Numbers, Numbers, Numbers …!
For more details, google “types of numbers”
What are the figurate numbers?
  These are the numbers that have shapes.
They were studied first and extensively by the
Greeks.!
  The most common shapes are triangular,
square, oblong, pentagonal, hexagonal, etc.
Which number systems require zero?
  Place value systems MUST have a zero !
Perfect numbers
Abundant, Deficient
Why care?
  Because they are cool !
Complex Numbers!
 Use i , −1
!
 Squared, it equals -1.!
 Written a + bi where a & b are real numbers. !
 i is the simplest ʻimaginaryʼ number.!
€
 i is a “made up” number. We make it up to
solve polynomial equations. !
 Just like negative numbers. They are made
up by mathematicians.!
What are complex numbers
good for?!
 Used a great deal in electronics.!
 Specifically, they are used for voltage, current
and resistance (impedance) in A.C. circuits !
 In these situations, a single complex number
works better than two real numbers.!
 Also used in electromagnetism to describe the
electric and magnetic field strength !
Types of REAL NUMBERS
  Counting numbers
  Natural (or Whole) numbers
  Integers
  Rational numbers
  Irrational numbers
  Real Numbers
Let’s make a Venn Diagram of the types of REAL NUMBERS
Types of REAL NUMBERS
 True or False: Every integer is a rational number.
 True or False: Every rational number is an integer.
 True or False: Every number is either a
rational or an irrational, but not both.
Classify according to number type; some numbers may be of more than one type.
 0.45!
 3.1415926535897932384626433
8327950288419716939937510...
 10!
 3.14159!
 5/3!
 1 2/3!
Classify according to number type
  !
−81
9
  !−
3
€
  0!
€
Types of numbers within the set of irrational numbers!
 Irrational constructible numbers !
 Irrational algebraic numbers!
 Transcendental numbers are the irrational
numbers that are not algebraic !
(They ʻtranscendʼ the power of algebra.)!
 So, every irrational number is either algebraic
or transcendental.!
Now, letʼs turn to this question:
How many numbers are there? !
 “Infinitely many”!
 IN-FINITE means NOT FINITE.!
 Are there transfinite numbers? !
 And could there be more than one level
of infinity?!
YES !!!
 But how could we tell one level of infinity !
without counting? !
 Put your hands together. *!
 One-to-one correspondence is the tool. We !
call this the “matching principle.”!
 Which set has more elements? !
* ODD counting numbers or!
* EVEN counting numbers?!
* Coincidences, Chaos, and All That Math Jazz, by Burger& Starbird!
HOW MANY ?!
Which set has “more” numbers? !
* ALL counting numbers or!
* EVEN counting numbers?!
 The subset principle. !
 Which set has “more” numbers? !
* ALL counting numbers or!
* the set of all Integers?!
€
HOW MANY ?!
 Here is the answer to the question: !
How many counting numbers are there?!
ω
  ℵ0
€
Aleph naught. (Aleph is the first letter Hebrew
ω
alphabet.) Sometimes we also use “omega” !
 What is meant by “more” numbers?!
€
HOW MANY ?!
 The set of Rational numbers is also
“countable”!
 So, we say there
ℵ0
are rational numbers.!
 Are you uncomfortable yet? !
Kronecker is !!
 These
€are Cantorʼs ideas and they were not
accepted when he first offered them.!
HOW MANY ?!
 It turns out that THERE IS a set of numbers
that has “more” numbers than the counting
numbers.!
 It is the set of Real numbers, the number line. !
 It is also called the CONTINUUM.!
The CONTINUUM!
 We say its cardinality is greater than ℵ0
 We usually use the symbol c to stand for
its cardinality. (c for continuum.)!
 It is “uncountable”.!
 Cantor proved this in 1873
€ and it shook !
up the world.!
 The Continuum Hypothesis.!
!
!
HOW MANY ?!
 The proof can easily be understood by smart
middle and high school kids. !
 It is called Cantorʼs “Diagonal Argument.” !
There are even higher infinities !!
 Cantor knew about these too.!
 We are not going there.!
 Thanks for your attention.!
As you can tell I love this subject, so feel free to !
ask about these ideas anytime.!
THANKS