chapter 2 · 3 3 integers a natural number, a negative number, zero examples: 249, 0, - 45645, - 32...
TRANSCRIPT
![Page 1: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/1.jpg)
Chapter 2
Binary Values and Number Systems
![Page 2: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/2.jpg)
Numbers
2 2
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
![Page 3: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/3.jpg)
3 3
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…
![Page 4: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/4.jpg)
2.2 Positional notation
4 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
![Page 5: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/5.jpg)
QUIZ
5 4
How many ones (units) are there in 6429
(642 in base nine)?
Use the base-10 model:
600 + 40 + 2
6 x 100 + 4 x 10 + 2 x 1
6 x 102 + 4 x 101 + 2 x 100
![Page 6: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/6.jpg)
QUIZ
6 4
How many ones (units) are there in 6429
(642 in base nine)?
6 x 92 + 4 x 91 + 2 x 90 = 12510
![Page 7: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/7.jpg)
Positional Notation
7 5
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, …
![Page 8: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/8.jpg)
Positional Notation
8 7
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
![Page 9: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/9.jpg)
Positional Notation reloaded
9 7
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:
![Page 10: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/10.jpg)
QUIZ
10 6 8
What is 642 in base 13?
![Page 11: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/11.jpg)
QUIZ
11 6 8
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
![Page 12: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/12.jpg)
Nota bene!
12
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
![Page 13: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/13.jpg)
Binary
13 9
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
![Page 14: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/14.jpg)
QUIZ: Converting Binary to Decimal
14
What is the decimal equivalent of the binary
number 110 1110?
110 11102 = ???10
13
![Page 15: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/15.jpg)
15
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
13
![Page 16: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/16.jpg)
Base 8 = octal system
16
What is the decimal equivalent of the octal
number 642?
6428 = ???10
11
![Page 17: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/17.jpg)
Converting Octal to Decimal
17
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
11
![Page 18: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/18.jpg)
Bases Higher than 10
18 10
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
![Page 19: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/19.jpg)
Converting Hexadecimal to Decimal
19
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
![Page 20: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/20.jpg)
QUIZ: 2AF16 = ???10
20
![Page 21: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/21.jpg)
Are there any non-positional number systems? Hint: Why did the Roman civilization have no contributions to mathematics?
21
![Page 22: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/22.jpg)
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
22
![Page 23: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/23.jpg)
QUIZ: Convert to decimal
1101 00112 =
AB716 =
5138 =
6928 = 23
![Page 24: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/24.jpg)
Addition in Binary
24
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
14
![Page 25: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/25.jpg)
Addition QUIZ
25
Carry values
go here
1 0 1 0 1 1 0
+1 0 0 0 0 1 1
14
Check in base ten!
![Page 26: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/26.jpg)
Subtraction in Binary
26
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
15
Borrow values
Check in base ten!
![Page 27: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/27.jpg)
Subtraction QUIZ
27
1 0 1 1 0 0 0
- 1 1 0 1 1 1
15
Check in base ten!
Borrow values
go here
![Page 28: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/28.jpg)
Direct conversions between bases that are powers of 2
28
binary
hexadecimal
octal
![Page 29: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/29.jpg)
Counting
29
![Page 30: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/30.jpg)
QUIZ: Add the hex column!
30
![Page 31: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/31.jpg)
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
31
![Page 32: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/32.jpg)
Converting Binary to Octal
32
• Mark groups of three (from right)
• Convert each group
10101011 10 101 011
2 5 3
10101011 is 253 in base 8
17
![Page 33: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/33.jpg)
Converting Binary to Hexadecimal
33
• Mark groups of four (from right)
• Convert each group
10101011 1010 1011
A B
10101011 is AB in base 16
18
![Page 34: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/34.jpg)
Converting Octal to Hexadecimal
34
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
18
![Page 35: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/35.jpg)
Extra-credit QUIZ: Convert from hex to octal 2AF16 = 8
Show all your work for credit!
35
?
![Page 36: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/36.jpg)
Today we’ve covered pp.39-42 of text
(stopped before Converting from Base 10 to
Other Bases)
Solve in notebook as individual work for
next class: 6 through 11
36
![Page 37: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/37.jpg)
Homework for ch.1 is due at the beginning of next class (this Friday)
37
![Page 38: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/38.jpg)
Subtraction QUIZ
38
1 0 1 0 0 0 0
- 1 0 0 1 0 1
15
Borrow values
go here
Check in base ten!
![Page 39: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/39.jpg)
Periodic patterns
39
![Page 40: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/40.jpg)
The inverse problem: Converting Decimal to Other Bases
40
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:
19
Known as repeated division (by the base)
![Page 41: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/41.jpg)
Converting Decimal to Binary
41
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
19
LSB MSB
![Page 42: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/42.jpg)
Repeated division QUIZ
42
Convert 4210 to binary
42 2 = rem.
4210 = 2
19
![Page 43: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/43.jpg)
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)
43
Read text example on p.43: Converting Decimal to Hex using repeated division
![Page 44: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/44.jpg)
Binary Numbers and Computers
44
Computers have storage units called binary digits or
bits
Low Voltage = 0
High Voltage = 1
All bits are either 0 or 1
22
![Page 45: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/45.jpg)
Binary and Computers
45
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?
23
![Page 46: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/46.jpg)
46
• Motivated by the 6-bit codes for printable graphic
patterns created by the U.S. Army and Navy
• 6, 18, 26, 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”?
![Page 47: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/47.jpg)
47
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”?
![Page 48: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/48.jpg)
Grace Murray Hopper
48
• Ph.D. in mathematics
• Wrote “A-0”, the world’s first
compiler, in 1952!
• Co-invented COBOL
• Found the first “bug” in Mark II
• Rear-admiral of the US Navy
• “Nanosecond” wires
![Page 49: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/49.jpg)
From the history of computing: bi-quinary
49 The front panel of the legendary
IBM 650 (source: Wikipedia)
Roman abacus (source: MathDaily.com)
![Page 50: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/50.jpg)
Ethical Issues →Tenth Strand
50
What do the following acronyms stand for:
• ACM ?
• IEEE ? What is the tenth strand? Why the “tenth”?
![Page 51: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/51.jpg)
51
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
![Page 52: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/52.jpg)
Ethical Issues →Tenth Strand
52
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!
![Page 53: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/53.jpg)
Chapter Review questions
• Describe positional notation
• 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
53 6 24
![Page 54: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/54.jpg)
Individual work for next class:
solve in notebook end-of-chapter exercises
22, 26
54
![Page 55: Chapter 2 · 3 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 ... End-of-chapter ex. 25: Explain how base 8 and base 16 are related ... Binary](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e6c258bd0150667ea71b9b4/html5/thumbnails/55.jpg)
Homework Due next Friday, Sept. 14:
End-of-chapter ex. 23, 24, 25, 28, 29, 30, 33, 38
The latest homework assigned is always available on the course webpage
55