Transcript Number Systems

Abstraction – Number Systems
and Data Representation
Memory
• Computer memory is used
to store data
• The smallest unit of
memory is a bit (Binary
digIT)
• A bit can be off (no voltage)
or on (has voltage) which
we interpret to be 0 or 1
• Memory is organized into 8
bit contiguous groups called
bytes. A megabyte is
1,048,576 bytes (over 1
million bytes). A gigabyte is
over 1 billion bytes.
How does Memory Represent Values?
• The different patterns of the on and off bits in a byte
determine the value stored
• Numbers are stored using binary numbers
– 101 is 1 * 20 + 0 * 21 + 1 * 22 = 5
• Characters are internally represented as numbers
– Different numbers represent different characters
– There are several systems for assigning numbers to
characters:
• ASCII, EBCDIC, and Unicode
Encodings Make Computers Powerful
• Voltages are interpreted as numbers
• Numbers can be interpreted as characters
• Characters can be interpreted to be part of a
link to Sun’s Java Site
<a href=http://java.sun.com>Sun’s Java Site </a>
a
0100 0001
off on off off off off off on
Notepad Exercise
• Open notepad and type a
sentence in it
• Save the file
• Check the size in bytes by
leaving the cursor over the
file name
– Or right click and check
properties
• Now count the number of
letters and spaces
Binary Numbers
• A bit is a binary digit with a value of 0 or 1
– A group of 8 bits is a byte
• Computer memory is allocated in bytes
• Numbers are stored using the binary number
system
– With digits of 0 or 1 and powers of 2
• Other number systems
– Decimal- digits of 0 to 9 and powers of 10
– Octal - digits of 0 to 7 and powers of 8
– Hexadecimal – digits of 0 to 9 and A, B, C, D, E,
F and powers of 16.
05-DrawingInJava
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Converting from Binary to Decimal
• Multiply the digit value
times the place value
and add up the
results to convert
from binary to decimal
• The place values start
with 20 on the right
(which is 1) and
increase to the left
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Converting from Decimal to Binary
• Subtraction Method
Keep subtracting out
largest power of two
until nothing remains
74
-64 1 ´ 2 6
10
-8 1 ´ 2 3
2
-2 1 ´ 21
0
result :1001010
full byte result : 01001010
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Converting from Decimal to Binary
• Division Method
0
2)1
2) 2
2) 4
2) 9
2)18
2) 37
2) 74
R1
R0
R0
R1
R0
R1
R0
result :1001010
full byte result : 01001010
Read result from top to
bottom.
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Binary Addition
• To add two decimal
numbers you add the digits
and if the total is greater
than ten you carry the one
into the next column
• To add two binary numbers
– 0+0=0
– 0 + 1 and 1 + 0 = 1
– 1 + 1 = 0 with a carry of 1 into
the next column to the left
00
01
---01
10
01
--11
00111001010
01010101101
------------------10001110111
111
001
-----1000
2’s Compliment Notation
• Computers actually only know how to add
– So, how do they handle subtraction?
• Computers subtract by adding a negative
number
• How do you represent a negative number in
memory?
– Positive numbers in 2’s compliment are just the
same as a binary number
– For negative numbers reverse 0s and 1s and then
add 1
2’s Compliment Example
• To subtract 3 from 7
• First represent both as a binary number
– 7 is 0000 0111
– 3 is 0000 0011
• Reverse the 0s and 1s and then add 1 to get -3
– 0000 0011 reversed is 1111 1100
– add 1
0000 0001
– The result is
1111 1101
Add the Negative Number
• To subtract 3 from 7
• Add -3 to 7
– 7 is
– -3 is
– The result is
0000 0111
1111 1101
1 0000 0100
• Throw away the leftmost 1
• The answer is 0000 0100 which is 4
Patterns Exercise
• How many different patterns of on and off bits
are there in 3 bits? How many in 4 bits? How
many in 8 bits?
• 000 is one pattern
• 001 is another pattern
• 010 is another pattern
Does the number of patterns matter?
• Some garage door
openers in the 70s used
8 bits to set the code to
use to open the door
– Giving 256 different
patterns
– Which is enough that
you won’t open your
neighbors door
– But small enough that
someone could try each
one
Remote Entry Systems
• With 8 bits for a code you
have a 1/256 chance of a
random code working
– You don’t want someone
opening your car in a place
with lots of cars (like a mall)
• There are also radio
scanners that can capture
your code
– So you want the code to
change each time
• Modern remote entry
systems use a 40 bit rolling
code