QUIZ: Generations of computer technology · 7 3 Integers A natural number, a negative number, zero...
Transcript of QUIZ: Generations of computer technology · 7 3 Integers A natural number, a negative number, zero...
QUIZ: Generations of computer technology
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QUIZ: Generations of computer technology
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Steampunk !
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Steampunk – Wikipedia article 4
The Telectroscope, 1878-2008
Chapter 2
Binary Values and Number Systems
Numbers
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Natural numbers, a.k.a. positive integers Zero and any number obtained by repeatedly
adding one to it.
Examples: 100, 0, 45645, 32
Negative numbers A value less than 0, with a – sign
Examples: -24, -1, -45645, -32
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Integers A natural number, a negative number, zero
Examples: 249, 0, - 45645, - 32
Rational numbers An integer or the quotient of two integers
Examples: -249, -1, 0, 3/7, -2/5
Real numbers In general cannot be represented as the quotient
of any two integers. They have an infinite # of
fractional digits.
Example: Pi = 3.14159265…
2.2 Positional notation
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How many ones (units) are there in 642?
600 + 40 + 2
6 x 100 + 4 x 10 + 2 x 1
6 x 102 + 4 x 101 + 2 x 100
10 is called the base
We also write 64210
This is called the
polynomial expansion of the
number
QUIZ
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Write the polynomial expansion of 6429
(642 in base nine), and convert it to
decimal.
Use the previous expansion as example:
600 + 40 + 2
6 x 100 + 4 x 10 + 2 x 1
6 x 102 + 4 x 101 + 2 x 100
QUIZ
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How many ones (units) are there
in 6429 (642 in base nine)?
6 x 92 + 4 x 91 + 2 x 90 = 52410
6429 = 52410
Positional Notation
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The base of a number determines how many
digits are used and the value of each digit’s
position.
To be specific:
• In base R, there are R digits, from 0 to R-1
• The positions have for values the powers of
R, from right to left: R0, R1, R2, …
Positional Notation
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dn * Rn-1 + dn-1 * R
n-2 + ... + d2 * R + d1
Formula:
R is the base
of the number
n is the number of
digits in the number
d is the digit in the
ith position
in the number
Positional Notation reloaded
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dn * Rn-1 + dn-1 * R
n-2 + ... + d2 * R + d1
… but, in CS, the digits are numbered from zero, to
match the power of the base:
dn-1 * Rn-1 + dn-2 * R
n-2 + ... + d1 * R1 + d0 * R
0
The text shows the digits numbered like this:
QUIZ
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What is 642 in base 13?
QUIZ
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What is 642 in base 13?
64213 = 106810
+ 6 x 132 = 6 x 169 = 1014
+ 4 x 131 = 4 x 13 = 52
+ 2 x 13º = 2 x 1 = 2
= 1068 in base 10
Nota bene!
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In a given base R, the digits range
from 0 up to R – 1
R itself cannot be a digit in base R
Trick problem:
Convert the number 473 from base 6 to base 10
Binary
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Decimal is base 10 and
has 10 digits:
0,1,2,3,4,5,6,7,8,9
Binary is base 2 and has 2
digits:
0,1
QUIZ: Converting Binary to Decimal
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What is the decimal equivalent of the binary
number 110 1110?
110 11102 = ???10
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What is the decimal equivalent of the binary
number 1101110?
1 x 26 = 1 x 64 = 64
+ 1 x 25 = 1 x 32 = 32
+ 0 x 24 = 0 x 16 = 0
+ 1 x 23 = 1 x 8 = 8
+ 1 x 22 = 1 x 4 = 4
+ 1 x 21 = 1 x 2 = 2
+ 0 x 2º = 0 x 1 = 0
= 110 in base 10
= 11010
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Base 8 = octal system
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What is the decimal equivalent of the octal
number 642?
6428 = ???10
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Converting Octal to Decimal
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What is the decimal equivalent of the octal
number 642?
