quiz: generations of computer technology · word = group of bits that the computer processes at a...
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QUIZ: Generations of computer technology
Hardware:
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QUIZ: Generations of computer technology
Software:
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6.2
Steampunk !
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Chapter 2
Binary Values and Number Systems
We are here!
Numbers
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Natural numbers, a.k.a. positive integersZero and any number obtained by repeatedly
adding one to it.
Examples: 100, 0, 45645, 32
Negative numbersA value less than 0, with a – sign
Examples: -24, -1, -45645, -32
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IntegersA natural number, a negative number,
zero
Examples: 249, 0, - 45645, - 32
Rational numbersAn integer or the quotient of two integers
Examples: -249, -1, 0, 3/7, -2/5
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Real numbersIn general cannot be represented as the
quotient of any two integers. They have an
infinite # of fractional digits, so they cannot
be represented exactly in a computer.
Examples: Pi = 3.1415926535…
e = 2.718281828…
2.2 Positional notation
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4
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
104
How many units are there in 6429 (642 in base nine)?
Hint: Convert it to decimal using the polynomial
expansion, as in the previous example:
600 + 40 + 2
6 x 100 + 4 x 10 + 2 x 1
6 x 102 + 4 x 101 + 2 x 100
Solution
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6 x 92 + 4 x 91 + 2 x 90 = 52410
Final answer: 6429 = 52410
Positional Notation
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The base (a.k.a. radix) 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 * Rn-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
147
dn * Rn-1 + dn-1 * Rn-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 * Rn-2 + ... + d1 * R1 + d0 * R0
The text shows the digits numbered like this:
QUIZ
1568
Convert 642 in base 13 to decimal.
Solution
1668
Final answer: 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
13
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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
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
2310
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 =
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The inverse problem: Converting Base 10 to Other Bases
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While (the quotient is not zero)Divide the decimal number by RMake the remainder the next digit to the left in the
answerReplace 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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LSBMSB
Repeated division QUIZ
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Convert 4210 to binary
42 2 = rem.
Final answer:
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
Today we’ve covered pp.35-41 and 42-45
of text (we skipped Arithmetic in Other
Bases, will cover it next time)
Solve in notebook quick work for next
class:
1 – 4 , 14 – 17, 21, 33a (only!)
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QUIZ: Convert to decimal using polynomial expansion:
1101 00112 =
AB716 =
5137 =
6828 =
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QUIZ
Explain in your own words the
difference between a positional and a
non-positional number system.
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Extra-credit QUIZ
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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 Values0 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
14
Addition QUIZ
37
Carry values
go here1 0 1 0 1 1 0
+1 0 0 0 0 1 1
14
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
SKIP:
Binary SUBTRACTION (with “borrow” bits)
• p.42 of text
• FYI, a few slides are provided at the end of this document.
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Direct conversions between bases that are powers of 2
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binary hexadecimal
octalbinary
Converting between Binary and Octal
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• Mark groups of three bits (R to L)
• Convert each group
10101011 10 101 011
2 5 3
10101011 is 253 in base 8
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Converting between Binary and Hex
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• Mark groups of four bits (R to L)
• Convert each group
10101011 1010 1011
A B
10101011 is AB in base 16
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End-of-chapter ex.37
4314
Perform the following octal additions:
a. 770 + 665
b. 101 + 707
End-of-chapter ex.38
4414
Perform the following hex additions:
a. 19AB6 + 43
Counting
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46
Hex
Conclusion:
• In order to represent any octal digit, we need ____ bits
• In order to represent any hex digit, we need ____ bits
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• If we have 3 bits, we can create _____ (how many?) different combinations of 0s and 1s.
• If we have 4 bits, we can create _____ (how many?) different combinations of 0s and 1s.
• If we have 5 bits, we can create _____ (how many?) different combinations of 0s and 1s.
• If we have N bits, we can create _____ (how many?) different combinations of 0s and 1s.
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• If we have 3 bits, we can create 23 = 8 different combinations of 0s and 1s.
• If we have 4 bits, we can create 24 = 16 different combinations of 0s and 1s.
• If we have 5 bits, we can create 25 = 32 different combinations of 0s and 1s.
• If we have N bits, we can create 2N different combinations of 0s and 1s.
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Source: AP CS Principles – Course and Exam Descriptions
Your turn:
Source: AP CS Principles – Course and Exam Descriptions
Solution
𝟐𝟔𝟒
𝟐𝟑𝟐= 𝟐𝟔𝟒−𝟑𝟐 = 𝟐𝟑𝟐
Binary and Computers
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Word = group of bits that the computer processes
at a time. Can differ from one comp. to another.
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 (on all computers!)
• 8, 16, 32, 64-bit computers
• 128? 256?
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Read bio: 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
54Harvard University Mark II Aiken Relay Calculator
Read: Ethical Issues →The FISA Court
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Quick workTo do by next class, before starting to work
on homework:
• Read the entire Chapter 2
• SKIP: Binary subtraction with borrow bits (p.42)
• Answer end-of-chapter questions 6 – 13 in your notebook
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This is not homework!Do not turn in for
grading!
Homework for Ch.2Due next Tue, Sep 18, at beginning of
lecture, hard-copy, not electronic!
• End-of-chapter ex. 23, 24, 25, 28, 29, 30, 33bcde, 38bc
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Addition QUIZ
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Carry values
go here1 1 1 0 1 1 0
+1 0 0 0 1 1 1
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Solution
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Carry values
1 1 1 0 1 1 0
+1 0 0 0 1 1 1
14
Check in base ten!
1 0 0 0 1 1 0
1 0 1 1 1 1 0 1
End-of-chapter ex.37
6014
Perform the following octal additions:
c. 202 + 667
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
15
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
15
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
15
Borrow values
go here
Check in base ten!
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, esp. binary
• Add and subtract in binary
• Convert between bases 2, 8, and 16 using groups of digits
• Count in any base, esp. binary
• Explain the importance in CS of the bases that are powers of 2
• What is the IEEE/ACM CS Curriculum and what are three issues in the “Tenth Strand”?
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