Transcript Introduction to Information Technology IT-101 Lecture #3

IT-101
Section 001
Introduction to Information
Technology
Lecture #3
Overview
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Chapter 3:
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Representing Information in Binary Form
(cont..)
Binary to decimal conversion
Decimal to binary conversion
Representing Information in
Binary Form
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BInary digiTal symbols (BITs) form a
universal language for any:
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Numbers
Text
Sound
Images
Video
Anything else you can imagine…
How is this possible????
How can numbers and text be
represented in binary code????
How Do We Normally Represent
Numbers?
Before we discuss binary code, let’s think about
the number system we use every day.
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We normally don’t use Binary Digits (Bits) (in which a single
placeholder can hold only 0 or 1) in everyday life.
We use Decimal Digits - a single placeholder can hold one of ten
numerical values between 0 and 9.
Digits are combined together into larger numbers.
For example: 8,234 is made up of 4 digits. The 4 holds the “1s
place,” the 3 holds the “10s place,” the 2 holds the “100s place” and
the 8 holds the “1000s place.”
The Decimal System
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Decimal digits are combined to create larger numbers
4,567 => (4 x 103) + (5 x 102) + (6 x 101) + (7 x 100)
10 raised to the power of …
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100 =1
101 =10
102 =10x10=100
103 =10x10x10=1,000
104 =10x10x10x10=10,000
and so on
We have ten fingers
and use ten digits!
Coincidence?
Also called Base-10 system
There are other ways of representing numbers other
than using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Comparing the Decimal Number
System to the Binary Number System
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While people routinely use decimal digits,
computers use binary digits.
The decimal system uses ten numbers (0,
1, 2, 3, 4, 5, 6, 7, 8, 9) to represent all
values. The binary system uses two
numbers (0 and 1) to represent all values.
In other words, computers use the “base-2”
system rather than the “base-10” system.
Counting in binary is simple (different, but
simple) because you use powers of two
instead of ten. Example follows.
Binary to Decimal Conversion
The same as calculating the value of a decimal system number
except use powers of two instead of powers of ten.
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The binary number 1101 can be converted to decimal as follows:
(1x23) + (1x22) + (0x21) + (1x20) = 8 + 4 + 0 + 1 = 13
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For understanding binary, it’s helpful to have a good command of
powers of 2:
20 = 1
21 = 2
22 = 2x2 = 4
23 = 2x2x2 = 8
24 = 2x2x2x2 = 16
25 = 2x2x2x2x2 = 32
26 = 2x2x2x2x2x2 = 64
27 = 2x2x2x2x2x2x2 = 128
28 = 2x2x2x2x2x2x2x2 = 256
29 = 2x2x2x2x2x2x2x2x2 = 512
210 = 2x2x2x2x2x2x2x2x2x2 = 1024
and so on...
Binary versus Decimal Numbers
Another Way to Think About It.
Decimal Number
9 5, 1 0 7
9 x 10,000 = 90,000
+ 5 x 1,000 = 5,000
+ 1 x 100 = 100
+ 0 x 10 = 0
+7x1=7
_______________
= 95,107 (10)
Binary Number
1 0 1 0 1
1 x 16 = 16
+0x8=0
+1x4=4
+0x2=0
+1x1=1
_______________
= 21 (10)
Another Example:
Converting Binary to Decimal
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A computer generates the following sequence of bits:
110101(2)
How do we convert 110101(2) into decimal?
(1x25) + (1x24) + (0x23) + (1x22) + (0x21) + (1x20)
=
32 + 16 + 0
+ 4
+ 0
+ 1
110101(2) = 53(10)
= 53(10)
Real World Example: The Internet
Address
Converting a 32-bit Internet address into dotted decimal format
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An Internet address, known as an IP address for “Internet Protocol”
is comprised of four binary octets, making it a 32-bit address.
IP addresses, difficult for humans to read in binary format, are often
converted to “dotted decimal format.”
To convert the 32-bit binary address to dotted decimal format,
divide the address into four 8-bit octets and then convert each octet
to a decimal number.
Each octet will have one of 256 values (0 through 255)
192.48.29.253
(IP address in dotted decimal form)
Real World Example: The Internet
Address
Convert the following 32-bit Internet address into dotted decimal format:
01011110000101001100001111011100
1) Divide the IP address into four octets
01011110
00010100
11000011
11011100
2) Convert each binary octet into a decimal number
01011110 = 64+16+8+4+2 = 94
00010100 = 16+4 = 20
11000011 = 128+64+2+1 = 195
11011100 = 128+64+16+8+4 = 220
3) Write out the decimal values separated by periods
94.20.195.220
Decimal to Binary
Conversion
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Sometimes it can be done intuitively.
For example:
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The decimal number 1 represented in 8-bit binary is:
 00000001.
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The decimal number 128 represented in 8-bit binary is:
 10000000.
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The decimal number 129 represented in 8-bit binary is:
 10000001.
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The decimal number 2 represented in 8-bit binary is:
 00000010.
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The decimal number 4 represented in 8-bit binary is:
 00000100.
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The decimal number 6 represented in 8-bit binary is:
 00000110.
But what are we really doing mathematically?
