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Section 2.1 Sets as a Basis for Whole Numbers
Introduction
Much of elementary school mathematics is devoted to the study of numbers. Children first learn to count using the natural number or counting numbers 1, 2, 3, …
This chapter develops the ideas that lead to the concepts central the system of whole numbers 0, 1, 2, 3, … (the counting numbers together with 0).
A collection of objects is called a set, and the objects are called elements or members.Examples
• a set of silverware• a set of color pencils• a set of tennis
Representations of sets• Diagram a diagram with circles (such as the one on the right) can be used to summarize the relationship between different collections of students in our class.
This kind of diagrams are called Venn Diagrams.
Brown Eyes Brown Hair
Curly hair
Students in our class
Venn Diagrams
Venn diagrams are illustrations used in the branch of mathematics known as set theory. They show the mathematical or logical relationship between different groups of things (sets). A Venn diagram shows all the possible logical relations between the sets.
John Venn (1834-1923) was a British philosopher and mathematician who introduced the Venn diagram in 1881.A stained glass window in Caius College, Cambridge, where he studied and spent most of his life, commemorates John Venn and represents a Venn diagram.
Origin
Venn Diagrams
can also be used to show the relationship between different categories of geometric objects.
Representations of sets• Diagram
• Listing: {Alaska, California, Hawaii, Oregon, Washington}
• Set-builder: { x | x is a U.S. state that borders the Pacific Ocean}
Operations on SetsSuppose that we have two sets A and B
If they overlap, then the overlapping part is called the intersection of A and B, the notation is AB
AB
A B
On the other hand, if we combine the two sets, we will get a bigger set called the union of A and B, the notation is AB.
A B
We can also created the difference of two sets. A – B is the elements that are in A but not in B.
A B
Why do we need numbers at elementary level?• To express quantity • To compare quantities i.e. to find out whether one set is larger than the other.
Well, can we compare sets without counting?
One-to-one correspondences
A one-to-one correspondence between two sets A and B is apairing of objects from A and B such that every object in set A is paired up with exactly one object in B and vice versa.
The above diagram is an example of a one-to-one correspondence. It is not hard to see that we can construct many other different one-to-one correspondences between these two sets.
Set A
Set B
Matching sets
If there is a one-to-one correspondence between the two sets A and B, we say that they are equivalent or they match each other.
Comparing sets
If we can construct a one-to-one correspondence from set B to a portion of set A such that there are some elements in set A left unpaired, then we say that set A has more elements.
Set A
Set B
So, do we really need numbers to compare sets?
Or, do we need them just for convenience?
• a number, being an abstract object, is more convenient and cost effective for - record keeping - transmission - duplication - comparison - manipulation.
What is a number anyway?
A number is the abstract attribute shared by sets that are equivalent (i.e. in one-to-one correspondence)
For example, the number three is the attribute shared among the following sets:
• to count the elements in the set A, we construct a 1-to-1 correspondence from A to an initial segment of the sequence of numbers {1, 2, 3, 4, 5, … … …}
Set A
• an initial segment is a portion that contains the first element and does not skip any element in between.
What are we doing when we count?
• the last number being used is the number of elements in the set.• the number of elements in the set should not depend on which 1-to-1 correspondence is used or the positions of the objects.• in other words, as long as you are using 1-to-1 correspondences, you should always come up with the same answer.
Set A
What are the benefits of knowing numbers and knowing how to count?
• the counting process converts a collection of physical objects into a number (usually represented by a symbol),• a number, being an abstract object, is more convenient and cost effective for - record keeping - transmission - duplication - comparison - manipulation.
Three common uses of numbers• Cardinal number - tells the size of a set - example: I have five brothers.• Ordinal number - tells the position of an object in a list
(or in other words, how far away the object is from the beginning of the list). - example: you are the 17th on the waiting list.• Identification number - gives an object a name or identity - example: The zip code here is
92020.
Three common uses of numbers• Cardinal number - tells the size of a set - example: I have five brothers.• Ordinal number - tells the position of an object in a list
(or in other words, how far away the object is from the beginning of the list). - example: you are the 17th on the waiting list.• Identification number - gives an object a name or identity - example: The zip code here is
92020.
Numbers and Numerals
A number is an abstract attribute.A numeral is a symbol (representing a number).
3 8
Which one represents a larger number? 8
Which one is a larger numeral? 3
Numeration system
• a systematic way to assign symbols to numbers.
Tally system• converts each object into a tally
Do you know what the following number is?
