Transcript Mathematical Ideas - Folsom Lake College

Chapter 1
Section 4-1
Historical Numeration Systems
Symbols
See word doc for complete list of symbols
used for ancient numeration systems.
2
Roman Numeration
Roman numerals are written as combinations of the seven
letters in the table below. The letters can be written as capital
(XVI) or lower-case letters (xvi).
Roman Numerals
I=1
C = 100
V=5
D = 500
X = 10
M = 1000
L = 50
Roman Numeration
Roman Numeral Table
1I
14 XIV
27 XXVII
150 CL
2 II
15 XV
28 XXVIII
200 CC
3 III
16 XVI
29 XXIX
300 CCC
4 IV
17 XVII
30 XXX
400 CD
5V
18 XVIII
31 XXXI
500 D
6 VI
19 XIX
40 XL
600 DC
7 VII
20 XX
50 L
700 DCC
8 VIII
21 XXI
60 LX
800 DCCC
9 IX
22 XXII
70 LXX
900 CM
10 X
23 XXIII
80 LXXX
1000 M
11 XI
24 XXIV
90 XC
1600 MDC
12 XII
25 XXV
100 C
1700 MDCC
13 XIII
26 XXVI
101 CI
1900 MCM
Roman Numeral Calculator
http://www.novaroma.org/via_romana/numbers.html
Note: see “Roman Numerals” on website for additional information
Roman Numeration (Additional HW
Problems)
Change to Roman Numerals
1. 215
2. 379
3. 1995
Change to Decimal Numbers
4. DCIV
5. CDXXIX
6. MCMXCVII
Ancient Egyptian Numeration –
Simple Grouping
The ancient Egyptian system is an example
of a simple grouping system. It used ten as
its base and the various symbols are shown
on the next slide.
Ancient Egyptian Numeration
7
Example: Egyptian Numeral
Write the number below in our system.
Solution
2 (100,000) = 200,000
3 (1,000) = 3,000
1 (100) =
100
4 (10) =
40
5 (1) =
5
Answer: 203,145
8
Example: Egyptian Numeral
Convert 427 to Egyptian.
Symbol
Inventory:
9
Example: Egyptian Numeral
Convert 427 to Egyptian.
Solution
10
Egyptian Numerals
See word doc from website
11
Traditional Chinese Numeration –
Multiplicative Grouping
A multiplicative grouping system involves
pairs of symbols, each pair containing a
multiplier and then a power of the base. The
symbols for a Chinese version are shown on
the next slide.
4-1-12
Chinese Numeration
13
Example: Chinese Numeral
Interpret each Chinese numeral.
a)
b)
14
Example: Chinese Numeral
Solution
a)
7000
400
80
2
Answer: 7482
b)
200
0 (tens)
1
Answer: 201
15
Example: Convert to Chinese
Numeral
Symbol Inventory
2018
16
Example: Convert to Chinese
Numeral
2018
2000
0 hundreds
10
8
17
Chinese Numerals
See word doc from website
18
Positional Numeration
The power associated with each multiplier can
be understood by the position that the
multiplier occupies in the numeral.
To work successfully, a positional system
must have a symbol for zero to serve as a
placeholder in case one or more powers of
the base are not needed.
19
Hindu-Arabic Numeration –
Positional
One such system that uses positional form is
our system, the Hindu-Arabic system.
The place values in a Hindu-Arabic numeral,
from right to left, are 1, 10, 100, 1000, and so
on. The three 4s in the number 45,414 all
have the same face value but different place
values.
20
Hindu-Arabic Numeration
7,
5
4
1,
7
2
5
.
21
Hindu-Arabic Numeration
In Expanded Notation
45, 414
= 4 x 10000 + 5 x 1000 + 4 x 100 + 1 x 10 + 4 x 1
= 4 x 104 + 5 x 103 + 4 x 102 + 1 x 101 + 4 x 100
5, 014 = 5 x 103 + 0 x 102 + 1 x 101 + 4 x 100
= 5 x 103 + 1 x 101 + 4 x 100
22
Hindu-Arabic Numeration
– Scientific Notation
Scientific Notation: Number written in
powers of ten such that one digit is left of the
decimal place.
45,414 = 4.5414 x 104
.045414 = 4.5414 x 10-2
23
Chapter 1
Section 4-2
More Historical Numeration Systems
Babylonian Numeration
The ancient Babylonians used a modified base 60
numeration system.
The digits in a base 60 system represent the
number of 1s, the number of 60s, the number of
3600s, and so on.
