Ok, first of all, you can't count the first event. The first 963 is "free." You didn't say, before the first hand, "What if the next flop is 963 rainbow?" It just happened, like billions of hands of poker dealt before it. Second of all, the SECOND event, despite being 1101:1 against, is free, in that, as a poker player, you sat through thousands of hands before seeing this 1101:1 shot happen, and then you noticed it, and went "oooh, pretty - I wonder if it happens again."

So, once you took note of this 1101:1 oddity, it happened again -- at 1101:1 odds against.

Eleven hundred and one to one, as you describe it.

Or, about 1:1, since it happened.

Only if he takes the odds on it happening before he sees the second flop. He's seen the second flop, and wants odds on the third. We're not counting the other billions of back-to-back flops where he didn't see this "phenomenon".

NO BUT WAT THE ODDS OF THREE TIMES 963 OFFSUIT LOL NOT TWICE

I'm not impressed till someone comes out with a thread where it happens four times.

Who really cares? With a shuffle master there are 2 decks in use, one is being used and the machine is shuffling the other deck for the next hand.

What are the odds of OP starting another useless thread?

in after 50-50 and bout tree fiddy

50/50, either the flop comes 963 rainbow three times in a row or any of not.

Isn't this kind of relative anyway?
Chances that I get dealt AA twice in a row are slim, but what if I play for the rest of my life, everyday, the odds on getting dealt AA twice in a row are significantly higher.
Doesn't this apply to the shuffledealer too? I think you can only measure it (as I think your question wants to be answered) by a certain stretch of time or hands to get a somewhat reliable number.

'bout tree fiddy in wan mirrion?

220:1 for AA

Say you play 5k hands in one day

= around 22 AA being dealt

so you have 22 220:1 shots of being dealt AA twice in a row at any table.

This thread is win. Whenever I think a poster can't make a bigger idiot out of themselves...

in after tree fiddy and close to 50:50

It is 0.78598478551445669754456476546876468774497% because ....
In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1]

i^2=-1.\,

Every complex number can be written in the form a + bi, where a and b are real numbers called the real part and the imaginary part of the complex number, respectively.

Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. These operations extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties, e.g., negative real numbers can be obtained by squaring complex (imaginary) numbers.

Complex numbers were first conceived and defined by the Italian mathematician Girolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.[2] The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, it is always possible to find solutions to polynomial equations of degree one or higher.

The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[3] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.

Complex numbers are used in many different fields including applications in engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial and complex Lie algebra.
Notation

The set of all complex numbers is usually denoted by C, or in blackboard bold by \mathbb{C}.

Although other notations can be used, complex numbers are very often written in the form

a + bi \,

where a and b are real numbers, and i is the imaginary unit, which has the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part.

For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a + ib, the real part a is denoted Re(z) or ℜ(z), and the imaginary part b is denoted Im(z) or ℑ(z).

The real numbers, R, may be regarded as a subset of C by considering every real number a complex number with an imaginary part of zero; that is, the real number a is identified with the complex number a + 0i. Complex numbers with a real part of zero are called imaginary numbers; instead of writing 0 + bi, that imaginary number is usually denoted as just bi. If b equals 1, instead of using 0 + 1i or 1i, the number is denoted as i.

In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j, so complex numbers are sometimes written as a + bj.
Formal development

In a rigorous setting, it is not acceptable to simply assume that there exists a number whose square is -1. The definition must therefore be a little less intuitive, building on the knowledge of real numbers. Write C for R2, the set of ordered pairs of real numbers, and define operations on complex numbers in C according to

(a, b) + (c, d) = (a + c, b + d)
(a, b)·(c, d) = (a·c − b·d, b·c + a·d)

Since (0, 1)·(0, 1) = (−1, 0), we have found i by constructing it, not postulating it. We can associate the numbers (a, 0) with the real numbers, and write i = (0, 1). It is then just a matter of notation to express (a, b) as a + ib.

[edit] Equality

Two complex numbers are said to be equal if and only if their real parts are equal and their imaginary parts are equal. In other words, if the two complex numbers are written as a + bi and c + di with a, b, c, and d real, then they are equal if and only if a = c and b = d. This is an equivalence relation.

[edit] Operations

Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i 2 = −1:

* Addition: \,(a + bi) + (c + di) = (a + c) + (b + d)i
* Subtraction: \,(a + bi) - (c + di) = (a - c) + (b - d)i
* Multiplication: \,(a + bi) (c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad)i
* Division: \,\frac{(a + bi)}{(c + di)} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i\,,

where c and d are not both zero. This is obtained by multiplying both the numerator and the denominator with the complex conjugate of the denominator.

