Transcript Hexidecimal Number System

Announcement!!!
• First exam next Thursday (I’m trying to
give you a first exam before the drop date)
I’ll post a sample exam over the weekend
and will try to go over it on Tuesday….
It will cover everything we’ve gone over so
far….some intro computers, data types;
I/O; if then else; Boolean operators;
number systems
A pause……number
systems..because I have to…
What you need to know …
1. binary number system
2. converting from decimal to binary and
binary to decimal
3. Hexadecimal system -- conversion
Why binary
Advantages
• The advantages of binary:
– Simple; easy to build.
– Unambiguous signals (hence noise immunity).
– Flawless copies can be made.
– Anything that can be represented with some
sort of pattern can be represented with
patterns of bits.
More advantages
• Since data of all kinds is stored in
computer memory (main and secondary)
using the same electronic methods, this
means that endless perfect copies can be
made of any type of data or program.
In order to understand the binary numbering
system lets first look at our decimal system.
•
The decimal numbering system consists of the numbers 0
through 9.
•
After nine we place a 1 in the tens column and start
again with 0. Which gives us 10.
•
The decimal system is also known as base 10 because it
is based on the 10 numbers 0 – 9.
0
1
2
3
4
5
6
7
8
9
10
9
1 0
Binary Numbers have only two digits 0 or 1
Binary is known as Base 2
Decimal
0
1
2
3
4
5
6
7
Binary
0
1
10
11
100
101
110
111
Converting binary – decimal
As you can see it would take a lot of time to create charts to represent
Binary numbers.
An easier way is to use the powers of 2
20 = 1
24 = 16
21 = 2
25 = 32
22 = 4
26 = 64
23 = 8
27 =128
Lets place the above calculations into a chart that will make it easy to
convert a binary number to a decimal number.
27
128
26
64
25
32
24
16
23
8
22
4
21
2
20
1
Converting Decimal to Binary
• Multiply each digit weight by the base
power (i.e. 2) at that unit position and add
up all the products
Remember how we “really” convert
decimal to decimal
125 10 =>
5 x 100 = 5
2 x 101 = 20
1 x 102 = 100
-------Base
125
Use the chart to convert the binary number to decimal.
27
128
26
64
25
32
0
0
1
24
16
0
1
x
1
=
1
2
x
0
=
0
4
x
0
=
0
8
x
1
=
8
16
x
0
=
0
32
x
1
= 32
64
x
0
=
0
128
x
0
=
0
Total
= 41
23
8
22
4
21
2
20
1
1
0
0
1
Note: The bit to
the far right is the
Least Significant
Bit (LSB) and will
determine if the
number is even or
odd.
Use the chart to convert the binary number to decimal.
27
128
0
26
64
1
25
32
24
16
23
8
22
4
21
2
20
1
1
1
1
1
1
1
1
x
1
=
1
2
x
1
=
2
4
x
1
=
4
8
x
1
=
8
16
x
1
= 16
32
x
1
= 32
64
x
1
= 64
128
x
0
=
Total
0
= 127
Note: if consecutive
bits from the right
are all 1’s
Then the answer is
the next power of 2
minus 1
In this case
128 – 1 = 127
Take a piece of paper and convert the binary number to decimal.
27
128
26
64
1
0
25
32
24
16
23
8
22
4
21
2
20
1
1
1
1
0
1
0
1
x
0
=
0
2
x
1
=
2
4
x
0
=
0
8
x
1
=
8
16
x
1
= 16
32
x
1
= 32
64
x
0
=
128
x
1
= 128
Total
0
= 186
Quick exercises
• Convert 1101100 from binary to decimal.
• Convert 101100 from binary to decimal.
• Convert 1110001 from binary to decimal.
The largest number that can be represented using an 8 bit binary
number is 255.
28
27
256 128
1
26
64
1
25
32
1
24
16
1
1
23
8
1
22
4
1
21
2
20
1
1
Remember the rule – if all the digits are 1 then the number is
the next power of 2 minus 1
256 – 1 = 255
So….we need something else
2 digits
Base 10
0
1
2
3
4
5
6
7
8
9
Add Placeholder
10
Base 2
0
1
10
Add Placeholder
Note: Base 16 is also called “Hexadecimal” or “Hex”.
16 digits
10 digits
Which Digits Are Available in which
Bases
Base 16
0
1
Base 16
2
Cheat Sheet
3
A16 = 1010
4
B16 = 1110
5
C16 = 1210
6
D16 = 1310
7
E16 = 1410
8
F16 = 1510
9
A
B
C
D
E
17
F
Add Placeholder
10
Hexadecimal Numbers - Example
Note:
162 = 256
256’s place
16’s place
1’s place
162 place
161 place
160 place
3AB16
This subscript denotes that this number is in Base 16 or “Hexadecimal” or “Hex”.
