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Chapter 2
Binary Values and Number
Systems
Numbers
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…
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2.2 Positional notation
How many ones (units) are there in 642?
600 + 40 + 2 ?
Or is it
384 + 32 + 2 ?
Or maybe…
1536 + 64 + 2 ?
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Positional Notation
642 is 600 + 40 + 2 in BASE 10
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, …
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Positional Notation
In our example:
642 in base 10 positional notation is:
6 x 102 = 6 x 100 = 600
+ 4 x 101 = 4 x 10 = 40
+ 2 x 10º = 2 x 1 = 2
= 642 in base 10
This number is in
base 10
The power indicates
the position of
the digit inside the
number
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Positional Notation
R is the base
of the number
Formula:
dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R + d1
n is the number of
digits in the number
d is the digit in the
ith position
in the number
642 is 63 * 102 + 42 * 10 + 21
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Positional Notation reloaded
The text shows the digits numbered like this:
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
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Positional Notation
What if 642 has the base of 13?
+ 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
642 in base 13 is equal to 1068 in base 10
64213 = 106810
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Positional Notation
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
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Binary
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
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Converting Binary to Decimal
What is the decimal equivalent of the binary
number 1101110?
11011102 = ???10
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Converting Binary to Decimal
What is the decimal equivalent of the binary
number 1101110?
1 x 26
+ 1 x 25
+ 0 x 24
+ 1 x 23
+ 1 x 22
+ 1 x 21
+ 0 x 2º
=
=
=
=
=
=
=
1 x 64
1 x 32
0 x 16
1x8
1x4
1x2
0x1
= 64
= 32
=0
=8
=4
=2
=0
= 110 in base 10
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QUIZ:
100110102 = ???10
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Bases Higher than 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
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Converting Hexadecimal to Decimal
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
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QUIZ:
2AF16 = ???10
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Converting Octal to Decimal
What is the decimal equivalent of the octal
number 642?
6428 = ???10
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Converting Octal to Decimal
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
= 418 in base 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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Today we’ve covered pp.33-39 of the text
(stopped before Arithmetic in Other Bases)
Solve in notebook for next class:
1, 2, 3, 4, 5, 20, 21, 22
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QUIZ: Convert to decimal
1101 00112 = ???10
AB716
= ???10
5138
= ???10
6928
= ???10
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Addition in Binary
Remember that there are only 2 digits in binary,
0 and 1
1 + 1 is 0 with a carry
011111
1010111
+1 0 0 1 0 1 1
10100010
Carry Values
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Addition QUIZ
1010110
+1 0 0 0 0 1 1
Carry values
go here
Check in base ten!
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Subtraction in Binary
Remember borrowing? Apply that concept
here:
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0202
1010111
- 111011
0011100
Borrow values
1010111
- 111011
0011100
Check in base ten!
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Subtraction QUIZ
Borrow values
1011000
- 110111
Check in base ten!
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Converting Decimal to Other Bases
Algorithm for converting number in base
10 to any other base R:
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
A.k.a. repeated division (by the base):
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Converting Decimal to Binary
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
MSB
LSB
 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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Repeated division QUIZ
Convert 4210 to binary
42  2 =
4210 =
rem.
2
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The repeated division algorithm
can be used to convert from any
base into any other base, but we
use it only for 10 → 2
SKIP
Converting Decimal to Octal
Converting Decimal to Hex
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Converting Binary to Octal
• 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
• Mark groups of four (from right)
• Convert each group
10101011
1010 1011
A
B
10101011 is AB in base 16
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Counting
Note the
patterns!
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Converting Octal to Hexadecimal
End-of-chapter ex. 25:
Explain how base 8 and base 16 are related
10 101 011
2 5 3
253 in base 8
1010 1011
A
B
=
AB in base 16
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Binary Numbers and Computers
Computers have storage units called binary digits or
bits
Low Voltage = 0
High Voltage = 1
All bits are either 0 or 1
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Binary and Computers
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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Individual work
• Read Grace Hopper’s bio, the trivia frames,
chapter review questions, and ethical issues
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Who am I?
I wrote the world’s
first compiler in
1952!
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From the history of computing: bi-quinary
Roman abacus (source: MathDaily.com)
The front panel of the legendary
IBM 650 IBM 650 (source: Wikipedia)
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Chapter Review questions
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•
•
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•
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
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Homework
Due next Friday, Feb. 3:
•
•
End-of-chapter ex. 23, 26, 28, 29, 30, 31, 38
End-of-chapter thought question 4
(paragraph-length answer required)
The latest homework assigned is always
available on the course webpage
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