Since the complex number a + bi is uniquely specified by the ordered pair (a, b), the complex numbers are in one-to-one correspondence with points on a plane. This complex plane is described below.

[edit] Elementary functions

There are also other elementary functions that can be applied to complex functions. Here are some of the most essential:

\sin(x+iy)=\sin(x)\cosh(y)+i(\cos(x)\sinh(y)).\,
\cos(x+iy)=\cos(x)\cosh(y)-i(\sin(x)\sinh(y)).\,
\ln(x+iy)=(\ln(x^2+y^2))/2+i(\arctan(y/x)).\,

Given that x and y are real numbers.

[edit] The field of complex numbers

A field is an algebraic structure with addition, subtraction, multiplication, and division operations that satisfy certain algebraic laws. The complex numbers form a field, known as the complex number field, denoted by C. In particular, this means that the complex numbers possess:

* An additive identity ("zero"), 0 + 0i.
* A multiplicative identity ("one"), 1 + 0i.
* An additive inverse of every complex number. The additive inverse of a + bi is −a − bi.
* A multiplicative inverse (reciprocal) of every nonzero complex number. The multiplicative inverse of a + bi is {a\over a^2+b^2}+ \left( {-b\over a^2+b^2}\right)i.

Examples of other fields are the real numbers and the rational numbers. When each real number a is identified with the complex number a + 0i, the field of real numbers R becomes a subfield of C.

The complex numbers C can also be characterized as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below.
Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = reiφ is defined as | z | = r. Algebraically, if z = x + yi, then |z|=\sqrt{x^2+y^2}.

The absolute value has three important properties:

| z | \geq 0, \, where | z | = 0 \, if and only if z = 0 \,

| z + w | \leq | z | + | w | \, (triangle inequality)

| z \cdot w | = | z | \cdot | w | \,

for all complex numbers z and w. These imply that | 1 | = 1 and | z / w | = | z | / | w | . By defining the distance function d(z,w) = | z − w | , we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.

The complex conjugate of the complex number z = x + yi is defined to be x − yi, written as \bar{z} or z^*\,. As seen in the figure, \bar{z} is the "reflection" of z about the real axis, and so both z+\bar{z} and z\cdot\bar{z} are real numbers. Many identities relate complex numbers and their conjugates:

\overline{z+w} = \bar{z} + \bar{w}

\overline{z\cdot w} = \bar{z}\cdot\bar{w}

\overline{(z/w)} = \bar{z}/\bar{w}

\bar{\bar{z}}=z

\bar{z}=z if and only if z is real

\bar{z}=-z if and only if z is purely imaginary

\operatorname{Re}\,(z) = \tfrac{1}{2}(z+\bar{z})

\operatorname{Im}\,(z) = \tfrac{1}{2i}(z-\bar{z})

|z|=|\bar{z}|

|z|^2 = z\cdot\bar{z}

z^{-1} = \frac{\bar{z}}{|z|^{2}} if z is non-zero.

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

That conjugation distributes over all the algebraic operations (and many functions; e.g. \sin\bar z=\overline{\sin z}) is rooted in the ambiguity in choice of i (−1 has two square roots). It is important to note, however, that the function f(z) = \bar{z} is not complex-differentiable (see holomorphic function).
Polar form
The angle φ and distance r locate a point on an Argand diagram.

Alternatively to the cartesian representation z = x+iy, the complex number z can be specified by polar coordinates. The polar coordinates are r = |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument or the angle of z. The representation of a complex number by its polar coordinates is called the polar form of the complex number.

For r = 0 any value of φ describes the same complex number z = 0. To get a unique representation, a conventional choice is to set φ = 0. For r > 0 the argument φ is unique modulo 2π; that is, if any two values of the complex argument differ by an exact integer multiple of 2π, they are considered equivalent. To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], i.e. −π < φ ≤ π. This choice of φ is sometimes called the principal value of arg(z).

[edit] Conversion from the polar form to the Cartesian form

x = r \cos \varphi
y = r \sin \varphi

[edit] Conversion from the Cartesian form to the polar form

r = |z| = \sqrt{x^2+y^2}
\varphi = \arg(z) = \operatorname{atan2}(y,x)

The value of φ can change by any multiple of 2π and still give the same angle. The function atan2 gives the principal value in the range (−π, +π]. If a non-negative value of φ in the range [0, 2π) is desired, add 2π to any negative value.

