L03 – Number Systems
8/31/2016
SESSION 3 – NUMBER
SYSTEMS
Reading: Chapter 2
(read 2.1-2.3)
© Robert F. Kelly, 2012-2016
2
Reading
• On-line tutorial (Modules I through III.b)
http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/index.html
• Exponents
http://www.mathsisfun.com/exponent.html
• Video – Binary Tutorial
http://www.youtube.com/watch?v=0qjEkh3P9RE
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L03 – Number Systems
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© Robert F. Kelly, 2012-2016
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Objectives
• Review your knowledge of exponents
• Understand the fundamentals of number systems
• Learn to convert between various radix systems
• Learn to add in binary
• Become familiar with typical computer measures
of space, time, and distance
© Robert F. Kelly, 2012-2016
4
Exponent Basics
• nm = n * n * n … (n multiplied by itself m times)
• Example: 103 = 1000 or (10 * 10 * 10)
• n1 = n
• n0 = 1
• n-1 = 1/n
• ni * nj = n(i+j)
• ni / nj = n(i-j)
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© Robert F. Kelly, 2012-2016
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Transformations
• Many times with computers we need to represent
numbers and characters in a very specific way
• To solve a problem, we often
• Transform the human readable number or character to
a computer representation
• Solve the problem using the computer representation
• Transform the computer representation of the answer
back to a human readable representation
© Robert F. Kelly, 2012-2016
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Transformation Example
Integer Arithmetic
• What is 80 * 12?
Exponent Representation
• What is (5*24) * (3*22)?
• = (5*24 * 3*22)
Transform the problem to
an exponent problem
• = 15 * 24 * 22
• = 15 * (24+2)
• = 15 * 26
• =15*64 = 960
Transform the answer to all
integer
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L03 – Number Systems
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© Robert F. Kelly, 2012-2016
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Which is Easier?
• 65,536 / 32 = 2,048
or
• 216 / 25 = 2(16-5) = 211 = 2,048
During the course you
will begin to memorize
some powers of 2
For example, 210 is 1024
Computers usually have only one
way to do a calculation, so they must
translate a number into their
representation
© Robert F. Kelly, 2012-2016
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Small Numbers
• We usually use negative exponents to represent
very small numbers
• .00001 = .1 * .1 * .1 * .1 * .1 = 10-5
= 10-1 * 10-1 * 10-1 * 10-1 * 10-1
= 10 (-1-1-1-1-1) = 10-5
We have lots of very
small numbers when
using computers (e.g.,
clock time of a computer
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L03 – Number Systems
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© Robert F. Kelly, 2012-2016
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Measures of Computer Memory
• A bit is the most basic unit of information in a
computer
• It is a state of “on” or “off” A byte is a group of eight bits.
• A byte (8 bits) is the smallest possible addressable unit
of computer storage
• The term, “addressable,” means that a particular byte
can be retrieved according to its location in memory
Think of “addressable” in terms of a
delivery person having an address
of your home, but not each room in
your home
© Robert F. Kelly, 2012-2016
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Introduction
• A word is a contiguous group of bytes
• Words can be any number of bits or bytes
• Word sizes of 16, 32, or 64 bits are most common
• In a word-addressable system, a word is the smallest
addressable unit of storage
• A group of four bits is called a nibble
Most systems are
byte addressable, not
word addressable
Word implications are
primarily with alignment
and path width
(not very significant)
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L03 – Number Systems
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© Robert F. Kelly, 2012-2016
Positional Numbering Systems
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Denotes
the base
• Our decimal system is the base-10 system. It
uses powers of 10 for each position in a number
• 40510 = 4*102 + 0*101 + 5*100 or 400 + 0 + 5
• Other number systems use a base other than 10
• 4058 = 4*82 + 0*81 + 5*80 = ?
• Bytes store numbers using the position of each
bit to represent a power of 2
• Example: 01110111
Base notation is sometimes
omitted when it is obvious
• The binary system is also called the base-2 system.
• You can (easily?) convert a number in one
system to a number in a different system
© Robert F. Kelly, 2012-2016
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Positional Numbering Examples
• The decimal number 947 in powers of 10 is:
9 × 10 2 + 4 × 10 1 + 7 × 10 0
100=1, 101=10, 102=100, etc.
This may seem obvious,
but it helps to understand
other number systems
• The decimal number 5836.47 in powers of 10 is:
5 × 10 3 + 8 × 10 2 + 3 × 10 1 + 6 × 10 0
+ 4 × 10 -1 + 7 × 10 -2
Note the representation
of decimal numbers
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L03 – Number Systems
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© Robert F. Kelly, 2012-2016
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Positional Numbering Systems
• The binary number 110012 in powers of 2 is:
1 × 24+ 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20
= 16
+
8
+
0
+
0
+
1
= 25 (base-10)
Example of conversion
of base 2 to base 10
• When the radix of a number is something other
than 10, the base is denoted by a subscript.
