1. Danville Senior Center May 5, 2016
Math Club Workshop
Danville Senior Center May 5, 2016

2. 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.

3. 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

4. i The plan…sort of… i 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 i i

5. Numbers π
7/15 X
697 i 2

6. Integers

7. Integers

8. Integers Symbols

9. Integers

10. Integers

11. Integers

12. Integers

13. Integers Group Symbols

14. Integers

15. Integers

16. Integers Roman NumeralsV X L

17. Integers
LXXXIII

18. Integers M D C L X V I ?
MMXVI

19. Integers M C L X V M D C L X V I X ˄ I
Etruscan X ↓
Egyptian

20. Integers ?

21. Integers Zero

22. 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.

23. 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.

24. Integers Decimal Numbering System
I II III IV V VI VII VIII IX Groups of … X C M

25. Integers
945,389,234,703,662

26. 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 public-key cryptography (RSA), which makes use of properties such as the difficulty of factoring large numbers into their prime factors.
Source: Wikipedia

27. Integers Binary, Octal, and Hexadecimal
Groups of 2
Octal Groups of 8
Hexadecimal A B C D E F
Groups of 16

28. 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?

29. More Numbers
Integers
Rationals (Fractions) 1/3 7/ /15 (includes Integers)
Reals π e (includes Rationals)
Complex i i 𝑒 2+3𝑖 (includes Reals)

30. Rationals (Fractions, including Integers)
1/ /3 7/ / /5872
Are there more Rationals than Integers?

31. Rationals
Are there more Rationals than Integers?
No.

32. Rationals All the possible Rationals

33. Rationals All the possible Rationals

34. Reals
Are there numbers that are not Rationals?

35. Reals
The set of Real numbers includes the Rationals, but also has numbers that are NOT fractions.
Examples are: π e … … Any Prime number …

36. 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 is NOT a Rational

37. Reals 1 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
Are there bigger infinities than the Reals?
Yes, but that is a little bit more “advanced” topic. 1

38. 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

39. Complex numbers
Are there more Complex Numbers than Rationals?

40. Complex numbers
Are there more Complex Numbers than Reals?
No.

41. Complex numbers 16273.4859 Example:
Take the Complex Number i
Map that to a Real number by stating at the decimal point and alternating between the digits in either half of the Complex Number i
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

42. Complex numbers
Extra credit
Who can calculate i

43. i i Complex numbers i Extra credit Who can calculate
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 i i
Substituting back into
We get (𝑒 𝑖π/2 ) i =
𝑒 −π/2 =
.207…

44. 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)

45. Numbers M X π
7/15 X
697 i 2 1
𝑒 𝑥+𝑦𝑖