The arg function is sometimes considered as multivalued taking as possible values atan2(y, x) + 2πk, where k is any integer.

[edit] Notation of the polar form

The notation of the polar form as

z = r\,(\cos \varphi + i\sin \varphi )\,

is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ. Using Euler's formula it can also be written as

z = r\,\mathrm{e}^{i \varphi}\,

which is called exponential form.

In electronics it is common to use angle notation to represent a phasor with amplitude A and phase θ as:

A \ang \theta = A e ^ {j \theta }

Where θ may be in either radians or degrees. In electronics j is used instead of i because i is used for electric current.

[edit] Multiplication, division, exponentiation, and root extraction in the polar form

Multiplication, division, exponentiation, and root extraction have simple formulas in polar form.

Using sum and difference identities it follows that

r_1\,e^{i\varphi_1} \cdot r_2\,e^{i\varphi_2} = r_1\,r_2\,e^{i(\varphi_1 + \varphi_2)} \,

and that

\frac{r_1\,e^{i\varphi_1}}{r_2\,e^{i\varphi_2}} = \frac{r_1}{r_2}\,e^{i (\varphi_1 - \varphi_2)}. \,

Exponentiation with integer exponents; according to De Moivre's formula,

(\cos\varphi + i\sin\varphi)^n = \cos(n\varphi) + i\sin(n\varphi),\,

from which it follows that

(r(\cos\varphi + i\sin\varphi))^n = (r\,e^{i\varphi})^n = r^n\,e^{in\varphi} = r^n\,(\cos n\varphi + \mathrm{i} \sin n \varphi).\,

Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation.

Multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching, in particular multiplication by i corresponds to a counter-clockwise rotation by 90 degrees (π/2 radians). The geometric content of the equation i 2 = −1 is that a sequence of two 90 degree rotations results in a 180 degree (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

If c is a complex number and n a positive integer, then any complex number z satisfying zn = c is called an n-th root of c. If c is nonzero, there are exactly n distinct n-th roots of c, which can be found as follows. Write c=re^{i\varphi} with real numbers r > 0 and φ, then the set of n-th roots of c is

\{ \sqrt[n]r\,e^{i(\frac{\varphi+2k\pi}{n})} \mid k\in\{0,1,\ldots,n-1\} \, \},

where \sqrt[n]{r} represents the usual (positive) n-th root of the positive real number r. If c = 0, then the only n-th root of c is 0 itself, which as n-th root of 0 is considered to have multiplicity n.

[edit] Some properties

[edit] Matrix representation of complex numbers

While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form

\begin{bmatrix} a & -b \\ b & \;\; a \end{bmatrix}

where a and b are real numbers. The sum and product of two such matrices is again of this form, and the product operation on matrices of this form is commutative. Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field, isomorphic to the field of complex numbers. Every such matrix can be written as

\begin{bmatrix} a & -b \\ b & \;\; a \end{bmatrix} = a \begin{bmatrix} 1 & \;\; 0 \\ 0 & \;\; 1 \end{bmatrix} + b \begin{bmatrix} 0 & -1 \\ 1 & \;\; 0 \end{bmatrix}

which suggests that we should identify the real number 1 with the identity matrix

\begin{bmatrix} 1 & \;\; 0 \\ 0 & \;\; 1 \end{bmatrix},

and the imaginary unit i with

\begin{bmatrix} 0 & -1 \\ 1 & \;\; 0 \end{bmatrix},

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.

The square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix.

|z|^2 = \begin{vmatrix} a & -b \\ b & a \end{vmatrix} = (a^2) - ((-b)(b)) = a^2 + b^2.

If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z.

If the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions. In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras.

It should also be noted that the two eigenvalues of the 2x2 matrix representing a complex number are the complex number itself and its conjugate.

[edit] Real vector space

C is a two-dimensional real vector space. Unlike the reals, the set of complex numbers cannot be totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. More generally, no field containing a square root of −1 can be ordered.

R-linear maps C → C have the general form

f(z)=az+b\overline{z}

with complex coefficients a and b. Only the first term is C-linear, and only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations.

The function

f(z)=az\,

corresponds to rotations combined with scaling, while the function

f(z)=b\overline{z}

corresponds to reflections combined with scaling.

