Download Numbers and Counting - Danville California Math and Science for

Math Club
Workshop
Danville Senior Center
May 5, 2016
The plan…sort of…
A seminar, not a class.
I have an “agenda”, but we can ignore it.
But let’s start with introductions-and maybe include your math
background and interests.
The plan…sort of…
My “agenda”, but we can ignore it
Starting with very basic numbers and counting
Move on to Rationals, Reals, and Complex numbers
On to operations on numbers
Then on to Algebra on numbers
Then “other” Algebras: Vectors, Matrices, and Groups
And if we have time/interest -Geometry
The plan…sort of…
Along the way we’ll
Solve some Linear Equations
Derive the Quadratic Formula
Do a Geometric proof
And for extra credit, calculate the value of
ii
Numbers
7/15
2
π
X
i
697
0
Integers
Integers
Integers
Symbols
Integers
Integers
Integers
Integers
Integers
Group Symbols
Integers
Integers
Integers
Roman Numerals
V
X
L
Integers
LXXXIII
Integers
M D C L XVI
MMXVI ?
Integers
MC L X V MD C L X V I
Etruscan X X˄ I
↓
Egyptian
Integers
?
Integers
Zero
0
Integers
The number zero as we know it arrived in the West circa 1200, most famously
delivered by Italian mathematician Fibonacci (aka Leonardo of Pisa), who brought
it, along with the rest of the Arabic numerals, back from his travels to north
Africa. But the history of zero, both as a concept and a number, stretches far
deeper into history.
http://www.scientificamerican.com/article/history-of-zero/
Integers
"There are at least two discoveries, or inventions, of zero," says Charles Seife, author
of Zero: The Biography of a Dangerous Idea(Viking, 2000).
"The one that we got the zero from came from the Fertile Crescent." It first came to be
between 400 and 300 B.C. in Babylon, Seife says, before developing in India, wending
its way through northern Africa and, in Fibonacci's hands, crossing into Europe via
Italy.
The second appearance of zero occurred independently in the New World, in Mayan
culture, likely in the first few centuries A.D. "That, I suppose, is the most striking
example of the zero being devised wholly from scratch," Kaplan says.
Integers
Decimal Numbering System
0 1 2 3 4 5 6 7 8 9
I II III IV V VI VII VIII IX
Groups of 10
10 100 1000 …
X C M
Integers
10 11 12 13
100 101 102
423
1000 1001
7135
945,389,234,703,662
Integers
A prime number (or a prime) is an Integer greater than 1 that has no
positive divisors other than 1 and itself.
2,3,5,7,11,…
You test for prime number “P” by trying to divide P by every Integer
between 2 and 𝑃 .
As of January 2016, the largest known prime number has 22,338,618 decimal
digits.
Primes are used in several routines in information technology, such as publickey cryptography (RSA), which makes use of properties such as the difficulty
of factoring large numbers into their prime factors.
Source: Wikipedia
Integers
Binary, Octal, and Hexadecimal
Binary
0 1
Groups of 2
10 100 1000
2 4
8
1000000000
1024
100000000000
4096
Octal 0 1 2 3 4 5 6 7
Groups of 8
10
8
100
64
1000
512
10000
4096
Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F
Groups of 16
10
16
100
256
1000
4096
Integers
How many Integers are there?
Infinite number of them: no largest Integer.
A particular kind of infinite named “countably infinite”
Countably infinite is the smallest kind of infinite.
The symbol for a countably infinite set is
(Pronounced Aleph Null)
Are there bigger kinds of infinite?
More Numbers
Integers 0 1 697
Rationals (Fractions) 1/3 7/15 34/15 (includes Integers)
Reals
Complex
2
5
7
i 3+5i
π
e
(includes Rationals)
𝑒 2+3𝑖 (includes Reals)
Rationals
(Fractions, including Integers)
1/10000 1/3 7/15 34/15
100089/5872
Are there more Rationals than Integers?
Rationals
Are there more Rationals than Integers?
No.
Rationals
All the possible Rationals
Rationals
All the possible Rationals
Reals
Are there numbers that are not Rationals?
Reals
The set of Real numbers includes the Rationals, but also has numbers
that are NOT fractions.
Examples are: π e
2
5
3 … 17 …
Any Prime number …
Reals
Example: 2 is NOT a Rational
Proof:
Suppose 2 is Rational
Then there exist integers P and Q such that P/Q = 2
And P, Q in lowest common denominator (in particular, both NOT even)
Then P2/Q2 = 2
P2 = 2Q2
So P2 must be even
But if P2 is even, then P must be even
But then P2 must be divisible by 4
But that means Q2 must be even
Which means that Q must be even
Which means both P and Q are both even
But they can’t be both even, so there is no such P/Q
Therefore 2 is NOT a Rational
Reals
There are more Reals than Rationals or Integers.
Reals are a bigger kind of Infinite than Countably Infinite
The “Continuum Hypothesis” says that there is no kind of infinity
between that of the Integers and the Reals.
The Cardinality of the Reals is Aleph One
1
Are there bigger infinities than the Reals?
Yes, but that is a little bit more “advanced” topic.
Complex numbers
i = −1
i2 = -1
Complex Numbers are usually written as either
x + iy
Or
𝑒 𝑥+𝑦𝑖
{ equal to 𝑒 𝑥 (cos[y] + i sin[y] ) }
Where both x and y are Real Numbers
Complex numbers
Are there more Complex Numbers than Rationals?
Complex numbers
Are there more Complex Numbers than Reals?
No.
Complex numbers
Example:
Take the Complex Number 123.45 + 67.89i
Map that to a Real number by stating at the decimal point
and alternating between the digits in either half of the
Complex Number
123.45 + 67.89i
16273.4859
Then you can get back to Complex number by stripping off
the digits in reverse order, starting from the decimal point.
So there are just as many Complex Numbers as Reals
Complex numbers
Extra credit
Who can calculate
i
i
Complex numbers
Extra credit
Who can calculate
i
i
𝑒 𝑖𝑥 = cos(x) + isin(x)
So 𝑒 𝑖π/2 = cos(π/2) + isin(π/2)
But cos(π/2) = 0
And isin(π/2) = i
So 𝑒 𝑖π/2 = i
Substituting back into
ii
We get (𝑒 𝑖π/2 ) i =
𝑒 −π/2 =
.207…
Large Numbers
Often represent large numbers in exponential notation, some number
raised to the power of another number.
100 = 1
101 = 10
102 = 100
103 = 1000
104 = 10,000
:
10100 = a Googol (More than the number of atoms in the observable Universe)
:
10Googol = a Googolplex (Largest number I know)
Numbers
M
7/15
1
𝑒
π
X
2
𝑥+𝑦𝑖
X
i
697
0