6 x 82 = 6 x 64 = 384
+ 4 x 81 = 4 x 8 = 32
+ 2 x 8º = 2 x 1 = 2
Add the above = 418 in base 10
= 41810
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Bases Higher than 10
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How are digits in bases higher than 10
represented?
Base 16 (hexadecimal, a.k.a. hex) has 16
digits:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, F
Converting Hexadecimal to Decimal
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What is the decimal equivalent of the
hexadecimal number DEF?
D x 162 = 13 x 256 = 3328
+ E x 161 = 14 x 16 = 224
+ F x 16º = 15 x 1 = 15
= 3567 in base 10
QUIZ: 2AF16 = ???10
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Are there any non-positional number systems? Hint: Why did the Roman civilization have no contributions to mathematics?
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QUIZ: Convert to decimal
0001 00112 =
C716 =
426 =
718 = 26
Today we’ve covered pp.33-39 of text
(stopped before Arithmetic in Other Bases)
Solve in notebook as individual work for
next class:
1, 2, 3, 4, 5, 20, 21
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QUIZ: Convert to decimal
1101 00112 =
AB716 =
5137 =
6928 = 28
The inverse problem: Converting Base 10 to Other Bases
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While (the quotient is not zero) Divide the decimal number by R Make the remainder the next digit to the left in the
answer Replace the original decimal number with the quotient
Algorithm for converting a number in base
10 to any other base R:
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Known as repeated division (by the base)
Converting Decimal to Binary
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Example: Convert 17910 to binary
179 2 = 89 rem. 1
2 = 44 rem. 1
2 = 22 rem. 0
2 = 11 rem. 0
2 = 5 rem. 1
2 = 2 rem. 1
2 = 1 rem. 0
17910 = 101100112 2 = 0 rem. 1
Notes: The first bit obtained is the rightmost (a.k.a. LSB)
The algorithm stops when the quotient (not the remainder!)
becomes zero
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LSB MSB
Repeated division QUIZ
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Convert 4210 to binary
42 2 = rem.
4210 = 2
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The repeated division algorithm can be used to convert from any
base into any other base (but normally we use it only for 10 → 2)
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For next time: Read text example on p.43: Converting Decimal to Hex using repeated division
Addition in Binary
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Remember that there are only 2 digits in binary,
0 and 1
1 + 1 is 0 with a carry
Carry Values 0 1 1 1 1 1
1 0 1 0 1 1 1
+1 0 0 1 0 1 1
1 0 1 0 0 0 1 0
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Addition QUIZ
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Carry values
go here
1 0 1 0 1 1 0
+1 0 0 0 0 1 1
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Check in base ten!
Compute in binary
Putting it all together!
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1
1001
+ 101
1110
Python
(or another
high-level language) Convert decimal
to binary
Convert binary to decimal
Computer hardware
Direct conversions between bases that are powers of 2
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binary
hexadecimal
octal
Converting Binary to Octal
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• Mark groups of three (from right)
• Convert each group
10101011 10 101 011
2 5 3
10101011 is 253 in base 8
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Converting Binary to Hexadecimal
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• Mark groups of four (from right)
• Convert each group
10101011 1010 1011
A B
10101011 is AB in base 16
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Extra-credit QUIZ:
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Converting Octal to Hexadecimal
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End-of-chapter ex. 25:
Explain how base 8 and base 16 are related
10 101 011 1010 1011
2 5 3 A B
253 in base 8 = AB in base 16
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End-of-chapter ex.37
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Perform the following octal additions:
a. 770 + 665
b. 101 + 707
SKIP:
Binary SUBTRACTION (with “borrow” bits)
• p.40 of text
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Today we’ve covered pp.36-43 of text
(stopped before Binary Values and
Computers)
Solve in notebook as individual work for
next class: 6 through 11
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Addition QUIZ
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Carry values
go here
1 1 1 0 1 1 0
+1 0 0 0 1 1 1
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Check in base ten!