[edit] Solutions of polynomial equations

A root of the polynomial p is a complex number z such that p(z) = 0. A surprising result in complex analysis is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and it shows that the complex numbers are an algebraically closed field. Indeed, the complex numbers are the algebraic closure of the real numbers, as described below.

[edit] Construction and algebraic characterization

One construction of C is as a field extension of the field R of real numbers, in which a root of x2+1 is added. To construct this extension, begin with the polynomial ring R[x] of the real numbers in the variable x. Because the polynomial x2+1 is irreducible over R, the quotient ring R[x]/(x2+1) will be a field. This extension field will contain two square roots of -1; one of them is selected and denoted i. The set {1, i} will form a basis for the extension field over the reals, which means that each element of the extension field can be written in the form a+ b·i. Equivalently, elements of the extension field can be written as ordered pairs (a,b) of real numbers.

Although only roots of x2+1 were explicitly added, the resulting complex field is actually algebraically closed – every polynomial with coefficients in C factors into linear polynomials with coefficients in C. Because each field has only one algebraic closure, up to field isomorphism, the complex numbers can be characterized as the algebraic closure of the real numbers.

The field extension does yield the well-known complex plane, but it only characterizes it algebraically. The field C is characterized up to field isomorphism by the following three properties:

* it has characteristic 0
* its transcendence degree over the prime field is the cardinality of the continuum
* it is algebraically closed

One consequence of this characterization is that C contains many proper subfields which are isomorphic to C (the same is true of R, which contains many subfields isomorphic to itself). As described below, topological considerations are needed to distinguish these subfields from the fields C and R themselves.

[edit] Characterization as a topological field

As just noted, the algebraic characterization of C fails to capture some of its most important topological properties. These properties are key for the study of complex analysis, where the complex numbers are studied as a topological field.

The following properties characterize C as a topological field:[citation needed]

* C is a field.
* C contains a subset P of nonzero elements satisfying:
o P is closed under addition, multiplication and taking inverses.
o If x and y are distinct elements of P, then either x-y or y-x is in P
o If S is any nonempty subset of P, then S+P=x+P for some x in C.
* C has a nontrivial involutive automorphism x→x*, fixing P and such that xx* is in P for any nonzero x in C.

Given a field with these properties, one can define a topology by taking the sets

* B(x,p) = \{y | p - (y-x)(y-x)^*\in P\}

as a base, where x ranges over the field and p ranges over P.

To see that these properties characterize C as a topological field, one notes that P ∪ {0} ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization.

Pontryagin has shown that the only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R by noting that the nonzero complex numbers are connected, while the nonzero real numbers are not.

0.78598478551445669754456476546876468774497% because ....
In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1]

i^2=-1.\,

Every complex number can be written in the form a + bi, where a and b are real numbers called the real part and the imaginary part of the complex number, respectively.

Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. These operations extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties, e.g., negative real numbers can be obtained by squaring complex (imaginary) numbers.

Complex numbers were first conceived and defined by the Italian mathematician Girolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.[2] The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, it is always possible to find solutions to polynomial equations of degree one or higher.

The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[3] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.

Complex numbers are used in many different fields including applications in engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial and complex Lie algebra.
Notation

The set of all complex numbers is usually denoted by C, or in blackboard bold by \mathbb{C}.

Although other notations can be used, complex numbers are very often written in the form

a + bi \,

where a and b are real numbers, and i is the imaginary unit, which has the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part.

For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a + ib, the real part a is denoted Re(z) or ℜ(z), and the imaginary part b is denoted Im(z) or ℑ(z).

The real numbers, R, may be regarded as a subset of C by considering every real number a complex number with an imaginary part of zero; that is, the real number a is identified with the complex number a + 0i. Complex numbers with a real part of zero are called imaginary numbers; instead of writing 0 + bi, that imaginary number is usually denoted as just bi. If b equals 1, instead of using 0 + 1i or 1i, the number is denoted as i.

In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j, so complex numbers are sometimes written as a + bj.
Formal development

In a rigorous setting, it is not acceptable to simply assume that there exists a number whose square is -1. The definition must therefore be a little less intuitive, building on the knowledge of real numbers. Write C for R2, the set of ordered pairs of real numbers, and define operations on complex numbers in C according to

(a, b) + (c, d) = (a + c, b + d)
(a, b)·(c, d) = (a·c − b·d, b·c + a·d)

Since (0, 1)·(0, 1) = (−1, 0), we have found i by constructing it, not postulating it. We can associate the numbers (a, 0) with the real numbers, and write i = (0, 1). It is then just a matter of notation to express (a, b) as a + ib.

