Section 4.3
Other Bases
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INB Table of Contents
2.3-2
Date
Topic
May 14, 2014
Section 4.3 Examples
8
May 14, 2014
Section 4.3 Notes
9
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Page #
What You Will Learn

Converting base 10 numerals to
numerals in other bases

Converting numerals in other bases
to base 10 numerals
4.3-3
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Positional Values
The positional values in the HinduArabic numeration system are
… 105, 104, 103, 102, 10, 1
The positional values in the Babylonian
numeration system are
…, (60)4, (60)3, (60)2, 60, 1
4.3-4
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Positional Values and Bases
10 and 60 are called the bases of the
Hindu-Arabic and Babylonian systems,
respectively.
Any counting number greater than 1
may be used as a base. If a positionalvalue system has base b, then its
positional values will be
…, b4, b3, b2, b, 1
4.3-5
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Positional Values
The positional values in a base 8
system are
…, 84, 83, 82, 8, 1
The positional values in a base 2
system are
…, 24, 23, 22, 2, 1
4.3-6
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Other Base Numeration Systems

Base 10 is almost universal.

Base 2 is used in some groups in
Australia, New Guinea, Africa, and South
America.

Bases 3 and 4 is used in some areas of
South America.

Base 5 was used by primitive tribes in
Bolivia, who are now extinct.

Base 6 is used in Northwest Africa.
4.3-7
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Other Base Numeration Systems

Base 6 also occurs in combination with
base 12, the duodecimal system.

Our society has remnants of other base
systems:

12: 12 inches in a foot, 12 months in a
year, a dozen, 24-hour day, a gross (12 ×
12)

60: Time - 60 seconds to 1 minute, 60
minutes to 1 hour; Angles - 60 seconds to
1 minute, 60 minutes to 1 degree
4.3-8
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Other Base Numeration Systems

Computers and many other electronic devices
use three numeration systems:

Binary – base 2
4.3-9

Uses only the digits 0 and 1.

Can be represented with electronic switches
that are either off (0) or on (1).

All computer data can be converted into a
series of single binary digits.

Each binary digit is known as a bit.
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Other Base Numeration Systems

4.3-10
Octal – base 8

Eight bits of data are grouped to form a
byte

American Standard Code for
Information Interchange (ASCII) code.

The byte 01000001 represents A.

The byte 01100001 represents a.

Other characters representations can
be found at www.asciitable.com.
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Other Base Numeration Systems

Hexadecimal – base 16

Used to create computer languages:
 HTML
 CSS
4.3-11
(Hypertext Markup Language)
(Cascading Style Sheets).

Both are used heavily in creating
Internet web pages.

Computers easily convert between
binary (base 2), octal (base 8), and
hexadecimal (base 16) numbers.
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Bases Less Than 10

A place-value system with base b has b
distinct objects, one for zero and one for
each numeral less than the base.

Base 6 system: 0, 1, 2, 3, 4, 5


Base 8 system: 0, 1, 2, 3, 4, 5, 6, 7

4.3-12
All numerals in base 6 are constructed
from these 6 symbols.
All numerals in base 8 are constructed
from these 8 symbols.
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Bases Less Than 10

A numeral in a base other than base 10
will be indicated by a subscript to the
right of the numeral.

1235 represents a base 5 numeral.

1236 represents a base 6 numeral.

The value of 1235 is not the same as the
value of 12310.

Base 10 numerals can be written without
a subscript: 123 means 12310.
4.3-13
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Bases Less Than 10

The symbols that represent the base
itself, in any base b, are 10b.

105 represents 5

105 = 1 × 5 + 0 × 1 = 5 + 0 = 5

To change a numeral from one base to
base 10
1.
multiply each digit by its respective
positional value
2.
find the sum of the products.
4.3-14
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Example 1: Converting from
Base 5 to Base 10
Convert 2435 to base 10.
4.3-15
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Units Digits in Different Bases

Notice that 35 has the same value as
310, since both are equal to 3 units.

That is,35 = 310.

If n is a digit less than the base b,
and the base b is less than or equal
to 10, then nb = n10.
4.3-17
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Example 3: Converting from
Base 2 to Base 10
Convert 1100102 to base 10.
4.3-18
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Converting Base 10
1.
Divide the base 10 numeral by the highest
power of the new base that is less than or
equal to the given base 10 numeral and record
this quotient.
2.
Then divide the remainder by the next smaller
power of the new base and record this
quotient.
3.
Repeat this procedure until the remainder is
less than the new base.
4.
The answer is the set of quotients listed from
left to right, with the remainder on the far
right.
4.3-20
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Example 5: Converting from
Base 10 to Base 3
Convert 273 to base 3.
4.3-21
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Example 5: Converting from
Base 10 to Base 3
Solution
4.3-23
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Bases Greater Than 10

We will need single digit symbols to
represent the numbers ten, eleven,
twelve, . . . up to one less than the base.

In this textbook, whenever a base larger
than ten is used we will use the capital
letter A to represent ten, the capital
letter B to represent eleven, the capital
letter C to represent twelve, and so on.
4.3-25
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Bases Greater Than 10

For example, for base 12, known as the
duodecimal system, we use the symbols
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, and B,
where A represents ten and B represents
eleven.

For base 16, known as the hexadecimal
system, we use the symbols 0, 1, 2, 3,
4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.
4.3-26
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Example 7: Converting to and from
Base 16
Convert 7DE16 to base 10.
4.3-27
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Example 7: Converting to and from
Base 16
Convert 6713 to base 16.
4.3-29
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Example 7: Converting to and from
Base 16
Solution
Thus 6713 = 1A3916.
4.3-31
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Convert 21023 to a Hindu-Arabic numeral.
a.
26
b.
65
c.
47
d.
59
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Slide 4 - 32
Convert 45126 to a Hindu-Arabic numeral.
a.
140
b.
502
c.
872
d.
1052
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Slide 4 - 33
Convert 1001112 to a Hindu-Arabic numeral.
a.
17
b.
33
c.
39
d.
57
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Slide 4 - 34
Convert 37512 to a Hindu-Arabic numeral.
a.
171
b.
207
c.
233
d.
521
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Slide 4 - 35
Convert 57 to base 2.
a.
1110012
b.
110012
c.
1110102
d.
1111002
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Slide 4 - 36
Convert 2034 to base 12.
a.
121212
b.
121612
c.
131212
d.
131612
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Slide 4 - 37
Convert 103 to base 7.
a.
100127
b.
5037
c.
1137
d.
2057
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Slide 4 - 38
Convert 798 to base 5.
a.
1435
b.
1111435
c.
11435
d.
111435
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Slide 4 - 39