Transcript Hexadecimal

Lecture 2
Bits, Bytes & Number systems
Representation of Numbers
 Different ways to say “how many”…
 Human: decimal number system


Radix-10 or base-10
Base-10 means that a digit can have one of ten possible values
• 0 through 9.
 Computer: binary number system



Radix-2 or base-2
Why binary?
Each digit can have one of two values
• 0 or 1
Bits and Bytes
 A binary digit is a single numeral in a binary
number.
 Each 1 and 0 in the number below is a binary digit:
10010101
 The term “binary digit” is commonly called a “bit.”
 Eight bits grouped together is called a “byte.”
Relationship between Decimal & Binary
Background:
 Number systems are positional
 There are 10 symbols that represent decimal quantities
 Multi-digit numbers are interpreted as in the following example
 79310
= 7 x 100 + 9 x 10 + 3
= 7 x 102 + 9 x 101 + 3 x 100
Remember that the
position index starts
from 0.
 Each place value in a decimal number is a power of 10.
 We can get a general form of this


ABCbase
A x (base)2 + B x (base)1 + C x (base) 0
Indicate
positions
Relationship between Decimal & Binary
 Binary numbers are represented using the digits 0 and 1.
 Multi-digit numbers are interpreted as in the following example
 101112
= 1 x 2 4 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 2 0
= 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 1 x 1
 Each place value in a binary number is a power of 2.
Converting binary numbers to decimal
Step 1: Starting with the 1’s place, write the binary place value
over each digit in the binary number being converted.
Step 2: Add up all of the place values that have a “1” in them.
Interpret the binary number 101012 in decimal
 TRY: Interpret the binary number 010101102 in decimal
Converting decimal numbers to binary
 We have now learnt how to convert from binary to
decimal

Using positional representation
 But how about decimal to binary
 Repeated division method
• Simply keep dividing it by 2 and record the remainder
• Repeat above step as many times as necessary until you get a
quotient that can’ t be divided by 2
• Remainders give the binary digits, starting from the last
remainder
 Let’s look at some examples…
Converting decimal numbers to binary
 Let’s convert decimal 23 to binary.
Step 1: 23/2 = 11 remainder 1
Step 2: 11/2 = 5 remainder 1
Step 3: 5/2 = 2 remainder 1
Step 4: 2/2 = 1 remainder 0
 The last quotient “1” cannot be divided by 2 any more. So the
process ends. The final binary number is read from the very
end including the last quotient:
10111
 Try: Convert decimal 73, 96, 127, 128 to binary.
Hexadecimal
 Computers use binary number system because of the electric
voltage (high or low voltage)

Very difficult to express for large number representation
 Hexadecimal to rescue
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
Hexadecimal system is interface between human brain and computer
brain
4 bits from binary are read together and represented using a single
digit
• Such 4-bits are known as nibble
• This gives a total of 16 different options
 The hexadecimal number system is a Base-16 number system:


There are 16 symbols that represent quantities:
Represented by the symbols 0-9 and A-F where the letters represent
values: A=10, B=11, C=12, D=13, E=14, and F=15
Numbering systems
Decimal
Hexadecimal
Binary
Decimal
Hexadecimal
Binary
0
0
0000
8
8
1000
1
1
0001
9
9
1001
2
2
0010
10
A
1010
3
3
0011
11
B
1011
4
4
0100
12
C
1100
5
5
0101
13
D
1101
6
6
0110
14
E
1110
7
7
0111
15
F
1111
Hexadecimal
 Thus each byte is two hex digits (shorthand representation for
human)
 EX: Binary: 110010102

How to represent this in hex representation?
110010102
 Separate them into nibbles
1100
1010
C
A
Hexadecimal representation: CA16
 Try: convert 11110101101011002 to Hex representation
Converting Hex number to Binary
 Converting hexadecimal numbers to binary is just the
reverse operation of converting binary to hexadecimal.
 Just convert each hexadecimal digit to its four-bit binary
pattern. The resulting set of 1s and 0s is the binary
equivalent of the hexadecimal number.
 Convert A5B916 to Binary.
Conversion: Hex and decimal
 Hex to decimal
Exact similar to binary to decimal
 Use base 16
 Try CA16

 Decimal to Hex
 Exactly similar to decimal to binary
 Divide by 16
 Try 20210
Notes on Bases
 Subscript is mandatory at least for a while.
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We use all three number bases.
When a number is written, you should include the correct subscript.
 Pronunciation

Binary and hexadecimal numbers are spoken by naming the digits
followed by “binary” or “hexadecimal.”
Ranges of Number Systems
System
Lowest
Highest
Number of
values
4-bit binary
00002
010
016
11112
1510
F16
1610
0000 00002
010
0016
1111 11112
25510
FF16
25610
1111 1111 1111 11112
6553510
FFFF16
6553610
(1-digit hex)
8-bit binary
(2-digit hex)
16-bit binary 0000 0000 0000
00002
(4-digit hex) 010
000016
Ranges of Unsigned Number Systems
Electronic Prefixes
 There is a set of terms used in electronics to represent very
large values and very small values.
 Kilo, Mega, Giga, Tera – used for representing very large values

E.g., KiloByte, MegaByte etc.
 milli, micro, nano, pico – used for representing very small values

E.g., milliseconds, microseconds etc.
Reading assignment
 In Blackboard:
reading02_08_29_2012.pdf
 reading03_08_29_2012.pdf

Practice problems
 Let us do some quick example problems …