• Sometimes, the subscript 10 is added for clarification
110012 = 2510
© Robert F. Kelly, 2012-2016
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Converting Between Bases
• Binary numbers are the basis for all data
representation in digital computer systems
• You should become proficient with the base-2
system
• Familiarity with base-2 enables you to understand
• computer components
• instruction set architectures
• Every integer value can be represented exactly
using any radix system
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L03 – Number Systems
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© Robert F. Kelly, 2012-2016
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Converting
• You will primarily convert base 2 numbers to base
10 – and the reverse
You will also need to be familiar
with base 16 (hexadecimal) as a
convenient way to represent base 2
© Robert F. Kelly, 2012-2016
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Converting Base 10 to Base 2
• Suppose we want to convert the decimal number 190 to base 2
• First look for the largest power of 2 that is less than or equal to the
•
•
•
•
•
•
•
target number (190) – in this case it is 128 (27)
Therefore, the 7th bit in the binary number is 1
Compute the remainder (190-128 = 62)
Repeat the process with the remainder, until the remainder is 0
190 = 27 + 62
The largest power of 2 less than 62 is 32 (25)
Therefore, the 5th bit in the binary number is 1
remainder
Partial result: 190=1*27+0*26+1*25+30
• Continue this process to determine that
• 190 = 1*27+0*26+1*25+1*24+1*23+1*22+1*21+0*20
• 190 = 101111102
• 190 = 128 + 0 + 32 + 16 + 8 + 4 + 2 + 0 (base 10)
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L03 – Number Systems
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© Robert F. Kelly, 2012-2016
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Converting Between Bases
• Fractional values can be approximated in all base
systems.
• Unlike integer values, fractions do not necessarily
have exact representations in all base systems
• The quantity
½ is exactly representable in the binary
and decimal systems, but is not in the ternary (base
3) numbering system.
© Robert F. Kelly, 2012-2016
Counting
Binary
0
1
10
11
100
101
110
111
1000
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Hexadecimal
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
10
11
12
13
14
15
16
17
18
19
1A
1B
1C
1D
1E
1F
20
21
22
23
...
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L03 – Number Systems
8/31/2016
© Robert F. Kelly, 2012-2016
Class Questions
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All questions refer to
non-negative numbers
• What is the smallest three digit
decimal number?
• What is the largest 3-digit decimal
number?
• What it the smallest 2-digit binary
number
• What is the largest 8 digit binary
number (in binary and in decimal)?
0 or 000
999
0
11111111
255
© Robert F. Kelly, 2012-2016
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Fractional Values
• Fractional decimal values have nonzero digits to the
right of the decimal point (e.g., .0123)
• Fractional values of other radix systems have nonzero
digits to the right of the radix point (e.g., .012)
• Numerals to the right of a radix point represent
negative powers of the radix:
Radix point is
the symbol that
separates the
integer part of a
number from the
fractional part
10-1 = 1/10 = .1
0.4710 = 4 × 10 -1 + 7 × 10 -2
0.112 = 1 × 2 -1 + 1 × 2 -2
= ½ + ¼
= 0.5 + 0.25 = 0.75
2-2 = 1/22 = ¼ = .2510
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L03 – Number Systems
8/31/2016
© Robert F. Kelly, 2012-2016
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Addition in Binary Number System
• Binary
•0+0=0
•0+1=1
•1+0=1
• 1 + 1 = 10 (or 0 with a carry of 1)
• Example:
0101
1000
1101
Check your work by converting
to decimal:
5+8=13 and 13 in binary is 1101
© Robert F. Kelly, 2012-2016
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Binary Addition With a Carry
• 1 + 1 = 0 with a carry of 1
• Example:
1
0101
1001
1110
It’s a good idea to check your
work by converting to decimal:
5+9=14 and 14 in binary is 1110
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L03 – Number Systems
8/31/2016
© Robert F. Kelly, 2012-2016
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Converting Between Bases
• It is difficult to read long strings of binary numbers
• Even a modestly-sized decimal number becomes a
very long binary number
• For example:
110101000110112 = 1359510
• For compactness and ease of reading, binary values
are usually expressed using the hexadecimal, or
base-16, numbering system
116 = 110
…
A16 = 1010
B16 = 1110
C16 = 1210
…
© Robert F. Kelly, 2012-2016
Hexadecimal Conversion
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116 = 110
…
• The hexadecimal system uses numerals
A16 = 1010
B16 = 1110
of
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
0-9 and A-F
• The decimal number 2610 is 1A16
• It is easy to convert between base 16
and base 2 because 16 = 24
• Thus, to convert from binary to hexadecimal, we
group the binary digits into clusters of four digits
000110102 = 1A16
1
A
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L03 – Number Systems
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© Robert F. Kelly, 2012-2016
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Hexadecimal Conversion
• Using groups of hextets, the binary number
110101000110112 (= 1359510) in hexadecimal is:
If the number of bits is not a multiple of
4, pad the left with zeros
© Robert F. Kelly, 2012-2016
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Measures of Capacity and Speed
• Kilo- (K) = 1 thousand = 103 and 210
• Mega- (M) = 1 million = 106 and 220
• Giga- (G) = 1 billion = 109 and 230
• Tera- (T) = 1 trillion = 1012 and 240
• Peta- (P) = 1 quadrillion = 1015 and 250
• Exa- (E) = 1 quintillion = 1018 and 260
• Zetta- (Z) = 1 sextillion = 1021 and 270
What are 103 and
210 in decimal
numbers?
