d) 15810 = ?
1 1 0 1 2
1 0 1 1 0 2
1 0 1 0 1 1 2
1 0 0 1 1 1 1 0 2
Department of CSE, Coimbatore 13
Binary → Decimal ……….Answers
a) 0110 2 = ?
b) 11010 2 = ?
c) 0110101 2 = ?
d) 11010011 2 = ?
6 10
26 10
53 10
211 10
Department of CSE, Coimbatore 14
The hexadecimal system (base 16)
• The word hexadecimal is derived from the Greek root hex (six) and the Latin root decem (ten).
• In this system the base b = 16 and we use sixteen symbols to represent a number.
• The set of symbols are:
Note that the symbols A, B, C, D, E, F are equivalent to 10, 11, 12, 13, 14, and 15 respectively.
The symbols in this system are often referred to as hexadecimal digits.
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,A, B, C, D, E, F}
Department of CSE, Coimbatore 15
• The number (2AE)16 in hexadecimal is equivalent to 686 in decimal is represented as…
• The equivalent decimal number is N = 512 + 160 + 14 = 686.Department of CSE, Coimbatore 16
Converting from base 10 to base 3
3 4 10 = __________3
33 32 3 1
3 3 4
1 1 - 13
3 - 23
1 - 0
1 0 2 1
Department of CSE, Coimbatore 17
Decimal to any base – Integral part
Department of CSE, Coimbatore 18
Decimal to any base – Fractional part
Department of CSE, Coimbatore 19
Binary-octal (base 8) conversion
• The binary bits are grouped into 3 and converted….
• Binary to octal and octal to binary conversion. Eg.
Department of CSE, Coimbatore 20
Binary-hexadecimal conversion
Department of CSE, Coimbatore
• The binary bits are grouped into 4 and converted…
• Binary to hexadecimal and hexadecimal to binary conversion. Eg.
21
Octal (base 8)-hexadecimal conversion
• Convert into binary first and then perform the required conversion.
• Octal to hexadecimal and hexadecimal to octal conversion. Eg.
Department of CSE, Coimbatore 22
Try it yourself✓Convert these binary numbers into octal numbers...
(a) 001011112 (8 bits) (b) 111101002 (8 bits)
✓Convert the following octal numbers into hexadecimal (16 bits)
(a) 658 (b) 1238
Can you find out .........
✓What is the biggest binary number one can write with n bits?
✓How many unique patterns does a sequence of 5 bits generate?
✓Write all the patterns of a sequence of 5 bits.Department of CSE, Coimbatore 23
Try it yourself
Department of CSE, Coimbatore
✓Convert these binary numbers into octal numbers...
(a) 001011112 (8 bits) (b) 111101002 (8 bits)
✓Convert the following octal numbers into hexadecimal (16 bits)
(a) 658 (b) 1238
Can you find out .........
✓What is the biggest binary number one can write with n bits?
✓How many unique patterns does a sequence of 5 bits generate?
✓Write all the patterns of a sequence of 5 bits.
24
Binary → Octal ........ Answers
Refer to the binary-octal
conversion table
000 101 111
= 578
0 5 7
Refer to the binary-octal
conversion table
011 110 100
= 3648
3 6 4
Department of CSE, Coimbatore 25
Refer to the binary-octal conversion table
68 58
110 101
0000 0000 0011 01012
0 0 3 5
= 3516
Refer to the binary-octal conversion table
18 28 38
001 010 011
0000 0000 0101 00112
0 0 5 3
= 5316
Octal → Hexadecimal Answers
Department of CSE, Coimbatore 26
Conversion from Hexadecimal to Binary
Replace each hex digit by the 4 equivalent bits, for examples,
• A3C5H = 1010 0011 1100 0101B
• 102AH = 0001 0000 0010 1010B
27Department of CSE, Coimbatore
Conversion from Binary to Hexadecimal
Starting from the right-most bit (least-significant bit), replace each group of 4 bits by
the equivalent hex digit (pad the left-most bits with zero if necessary), for example,
• 1001001010B = 0010 0100 1010B = 24AH
• 10001011001011B = 0010 0010 1100 1011B = 22CBH
28Department of CSE, Coimbatore
Exercises (Number Systems Conversion)
Convert the following decimal numbers into binary and hexadecimal numbers:
• 108
• 4848
• 9000
29
Answers……………
• 108 = 1101100 = 6C
• 4848 = 1001011110000 = 12F0
• 9000 = 10001100101000 = 2328
Depart
ment
of
CSE,
Coimb
atore
30
Exercises (Number Systems Conversion)
Convert the following binary numbers into hexadecimal and decimal numbers:
• 1000011000
• 10000000
• 101010101010
31
Answers……………
• 1000011000 = 218H = 536D
• 10000000 = 80H = 128D
• 101010101010 = AAAH = 2730D
Depart
ment
of
CSE,
Coimb
atore
32
Exercises (Number Systems Conversion)
Convert the following hexadecimal numbers into binary and decimal numbers:
• ABCDE
• 1234
• 80F
33
Answers……………
• ABCDE = 10101011110011011110 = 703710D
• 1234 = 1001000110100 = 4660D
• 80F = 100000001111 = 2063D
34
Exercises (Number Systems Conversion)
Convert the following decimal numbers into binary equivalent:
• 19.25D
• 123.456D
35
Answers……………
• 19.25 = 10011.01
• 123.456 = 1111011.01110100…
Depart
ment
of
CSE,
Coimb
atore
36