Improve clarity by grouping
The base of a system is the size of each basic group.
The base of the tally system is five and • five tallies = one bundle• five bundles = one superbundle
What are the short comings of the Tally system?
• five superbundles = one super-superbundle • etc.
Egyptian System• a base ten system• with a special symbol for each power of ten
one ten hundredthousand
ten- thousand
hundred- thousand
million
• an additive system
= One thousand one hundred twenty one.
• a non-positional system
has the same value as
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The Egyptian System
Drawbacks of this system• new symbols must be invented whenever larger powers of ten are encountered.• lots of symbols may be needed for a relatively small number.• the length of an expression is not proportional to the size of the number it represents;
Advantages of this system• simplicity in structure, easy to learn• intuitive approach
- this leads to confusion and makes calculations very cumbersome.
means thirty two means one thousand
The Roman System
• a base ten system• positional eg. VI is different from IV• additive and subtractive eg. VI = 5 + 1 and IV = 5 – 1• multiplicative as well eg. XII = 12 and XII = 12,000
I V X L Cone five ten fifty hundred
D Mfive hundred thousand
• has no place value
Here is an old building in Macau, the Roman numerals printed above the front entrance states which year it was built.
(see next slide for enlargement)
The Roman System
Almost no advantage.Disadvantages• new symbols are needed for higher powers of ten• subtractive system is very unnatural to use• pattern is too complex to be practical• length of expression is not proportional to the number being represented
Babylonian System
• a base sixty system – such as the clock system• requires only three symbols
one ten place-holder
• is positional: is different from
• has a very sophisticated feature – place value, i.e. the value of a symbol depends on its position in the whole expression.
Babylonian System
Examples:…
…
two
three
four
five
six
seven
eight
ten
eleven
twelve
fifty nine
sixty
sixty one
sixty two
nine
Babylonian System
Advantage of the system• a very small set of symbols is used, and this set will never need to be expanded.
Disadvantages of the system• the base is too large• ambiguity exists• place value is difficult to learn• length of the whole expression is still not proportional to the number it represents.
Hindu-Arabic System
Set of symbols
1 2 3 4 5 6 7 8 9 0
Characteristics• base ten• additive• has place value• has a unique symbol for each number less than the base.
place holder
Example:
2 0 2 3
ones
tenshundreds
thousands
The same symbol means two different quantities when placed in two different positions.
Hindu-Arabic System
Example:
5 6 1 4
ones
tenshundreds
thousands
For this particular symbol, the• face value is six• place value is one hundred• actual value is six hundred
Hindu-Arabic System
Advantages• a relatively small finite set of symbols is used• very concise (a few symbols rep. a big number)• the length of an expression is proportional to the size of the number being represented. This facilitates calculations.
Disadvantages• the place value structure is sophisticated and difficult to master.• students have to learn many symbols to read or write even relatively small numbers (eg. compare this with the Roman system)
Hindu-Arabic System
To understand potential difficulties in learning Hindu-Arabic system andto assist the development of better teaching methods,
we introduce the
Base Five Hindu-Arabic system
Symbols required: 1, 2, 3, 4, 0
Base Five Hindu-Arabic system
Symbols required: 1, 2, 3, 4, 0
A list of small numbers
one 1five
two 2five
three 3five
four 4five
five 10five
six 11five
seven 12five
eight 13five
nine 14five
ten 20five
eleven 21five
twelve 22five
... ... ... ...twenty four 44five
twenty five 100five
Physical representations of numbers in base Five
Sticks and bundles
a stick= one
a bundle= five
a superbundle = five bundles= twenty five
a super-superbundle= five superbundles= twenty five bundles= one hundred twenty five
Examples:
What is the base five numeral for the following number of sticks?
Ans: 13five
How about the following?
Ans: 102five
Why do we need the 0?
Examples:
What is the physical representation for 132five ?
How many sticks are there?Ans: twenty five + three×five + two = forty two.
Convert thirty two to a base five numeral.Ans: 112five.
Convert eighty three to a base five numeralAns: 313five.
Base Five Dienes Blocks a (small) cube = one
a long = five
a flat = twenty five
a block= one hundred twenty five
Base Five Dienes Blocks
Examples:
23five =
= five + five + three= thirteen
1204five =
= hundred twenty five + twenty five +twenty five + four= one hundred seventy nine.
TradesA trade is an exchange of objects of equal value.Example:• Trade five small cubes for one long
• Trade five longs for one flat