Quote:
Originally Posted by quexos
Ponder the origin of the Universe, life, sentience, the anthropic principle, the nature of matter and energy, the nature of the time-space continuum, the (probably quantum) physics behind consciousness and sometimes the nature and motives of whatever created the universe if anyone ...
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Set Cardinality.
Some
number conventions:
- P - Prime Numbers: {2,3,5,7,11,13,17,19,...}. Countably Infinite
- N+ - Natural Numbers: {|a| > 0: a ∈ Z}. The positive integers. {1,2,3,...}. Closed under addition and multiplication. Countably Infinite
- N - Whole Numbers: {|a| : a ∈ Z} or {0,1,2,3,...}. Countably Infinite
- Z - Integer Numbers: {p, −p : p ∈ N ∪ {0}} The set of positive and negative integers and zero. {...,-3,-2,-1,0,1,2,3,...} or 0,ą1,ą2,ą3. Closed under addition, multiplication and subtraction. Countably Infinite (die Zahlen)
- Q - Rational Numbers: {p/q : p ∈ Z, q ∈ N} or {a/b|a,b ∈ Z, b ≠ 0} where a is some Integer Number, b some non-zero Natural Number. Closed under addition, subtraction, multiplication and division. 'Ratio'nal comes from the word ratio. Countably Infinite.
- R - Real Numbers. {x : -∞ < x < ∞}. The set of rational numbers Q and the irrational numbers such as pi. It is any number you will find on the number line. Almost all real numbers are irrational as the irrational numbers are uncountable.
- C - Complex Numbers z = a + bi, z = x + iy, in component notation(x,y). {Z : Z = a + bi, -∞ < a < ∞, -∞ < b < ∞} or {a + bi : a,b ∈ R} i = √(−1) ∈ C. They are numbers not found on the number but are viewed as points on a plane.
- H - Hamiltonian Quatermions: {(a,b,c,d)|a,b,c,d ∈ R}. Each number system is a proper subset of the next number system. a1 + bi + cj + dk, where a, b, c, and d are real numbers. i^2 = j^2 = k^2 = ijk = − 1. The basis elements are: 1 = (1,0,0,0), i = (0,1,0,0,), j = (0,0,1,0), k = (0,0,0,1).
Irrational Numbers
Numbers that cannot be written as a ratio of two integers. Cantor defined irrational numbers as limits of convergent sequences of rational numbers; Periodic expansions correspond to rational numbers whereas the non-periodic ones correspond to the irrational numbers. For any integer n, which is not a square of another integer, √n is irrational.
Examples:
- π or pi = 3.1415926535897932384626433832795...
- e or Euler's Number = 2.7182818284590452353602874713527...
The value of the derivative of the function f(x) = e^x at the point x = 0 is equal to 1.
- φ or Golden Ratio = 1.61803398874989484820...
- ζ(3) or Apéry's constant = 1.20205690315959...
- √2 = 1.4142135623730950488016887242097...
- √3 = 1.7320508075688772935274463415059...
Lemma
Quote:
1. Between any two different rational numbers a and b there is at least one other rational number.
2. Between any two different irrational numbers a and b there is at least one other rational number.
3. Between any two different rational numbers a and b there is at least one other irrational number.
4. Between any two different irrational numbers a and b there is at least one other irrational number.
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ln(Z)
Imaginary Numbers
The square root of a negative number and has the form bi where b is a non-zero real number and i is the imaginary unit, defined as i = √-1. i^2 or i × i = −1. √(-x) = i√x. Higher integral powers of i are replaced with −i, 1, i, or −1. Provides at least one root for every polynomial P(x). i^0 = 1, i^1 = i, i^2 = -1, i^3 = -√-1 = -i, i^4 = 1, so i^n = i^n mod 4. bi can be added to a real number a to form a complex number of the form a + bi.
Sets A and B are equivalent, A~B, if there is a one-to-one correspondence or bijection between the two sets. A Set is infinite if it can be placed in a one-to-one correcpondence with a proper subset of itself.
I once had to show that there are different orders or sizes to infiniti according to the
cardinality of Infinite Sets
The Natural numbers {0,1,2,3,...} or
{N, +, .}, where
|N| is a countably infinite set, (א0) aleph-null or
aleph-0 and is one-to-one with the set of Integers.
|Z|=|N|.
The Rational numbers
|Q| =|N| because there exists a
bijective function. There are also the same amount of rational numbers between 0 and 1 or unit fractions
1/n as
|N|. The Real numbers
R is not countably infinite, as we can't construct a bijective function that maps eather
N or
Q to
R. Its cardinality is
C, the
continuum infinity or
aleph-1 or א1. See
Cantor's Diagonal Argument.
My problem: 'How many circles can we draw whose center can be any point on a
cartesian plane Q, having some radius
Q? Short answer:
|Q| = א0. |Q|x|Q|x|Q| = א0 x א0 x א0 = |N|. The answer, in what appreared to be a smaller set then what we started out with, ruffled a lot of feathers, but this is one of the pitfalls when talking about infinity.
The Continuum Hypothesis
Euclid's 3rd postulate:
A circle may be drawn with any given center and any given radius.
Equation of a circle: x^2 + y^2 = r^2.
Conic sections
Glossery
Set Notation
∅ or {} empty set
∈ is an element of
∉ not an element of
⊆ subset
⊂ proper subset
{ : } or { | } the set of … such that
Complex Numbers
Axiomatic Foundations:
Equality (a,b) = (c,d) iff a=c, b=d
Sum (a,b)+(c,d) = (a+c, b+d)
Product (a,b)(c,d) = (ac-bd, ad+bc). m(a,b)
If z1,z2,z3 belong to set S, then:
Closure law z1+z2 and z1z2 belong to S
Identity with respect to addition z1+0 = 0+z1
Commutative law of addition z1+z2 = z2+z1
Associative law of addition z1+(z2+z3) = (z1+z2)+z3
Identity with respect to multiplication 1(z1) = (z1)1
Commutative law of multiplication z1z1 = z2z1
Associative law of multiplication z1(z2z3) = (z1z2)z3
Distributive law of multiplication z1(z2 + z3) = z1z2 + z1z3
Fundamental Operations:
Addition (a + bi) + (c + di) = a + bi + c + di = (a + c) + (b + d)i
Subtraction (a + bi) - (c + di) = a + bi - c - di = (a - c) + (b - d)i
Multiplication (a + bi)(c + di) = ac + adi + bci + adi^2 = (ac - bd) + (ad + bc)i
See also
360° Circle.
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United Arab Emirates
Nerdiest Thing Ever
Get a cup or somethin', man!
).
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