The Babylonians used only two symbols to create
all the numbers between 1 and 59.
▼ = 1 and ‹ =10
Example: Babylonian Numeral
Interpret each Babylonian numeral.
a)
‹‹‹‹▼▼
b) ‹ ‹
‹‹▼▼‹ ‹ ▼▼▼▼
c) ▼ ▼ ‹ ‹
‹‹▼▼‹ ‹ ▼▼▼▼
26
Example: Babylonian Numeral
Convert to Babylonian numeral.
a)
47, 094
b) 7,241
27
Mayan Numerals
See word doc from website
28
Mayan Numeration
The ancient Mayans used a base 20 numeration
system, but with a twist.
Normally the place values in a base 20 system
would be 1s, 20s, 400s, 8000s, etc. Instead, the
Mayans used 360s as their third place value but
multiplied by 18 in one case.
1, 20, 20 x 18 = 360, 360 x 20 = 7200,
7200 x 20 = 144,000, and so on
29
Mayan Numeration
Mayan numerals are written from top to bottom.
30
Example: Mayan Numeral
Write the number below in our system.
31
Example: Mayan Numeral
Write the number below in our system.
Solution
10  360
0  20
Answer: 3619
19 1
32
Greek Numeration
The classical Greeks used a ciphered counting
system.
They had 27 individual symbols for numbers,
based on the 24 letters of the Greek alphabet, with
3 Phoenician letters added.
Multiples of 1000 are indicated with a small
stroke next to a symbol.
Multiples of 10000 are indicated by the letter M.
33
Greek Numerals
See word doc from website
34
Greek Numeration
Table 2
Table 2
(cont.)
35
Example: Greek Numerals
Interpret each Greek numeral.
a) ma
b) cpq
36
Example: Greek Numerals
Solution
a)
ma
b) cpq
Answer: 41
Answer: 689
37
Chapter 1
Section 4-3
Arithmetic in the Hindu-Arabic System
Historical Calculation Devices
One of the oldest devices used in calculations
is the abacus. It has a series of rods with
sliding beads and a dividing bar. The abacus
is pictured on the next slide.
Abacus
Reading from right to left, the rods have values of 1,
10, 100, 1000, and so on. The bead above the bar has
five times the value of those below. Beads moved
towards the bar are in “active” position.
Example: Abacus
Which number is shown below?
Solution
104 103 102 101 100
1000 + (500 + 200) + 0 + (5 + 1) = 1706
Example: Abacus
Which number is shown below?
Solution
104 103 102 101 100
10000 + (5000 + 1000) + (500 + 200) + 30 +
(5 + 2) = 16737
Example: Abacus
Use an abacus to show a number?
Lattice Method
The Lattice Method was an early form of a
paper-and-pencil method of calculation. This
method arranged products of single digits into
a diagonalized lattice.
Example: Lattice Method
Find the product 32 x 741 by the lattice
method.
Solution
Set up the grid
to the right.
7
4
1
3
2
Example: Lattice Method
Fill in products
7
4
2
1
1
1
1
0
2
0
4
3
0
8
2
3
2
Example: Lattice Method
Add diagonally right to left and carry as
necessary to the next diagonal.
1
2
2
1
1
1
2
0
4
3
7
0
3
0
8
1
2
2
Example: Lattice Method
1
2
2
1
1
1
2
0
4
3
0
7
3
0
8
1
Answer: 23,712
See Lattice Template on Website
2
2
Napier’s Rods (Napier’s Bones)
John Napier’s invention, based on the
lattice method of multiplication, is often
acknowledged as an early forerunner to
modern computers.
Refer to figure 2 on page 155
Russian Peasant Method
Similar to the Egyptian Method of
multiplication but dividing one column by 2
instead of doubling.
Chapter 1
Extension
Clock Arithmetic and Modular
Systems
Clock Arithmetic and Modular
Systems
• Finite Systems and Clock Arithmetic
• Modular Systems
Finite Systems
Because the whole numbers are infinite,
numeration systems based on them are
infinite mathematical systems. Finite
mathematical systems are based on finite
sets.
12-Hour Clock System
The 12-hour clock system is based on an
ordinary clock face, except that 12 is
replaced by 0 so that the finite set of the
system is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
Clock Arithmetic
As an operation for this clock system, addition
is defined as follows: add by moving the hour
hand in the clockwise direction.
11 0 1
10
2
9
8
3
4
7 6
5
5+3=8
Example: Finding Clock Sums by
Hand Rotation
Find the sum: 8 + 7 in 12-hour clock
arithmetic
Solution
Start at 8 and move
the hand clockwise
through 7 more hours.