Hexadecimal Numbers - Example
256’s place
16’s place
1’s place
3AB16
So this number represents
• 3 two-hundred fifty-sixes
• 10 sixteens
• 11 ones
Mathematically, this is
(3 x 256) + (10 x 16) + (11 x 1)
= 768 + 160 + 11 = 93910
Base 16
Cheat Sheet
A16 = 1010
B16 = 1110
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
Converting Hex to Decimal –
Again we use the technique:
• Multiply each bit by 16 n, where n is the
weight (or power) of the bit
• The weight is the position of the bit,
starting from 0 on the right
• Add the results
Example
ABC16
=>
C x 160 = 12 x 1 =
12
B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
------2748
Exercises
• Convert 3F3 to decimal
• Convert AA1 to decimal
• Convert 11A to decimal
Why Hexadecimal Is Important
What is the largest
number you can
represent using four
binary digits?
_ _ _ _
1
1
1
1
23
22
21
20
=
=
=
=
8
4
2
1
What is the largest
number you can
represent using a
single hexadecimal
digit?
2
22
21
16
_
= 010
0
0
_ _ _ _
23
= 1510
F
20
0 + 0 + 0 + 0 = 010
Cheat Sheet
A16 = 1010
B16 = 1110
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
… the smallest
number?
8 + 4 + 2 + 1 = 1510
… the smallest
number?
0
0
0
_
Base 16
2
16
Note: You can represent the
same range of values with a
single hexadecimal digit that
you can represent using four
binary digits!
Why Hexadecimal Is Important
It can take a lot of
digits to represent
numbers in binary.
Example:
5179410 = 11001010010100102
Long strings of digits
can be difficult to
work with or look at.
Also, being only 1’s
and 0’s, it becomes
easy to insert or
delete a digit when
copying by hand.
Converting 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.
Converting Decimal to Binary
Example:
We want to convert 12510 to binary.
125
62
31
15
7
3
1
/
/
/
/
/
/
/
2
2
2
2
2
2
2
= 62 R 1
= 31 R 0
= 15 R 1
= 7 R 1
= 3 R 1
= 1 R 1
= 0 R 1
12510 = 11111012
Exercises
• Convert 33 decimal to Binary.
• Convert 51 decimal to Binary.
• Convert 19 decimal to Binary.
Converting Decimal to HEX
• Technique
– Divide by 16, keep track of the remainder
– First remainder is bit 0 (LSB, least significant
bit)
– Second remainder is bit 1
– Etc
Converting Decimal to Hex
Example:
We want to convert 12510 to hex.
125 / 16 = 7 R 13
7 / 16 = 0 R 7
12510 = 7D16
Base 16
Cheat Sheet
A16 = 1010
B16 = 1110
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
Exercises
• Convert 28 to Hex
• Convert 346 to Hex
• Convert 117 to Hex
Converting Hex to Binary
• Technique
– Convert each hexadecimal digit to a 4 bit
equivalent binary representation (chop it into
4 digit representation
Converting Binary Numbers to Hex
Recall the example binary number from the previous slide:
11001010010100102
1100 1010 0101 00102
C
A
5
2
16
First, split the
binary number
into groups of
four digits,
starting with the
least significant
digit.
Next, convert
each group of
four binary digits
to a single hex
digit.
Put the single
hex digits
together in the
order in which
they were found,
and you’re done!
Base 16
Cheat Sheet
A16 = 1010
B16 = 1110
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
Example
1
0
A
F
0001 0000 1010 1111
10AF16 = 0001 0000 1010 1111
Exercise
• Convert 1011100 to hex
• Convert AA1 (hex) to binary
Summary
• All programs and data are stored in binary as it maps
directly to on/off signals
• Hexadecimal is base 16, every four binary digits can be
represented by one Hex digit (Shorthand for computers)
• To convert any number base to decimal
– Multiply each digit weight by the base power at that
unit position and add up all the products
• To convert decimal to any base(2,16..)
– Keep Dividing the decimal number by the base until
you reach zero, keeping the remainders each time.
Read from the bottom up.
Problems:
• Convert 101 to binary
• Convert 1234 16 to decimal
• Convert 0000 1010 1011 1100 1101 to
hexadecimal
Real-world problems
• Convert the following MAC address to
decimal, keeping a colon between each
byte.
fe:fd:00:00:5c:a4
• Convert the following IP address to binary
and hexadecimal, keeping a dot between
each byte (remember a byte is 8 bits)
131.247.168.48
More real-world problems
• Convert the following subnet mask to
binary, keeping a dot between each byte.
255.255.128.0