• Yotta- (Y) = 1 septillion = 1024 and 280
Whether a metric refers to a power of ten or a power of
two typically depends upon what is being measured
(e.g., money - power of 10, bits – power of 2
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L03 – Number Systems
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© Robert F. Kelly, 2012-2016
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Examples of Download Speed
• Pandora on the Web streams
• 64Kbs-128Kbs – free listener
• 192Kbs - Pandora
• Netflix streams
• SD – 3 Mbs
• HD – 5 Mbs
• Ultra HD – 25 Mbs
More on screen resolution
when we get to displays
• Cablevision Ultra 50 = 50 Mbs (download)
• Try out your download and upload speed
• www.speedtest.net
© Robert F. Kelly, 2012-2016
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Units of Storage
• Bit – an electronic component that has 2 states
(usually represented as 0 and 1)
• Byte – an addressable 8 bit sequence
What does it mean to have “state”?
Do we care how it is implemented in the computer?
• Typical byte units
• 1KB = 210 = 1,024 Bytes
• 1MB = 220 = 1,048,576 Bytes
• 1GB = 230 = 1,073,741,824 Bytes
“B” denotes bytes
and “b” denotes bits
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L03 – Number Systems
8/31/2016
© Robert F. Kelly, 2012-2016
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Measures of Storage
• Main memory (RAM) is measured in GB
• Disk storage is measured in GB for small
systems, TB (240) for large systems.
• Examples
• Flash drive – 8-128GB are typical
• Data rate
• USB 2.0 – 60 MBs
• USB 3.0 – 625MBs or 5Gbs
© Robert F. Kelly, 2012-2016
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Fractional Measures
• Milli- (m) = 1 thousandth = 10-3
• Micro- (μ) = 1 millionth = 10-6
• Nano- (n) = 1 billionth = 10-9
• Pico- (p) = 1 trillionth = 10-12
These symbols
are important to
remember
• Femto- (f) = 1 quadrillionth = 10-15
• Atto- (a) = 1 quintillionth = 10-18
• Zepto- (z) = 1 sextillionth = 10-21
• Yocto- (y) = 1 septillionth = 10-24
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L03 – Number Systems
8/31/2016
© Robert F. Kelly, 2012-2016
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Typical Units
• Millisecond = 1 thousandth of a second
• Hard disk drive access times are often 10 to 20
milliseconds.
• Nanosecond = 1 billionth of a second
• Main memory access times are often 50 to 70
nanoseconds.
• Micron (micrometer) = 1 millionth of a meter
• Circuits on computer chips are measured in microns.
How many nanoseconds
in a millisecond?
10-9 x ? = 10-3
© Robert F. Kelly, 2012-2016
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Intuitive Measure of a Nanosecond
• Light travels 11.8 inches in a nanosecond
• The speed of an electron is close to, but less than
the speed of light, so an electron travels no more
than 11.8 inches in a second
• Clock speeds are measured in nanoseconds, so
this gives you a sense of the importance of
packaging (smaller is faster)
Speed of network travel is
hugely important to high
frequency traders (FT)
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L03 – Number Systems
8/31/2016
© Robert F. Kelly, 2012-2016
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More Measures
• Hertz = clock cycles per second (frequency)
• 1MHz = 1,000,000Hz
• Processor speeds are measured in MHz or GHz.
Computer operations are
controlled by a clock
(like playing a musical instrument)
© Robert F. Kelly, 2012-2016
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Cycle Time
• Note that cycle time is the reciprocal of clock
frequency
• A bus operating at 133MHz has a cycle time of
about 7.52 nanoseconds:
133,000,000 cycles/second corresponds to 7.52ns/cycle
or
133,000,000 cycles/second * 7.52 ns/cycle =
1,000,160,000 ns/second (a billion nanoseconds in a second)
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L03 – Number Systems
8/31/2016
© Robert F. Kelly, 2012-2016
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Have You Satisfied the Objectives?
• Review exponents
• Understand the fundamentals of number systems
• Learn to convert between various radix systems
• Learn to add in binary
• Become familiar with typical computer measures
of space, time, and distance
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