Answer: 3
11 0 1
10
2
9
8
3
4
7 6
5
Example: Finding Clock Sums by
Hand Rotation
Find the sum: 8 + 7 in 12-hour clock
arithmetic
Solution
Start at 8 and move
the hand clockwise
through 7 more hours.
Answer: 3
11 0 1
10
2
9
8
3
4
7 6
5
Example: Clock Arithmetic
If is 3 o’clock, what time will it be 94
hours from now?
Solution
Add 3 and 94. Divide
by 12. The remainder
is the answer.
Answer: 1 o,clock
11 0 1
10
2
9
8
3
4
7 6
5
12-Hour Clock Addition Table
Note the 12-hour is symmetric therefore commutative
under addition
Properties of Real Numbers
Let a, b, and c be real numbers.
Addition
Closure
a + b is in the set
Multiplication
ab is in the set
Commutative
a+b=b+a
ab = ba
Associative
a + (b + c) = (a + b) + c
a(bc) = (ab)c
Identity
a+0=a
a1 = a
Inverse
a + (-a) = 0
a (1/a) = 1
Notes: A set is closed if any possible combination of elements
under addition or multiplication must be in the set
12-Hour Clock Addition Properties
Closure The set is closed under addition.
Commutative For elements a and b, a + b = b + a.
Associative For elements a, b, and c,
a + (b + c) = (a + b) + c.
Identity The number 0 is the identity element.
Inverse Every element has an additive inverse.
Inverses for 12-Hour Clock Addition
Clock
value a
0
Additive
Inverse -a
0 11 10 9 8 7 6 5 4 3
1
2
3 4 5 6 7 8 9 10 11
2
Note these pairs are inverses because they add
to the identity which is 0.
1
Modular Systems
In this area the ideas of clock arithmetic are
expanded to modular systems in general.
Example: Modular Arithmetic
Work each modular arithmetic problem.
a) (16  14)(mod 7)
b) (82  45)(mod 3)
c) (4 x14)(mod 4)
Solution
a) 2
b) 1
c) 0
© 2008 Pearson Addison-Wesley. All rights reserved
4-4-64
Example: Truth of Modular Equations
Decide whether each statement is true or false.
a) 12  4(mod 2)
b) 35  4(mod 7)
c) 11  44(mod 3)
Solution
a) True. 12 – 4 = 8 is divisible by 2.
b) False. 35 – 4 = 31 is not divisible by 7.
c) True. 11 – 44 = –33 is divisible by 3.
© 2008 Pearson Addison-Wesley. All rights reserved
4-4-65
Example: Mod 4 Multiplication Table
Determine which are satisfied by the system.
x
0
1
2
3
0
0
0
0
0
1
0
1
2
3
2
0
2
0
2
3
0
3
2
1
Solution
Closed? Yes
Commutative? Yes
Identity = 1
Inverses: 1 & 3 are own inverses
© 2008 Pearson Addison-Wesley. All rights reserved
4-4-66
Chapter 1
Extension
Properties of Mathematical Systems
An Abstract System
The focus will be on elements and operations that
have no implied mathematical significance. We can
investigate the properties of the system without
notions of what they might be.
Operation Table
Consider the mathematical system with elements
{a, b, c, d} and an operation denoted by ☺.
The operation table on the next slide shows how
operation ☺ combines any two elements. To use
the table to find c ☺ d, locate c on the left and d
on the top. The row and column intersect at b, so
c ☺ d = b.
Operation Table for ☺
☺
a
b
c
d
a
a
b
c
d
b
b
d
a
c
c
c
a
d
b
d
d
c
b
a
Find
a) b ☺ c
b) d ☺ a
c) (c ☺ a) ☺ b
Solutions
a) a
b) d
c) a
Potential Properties of a Single
Operation Symbol
Let a, b, and c be elements from the set of any
system, and ◘ represent the operation of the system.
Closure
a ◘ b is in the set
Commutative
a ◘ b = a ◘ b.
Associative
a ◘ (b ◘ c) = (a ◘ b) ◘ c
Identity The system has an element e such that
a ◘ e = a and e ◘ a = a.
Inverse there exists an element x in the set such
that
a ◘ x = e and x ◘ a = e.
Operation Table for ☺
☺
a
b
c
d
a
a
b
c
d
b
b
d
a
c
c
c
a
d
b
d
d
c
b
a
Closure Property
For a system to be closed under an operation, the
answer to any possible combination of elements
from the system must in the set of elements.
