Chapter 2

Sets, Whole Numbers and Numeration

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

Exercise from textbook

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?

Comparing without counting

Are there enough bananas for the monkeys?

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.

The followings are not one-to-one correspondences, do you know why?

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

Million Dollar Display - This venue closed shortly after the death of Ted Binion. Rumor has it that his sister removed it from the casino. You used to be able to get your picture taken next to Binion's Million! It was located downtown in Binion's Horseshoe Hotel & Casino you'd have found this cool display of 100, discontinued, yet entirely legal tender $10,000 bills (A genuine US$1,000,000 in all!) which you could have walked right up and stood next to. For many of us it was the closest we'll ever get to that much cash in Las Vegas.

The Horseshoe Casino in downtown L.V., on Fremont Street Experience.

From 1966 to 1999 one of the legendary icons of Las Vegas was the Million Dollar Display at Binion’s Horseshoe Casino.  This collection of one hundred 1934 $10,000 bills, encased in Plexiglas and framed by an inverted golden horseshoe, was the backdrop for over 5 million photographs.  Tourists, celebrities, even royalty came to have their pictures taken next to the million dollars in cash.  When the casino decommissioned the display in the late 1990’s, Jay Parrino’s The Mint was there and bought all 100 notes.$10,000 bills are rare in their own right, being the largest denomination note issued for general circulation ($100,000 notes were printed, but used only for transfers within the Federal Reserve Bank system – private ownership is prohibited).  This particular note has the dual distinction of being not only the best of the 100 note hoard, but the finest known example in the world. 

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)

Which year was it built?

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

Trades

Or• Trade five flats for one block