End-of-chapter ex.37
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Perform the following octal additions:
c. 202 + 667
Counting
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Basic skill: counting (in any base!)
• 0, 1, 2, 3, 4, 5, …
• 0, 1, 10, 11, 100, 101, …
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Hex
Conclusion:
• In order to represent any octal digit, we need at most ______ bits
• In order to represent any hex digit, we need at most ______ bits
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Binary and Computers
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Word = group of bits that the computer processes
at a time
The number of bits in a word determines the
word length of the computer. It is usually a
multiple of 8.
1 Byte = 8 bits
• 8, 16, 32, 64-bit computers
• 128? 256?
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• Motivated by the 6-bit codes for printable graphic
patterns created by the U.S. Army and Navy
• 6, 18, 24, 36, 48-bit words
• Some history: – 18-Bit Computers from DEC
– 36-bit Wikipedia
• Edged out of the market by the need for floating-point numbers – IBM System/360 (1965)
– 8-bit microprocessors (1970s)
6-bit computers: an “Evolutionary dead-end”?
Not in text
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Unisys is still successful with their 36- and 48-bit machines :
• Clearpath Dorado line of 36-bit CISC high-end servers
• Clearpath Libra line of 48-bit mainframes
… although they are being transitioned to Intel Xeon chips (64-bit): see article
6-bit computers: an “Evolutionary dead-end”?
Not in text
Grace Murray Hopper
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• Ph.D. in mathematics
• Wrote “A-0”, the world’s first
compiler, in 1952!
• Co-invented COBOL
• Rear-admiral of the US Navy
• “Nanosecond” wires
Grace also liked telling this story
54 Harvard University Mark II Aiken Relay Calculator
Ethical Issues →Tenth Strand
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What do the following acronyms stand for:
• ACM ?
• IEEE ? What is the tenth strand? Why the “tenth”?
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Why the “tenth”? A: In the 1989 ACM report (“Computing as a Discipline” p.12), the following 9 areas (strands) of CS were defined:
• Algorithms and data structures • Programming languages • Computer Architecture • Numerical and symbolic computations • Operating systems (OS) • Software engineering • Databases • Artificial intelligence (AI) • Human-computer interaction
Ethical Issues →Tenth Strand
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The latest official release of the IEEE/ACM CS Curriculum was in 2001:
• 3 levels of organization: areas → knowledge units → topics
• There are now 14 areas, and the “tenth strand” is listed in position 12
• What does “SP” stand for?
• How many knowledge units does the “SP” area have?
• Name two of these units!
Chapter Review questions
• Describe positional notation (polynomial in the base)
• Convert numbers in other bases to base 10
• Convert base-10 numbers to numbers in other bases
• Add and subtract in binary
• Convert between bases 2, 8, and 16 using groups of digits
• Count in binary
• Explain the importance to computing of bases that are powers of 2
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Chapter Review questions
• IEEE/ACM CS Curriculum and the “Tenth Strand”
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Homework for Ch.2
End-of-ch. 23, 24, 25, 28, 29, 30, 33, 38
Due next Wednesday, Sep.9
The latest homework assigned is always available on the course webpage
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FYI: Subtraction in Binary
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Remember borrowing?
1 2
0 2 0 2
1 0 1 0 1 1 1
- 1 1 1 0 1 1
0 0 1 1 1 0 0
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Borrow values
Check in base ten!
Subtraction QUIZ
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1 0 1 0 0 0 0
- 1 0 0 1 0 1
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Borrow values
go here
Check in base ten!
Subtraction QUIZ
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1 1 1 0 1 0 0
- 1 1 0 1 1 1
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Check in base ten!
Borrow values
go here
Another subtraction QUIZ
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1 0 1 0 0 0 1
- 1 0 0 1 1 1
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Borrow values
go here
Check in base ten!