This system is
closed.
☺ a
b
c
d
a
a
b
c
d
b
b
d
a
c
c
c
a
d
b
d
d
c
b
a
Identity Property
For the identity property to hold, there must be an
element E in the set such that any element X in the set,
X ☺ E = X and E ☺ X = X.
a is the identity
element of the set.
☺
a
b
c
d
a
a
b
c
d
b
b
d
a
c
c
c
a
d
b
d
d
c
b
a
Inverse Property
If there is an inverse in the system then for any
element X in the system there is an element Y (the
inverse of X) in the system such that
X ☺ Y = E and Y ☺ X = E, where E is the identity
element of the set.
☺ a b c d
You can inspect the
table to see that every
element has an
inverse.
a
a
b
c
d
b
b
d
a
c
c
c
a
d
b
d
d
c
b
a
Commutative Property
For a system to have the commutative property, it
must be true that for any elements X and Y from the
set, X ☺ Y = Y ☺ X.
This system has the
commutative property.
The symmetry with
respect to the diagonal
line shows this
property
☺
a
b
c
d
a
a
b
c
d
b
b
d
a
c
c
c
a
d
b
d
d
c
b
a
Associative Property
For a system to have the associative property, it must
be true that for any elements X, Y, and Z from the set,
X ☺ (Y ☺ Z) = (X ☺ Y) ☺ Z.
This system has the
associative property.
There is no quick
check – just work
through cases.
☺
a
b
c
d
a
a
b
c
d
b
b
d
a
c
c
c
a
d
b
d
d
c
b
a
Example 1: Identifying Properties
Consider the system
shown with elements
{0, 1, 2, 3} and
operation #. Which
properties are
satisfied by this
system?
#
0
1
2
3
0
0
1
2
3
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
Example 1: Identifying Properties
Solution
The system satisfies
the closure,
associative,
commutative, and
identity properties, and
inverse property.
#
0
1
2
3
0
0
1
2
3
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
Example 2: Identifying Properties
Consider the system
shown with elements
{0, 1, 2, 3, 4} and
operation . Which
properties are
satisfied by this
system?

0
1
2
3
4
0
0
0
0
0
0
1
0
1
2
3
4
2
0
2
4
0
2
3
0
3
0
3
0
4
0
4
2
0
4
Example 2: Identifying Properties
Solution

The system satisfies
the closure,
associative,
commutative, and
identity properties, but
not the inverse
property.
0
0
0
1
0
2
0
3
0
4
0
1
2
3
0
0
0
1
2
3
2
4
0
3
0
3
4
2
0
4
0
4
2
0
4
Example 3: Identifying Properties
Construct a base 5
addition system of
remainders. Which
properties are
satisfied by this
system?
+
0
1
2
3
4
0
1
2
3
4
Example: Identifying Properties
Which properties are
satisfied by this
system?
J m n p
m n p n
n p m n
p n n m
Distributive Property (not covered)
Let ☺ and ◘ be two operations defined for
elements in the same set. Then ☺ is
distributive over ◘ if
a ☺ (b ◘ c) = (a ☺ b) ◘ (a ☺ c)
for every choice of elements a, b, and c from
the set.
Example: Testing for the Distributive
Property (not covered)
Is addition distributive over multiplication on
the set of whole numbers?
Solution
We check the statement below:
a  (b  c)  (a  b)  (a  c).
Notice, it fails when using 1, 2, and 3:
1  (2  3)  (1  2)  (1  3).
This counterexample shows that addition is not
distributive over multiplication.
Chapter 1
Extension
Groups
Group
A mathematical system is called a group if,
under its operation, it satisfies the closure,
associative, identity, and inverse properties.
Note: the inverse property has to be more
than an inverse of itself to be a group.
Example 1: Checking Group
Properties
Does the set {–1, 1} under the operation of
multiplication form a group?
Solution
All of the properties to be a
group (closure, associative,
identity, inverse) are
satisfied as can be seen by
the table.
x –1 1
–1 1 –1
1 –1 1
Example 2: Checking Group Properties
Does the set {–1, 1} under the operation of
addition form a group?
Solution
No, right away it can be
seen that closure is not
satisfied.
+ –1
–1 –2
1 0
1
0
2
Example 3: Identifying Group
Is the system of
integers a group
under subtraction?
No, there is no
identity
- -2 -1 0 1 2
-2 0 -1 -2 -3 -4
-1 1 0 -1 -2 -3
0 2 1 0 -1 -2
1 3 2 1 0 -1
2 4 3 2 1 0