Download Binary Numbers - Computer Science

CPSC 171 Introduction to
Computer Science
Binary
Announcements
Read Chapter 4
Lab 4 & 5 due tomorrow at beginning of
Lab
Homework 3 due this Friday at
beginning of Lecture
EXAM Friday October 2nd in class
Example Representation
Real World
To be, or not to be: that is the
question: Whether 'tis nobler in the
mind to suffer The slings and
arrows of outrageous fortune, Or to
take arms against a sea of troubles,
And by opposing end them? To die:
to sleep; No more; and by a sleep
to say we end The heart-ache and
the thousand natural shocks That
flesh is heir to, 'tis a consummation
Devoutly to be wish'd.
-- William Shakespeare - (from
Hamlet Act 3, Scene 1)
Computer World
10101000110111110000001100010101
01000110110011001100110001011101
11011111100010101011010100010110
11101100010101011000101101110100
10001000101010101110101010110110
10010111101010001101111100000011
00010101010001101100110011001100
01011101110111111000101010110101
00010110111011000101010110001011
01110100100010001010101011101010
10110110100101111010100011011111
00000011000101010100011011001100
11001100010111011101111110001010
10110101000101101110110001010101
10001011011101001000100010101010
111010101011011010010111…
Internal and External
Representation of Data
Real World
Integers: 34
Signed Integers:
-156
Decimal Numbers:
-23.431
Text: Hello
Music: Hey Jude
Pictures:
Computer World
Zeros and Ones:
110101
Integer Representation
We use a base 10 number system (Decimal)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
2,359
Computers use a base 2 number system (Binary)
0, 1
110101
Conversion from Binary to Decimal
110101
1x25+1x24+0x23+1x22+0x21+1x20=53
You Try it:
What are the following binary numbers in decimal?
11011
101100
110111
Conversion from Decimal to Binary
Perform repeated divisions by 2
Keep track of the remainders
19 / 2
quotient = 9
remainder = 1
9/2
quotient = 4
remainder = 1
4/2
quotient = 2
remainder = 0
2/2
quotient = 1
remainder = 0
1/2
quotient = 0
remainder = 1
Stop when the quotient is 0
Decimal number 19 in binary is 10011
You Try it
Convert the following decimal numbers to binary
12
31
53
Addition on Binary
0
1
0
1
+
+
+
+
0
0
1
1
=
=
=
=
0
1
1
10 (carry the 1)
1101
11010
+1001
+10011
Fixed Sizes for Numbers
On computers a fixed number of digits are
typically used to store a number
(8, 16, 32, or 64 bits are common)
The decimal number 3 in binary is 11, but
using a fixed size of 8 bits it would be
represented as 00000011
Try adding the binary numbers using a fixed
size of 8 bits:
11011001
+10001011
Internal and External
Representation of Data
√
Real World
Integers: 34
Signed Integers:
-156
Decimal Numbers:
-23.431
Text: Hello
Music: Hey Jude
Pictures:
Signed Integers
-134
Sign/Magnitude Notation
110000110
magnitude
Sign
0 = positive
1 = negative
Not frequently used on computers
•2 numbers for zero
•Not easy to add/subtract
Signed Integers
-134
1.
2.
3.
Two’s Complement Notation (for fixed size window
16)
Calculate the magnitude in binary
0000000010000110
Flip the bits
1111111101111001
Add one
1111111101111010
You Try it
-129
-151
Internal and External
Representation of Data
√
√
Real World
Integers: 34
Signed Integers:
-156
Decimal Numbers:
-23.431
Text: Hello
Music: Hey Jude
Pictures:
Decimal Numbers
5.75
Write the 5 in binary and the 0.75 in binary
5
– 101
0.75
– 0.11
Normalize the number, keeping track of Mantissa and Exponent:
±MxB±E
M – Mantissa
B – Base (we use base 2)
E – Exponent
Used fixed size window (16 bits)
First bit is sign
Next 9 bits are Mantissa
Next bit is sign
Last 5 bits are Exponent
You Try It: -8.25
11.5
Text
Fixed Size Window represents a
character
ASCII (8 bits)
pg 141 in text
Unicode (16 bits) represents 65,636
characters
Binary Representation of
Sound and Images
Multimedia data is sampled to store a
digital form with or without detectable
differences
Representing sound data


Sound data must be digitized for storage in
a computer
Digitizing means periodic sampling of
amplitude values
Binary Representation of Sound
and Images (continued)


From samples, original sound can be
approximated
To improve the approximation
 Sample more frequently
 Use more bits for each sample value
Figure 4.5
Digitization of an Analog
Signal
(a) Sampling the Original
Signal
(b) Recreating the
Signal from the Sampled
Values
Binary Representation of Sound
and Images (continued)
Representing image data



Images are sampled by reading color and
intensity values at even intervals across the
image
Each sampled point is a pixel
Image quality depends on number of bits at
each pixel
Pictures
For each pixel keep
track of:


RGB values
0-255 (8-bit)
Why Binary Representation
Electronic devices are most reliable in a
bistable environment
Bistable environment

Distinguishing only two electronic states
 Current flowing or not
 Direction of flow
Computers are bistable: binary
representations
Binary Storage Devices
Magnetic core



Historic device for computer memory
Tiny magnetized rings; flow of current sets
the direction of magnetic field
Binary values 0 and 1 are represented
using the direction of the magnetic field
Figure 4.9
Using Magnetic Cores to Represent Binary Values
Binary Storage Devices
(continued)
Transistors

Solid-state switches; either permit or block
current flow

A control input causes state change

Constructed from semiconductors
Figure 4.11
Simplified Model of a Transistor