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Convert the Decimal Number 174 to a binary octet
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128s
place
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64s
place
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32s
place
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16s
place
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8s
place
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4s
place
2s
place
1s
place
Step 1: Compare 174 to 128. 174>128 so place a 1 in the 128s place and subtract 174-128 = 46
1
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128s
place
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64s
place
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32s
place
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16s
place
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8s
place
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4s
place
2s
place
1s
place
Step 2: Compare 46 to 64. 46<64 so place a 0 in the 64s place and continue with 46.
1
0
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128s
place
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64s
place
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32s
place
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16s
place
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8s
place
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4s
place
2s
place
1s
place
Reversing the Process:
Converting a Decimal Number to
Binary
1
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0
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128s
place
64s
place
32s
place
16s
place
8s
place
4s
place
2s
place
1s
place
Step 3: Compare 46 to 32. 46>32 so place a 1 in the 32s place and subtract 46-32 = 14
1 ____
0
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128s
place
64s
place
1
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32s
place
16s
place
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8s
place
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4s
place
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2s
place
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1s
place
Step 4: Compare 14 to 16. 14<16 so place a 0 in the 16s place and continue with 14.
1 ____
0
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128s
place
64s
place
1
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32s
place
0
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16s
place
8s
place
4s
place
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2s
place
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1s
place
Step 5: Compare 14 to 8. 14>8 so place a 1 in the 8s place and subtract 14-8=6.
1 ____
0
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128s
place
64s
place
1
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32s
place
0
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1
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16s
place
8s
place
4s
place
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2s
place
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1s
place
Step 6: Compare 6 to 4. 6>4 so place a 1 in the 4s place and subtract 6-4=2.
1 ____
0
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128s
place
64s
place
1
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32s
place
0
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1
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1
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16s
place
8s
place
4s
place
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2s
place
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1s
place
Step 7: Compare 2 to 2. 2=2 so place a 1 in the 2s place and subtract 2-2=0.
There is no remainder left to convert, so also place a 0 in the 1s place.
1 ____
0
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128s
place
64s
place
1
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32s
place
0
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1
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1
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16s
place
8s
place
4s
place
1
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2s
place
0
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1s
place
The decimal number 174 has been converted to the binary number 10101110
Another Approach: Converting from
Decimal to Binary Using BCD
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We can also simply represent one number at a time.
How can we represent the ten decimal numbers (0-9) in binary code?
Numeral
BCD Representation
0
0000
1
0001
2
0010
3
0011
4
0100
5
0101
6
0110
7
0111
8
1000
9
1001
We can represent any integer by a string of binary digits.
For example, 749 can be represented in binary as: 011101001001
Binary Conventions
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Most Significant Bit (MSB) and Least Significant Bit (LSB)
 Decimal Example: 64
 6 is the Most Significant Digit
 4 is the Least Significant Digit
 Binary: 1000000
 1 is the MSB
 0 on the right is the LSB
Subscripts: Note that the subscript “2” makes it clear a number
is in binary format and the subscript “10” makes it clear a
number is in decimal format.
 This avoids confusion between a number like 110101
which can either be binary, written as 110101(2) or
decimal, written as 110,101(10)
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If there is a “1” in the LSB of a binary number, then its decimal
equivalent is an odd number
If there is a “0” in the LSB of a binary number, then its decimal
equivalent is an even number
In-Class Examples
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Convert 12(10) to binary
representation
Convert 1010101(2) to decimal
Convert 6234(10) to “binary
coded decimal” (BCD)
representation
How Many Bits Are Necessary to
Represent Something?
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1 bit can represent two (21) symbols
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either a 0 or a 1
2 bits can represent four (22) symbols
 00 or 01 or 10 or 11
3 bits can represent eight (23) symbols
 000 or 001 or 011 or 111 or 100 or 110 or 101 or 010
4 bits can represent sixteen (24) symbols
5 bits can represent 32 (25) symbols
6 bits can represent 64 (26) symbols
7 bits can represent 128 (27) symbols
8 bits (a byte) can represent 256 (28) symbols
n bits can represent (2n) symbols!
So…how many bits are necessary for all of us in class to have a unique
binary ID? Are two bits enough? Three? Four? Five? Six? Seven?
To think about..
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Can 64 bits represent twice as many symbols as 32 bits?
32 = 4,294,967,296 symbols
 32 bit = 2
64 = 1.8 x 1019 symbols
 64 bit = 2
128 = 3.4 x 1038 symbols
 128 bit = 2
Can 8 bits represent twice as many symbols as 4 bits?
8
 8 bit = 2 = 256 symbols
4
 4 bit = 2 = 16 symbols
Remember that we’re dealing with exponents!
 8 bit is twice as big as __________?
 7 bit!
7
 7 bits can represent 2 possible symbols or 2x2x2x2x2x2x2
= 128
8
 8 bits can represent 2 possible symbols or
2x2x2x2x2x2x2x2 = 256
Exercises
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Convert the following to binary form:
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810
4010
10110
Convert the following to decimal form:
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11002
001100102
011112
Comments for next class
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Do the assigned exercises
Topics to be covered next class:
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Bits vs. Bytes
Representing real numbers in binary form
Representing negative numbers in binary form
Octal numbering system
Hexadecimal numbering system
Conversion between different numbering systems
Representing alphanumeric characters in binary form