[edit] Equality

Two complex numbers are said to be equal if and only if their real parts are equal and their imaginary parts are equal. In other words, if the two complex numbers are written as a + bi and c + di with a, b, c, and d real, then they are equal if and only if a = c and b = d. This is an equivalence relation.

[edit] Operations

Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i 2 = −1:

* Addition: \,(a + bi) + (c + di) = (a + c) + (b + d)i
* Subtraction: \,(a + bi) - (c + di) = (a - c) + (b - d)i
* Multiplication: \,(a + bi) (c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad)i
* Division: \,\frac{(a + bi)}{(c + di)} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i\,,

where c and d are not both zero. This is obtained by multiplying both the numerator and the denominator with the complex conjugate of the denominator.

Since the complex number a + bi is uniquely specified by the ordered pair (a, b), the complex numbers are in one-to-one correspondence with points on a plane. This complex plane is described below.

[edit] Elementary functions

There are also other elementary functions that can be applied to complex functions. Here are some of the most essential:

\sin(x+iy)=\sin(x)\cosh(y)+i(\cos(x)\sinh(y)).\,
\cos(x+iy)=\cos(x)\cosh(y)-i(\sin(x)\sinh(y)).\,
\ln(x+iy)=(\ln(x^2+y^2))/2+i(\arctan(y/x)).\,

Given that x and y are real numbers.

[edit] The field of complex numbers

A field is an algebraic structure with addition, subtraction, multiplication, and division operations that satisfy certain algebraic laws. The complex numbers form a field, known as the complex number field, denoted by C. In particular, this means that the complex numbers possess:

* An additive identity ("zero"), 0 + 0i.
* A multiplicative identity ("one"), 1 + 0i.
* An additive inverse of every complex number. The additive inverse of a + bi is −a − bi.
* A multiplicative inverse (reciprocal) of every nonzero complex number. The multiplicative inverse of a + bi is {a\over a^2+b^2}+ \left( {-b\over a^2+b^2}\right)i.

Examples of other fields are the real numbers and the rational numbers. When each real number a is identified with the complex number a + 0i, the field of real numbers R becomes a subfield of C.

The complex numbers C can also be characterized as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below.
Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = reiφ is defined as | z | = r. Algebraically, if z = x + yi, then |z|=\sqrt{x^2+y^2}.

The absolute value has three important properties:

| z | \geq 0, \, where | z | = 0 \, if and only if z = 0 \,

| z + w | \leq | z | + | w | \, (triangle inequality)

| z \cdot w | = | z | \cdot | w | \,

for all complex numbers z and w. These imply that | 1 | = 1 and | z / w | = | z | / | w | . By defining the distance function d(z,w) = | z − w | , we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.

The complex conjugate of the complex number z = x + yi is defined to be x − yi, written as \bar{z} or z^*\,. As seen in the figure, \bar{z} is the "reflection" of z about the real axis, and so both z+\bar{z} and z\cdot\bar{z} are real numbers. Many identities relate complex numbers and their conjugates:

\overline{z+w} = \bar{z} + \bar{w}

\overline{z\cdot w} = \bar{z}\cdot\bar{w}

\overline{(z/w)} = \bar{z}/\bar{w}

\bar{\bar{z}}=z

\bar{z}=z if and only if z is real

\bar{z}=-z if and only if z is purely imaginary

\operatorname{Re}\,(z) = \tfrac{1}{2}(z+\bar{z})

\operatorname{Im}\,(z) = \tfrac{1}{2i}(z-\bar{z})

|z|=|\bar{z}|

|z|^2 = z\cdot\bar{z}

z^{-1} = \frac{\bar{z}}{|z|^{2}} if z is non-zero.

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

That conjugation distributes over all the algebraic operations (and many functions; e.g. \sin\bar z=\overline{\sin z}) is rooted in the ambiguity in choice of i (−1 has two square roots). It is important to note, however, that the function f(z) = \bar{z} is not complex-differentiable (see holomorphic function).
Polar form
The angle φ and distance r locate a point on an Argand diagram.

Alternatively to the cartesian representation z = x+iy, the complex number z can be specified by polar coordinates. The polar coordinates are r = |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument or the angle of z. The representation of a complex number by its polar coordinates is called the polar form of the complex number.

For r = 0 any value of φ describes the same complex number z = 0. To get a unique representation, a conventional choice is to set φ = 0. For r > 0 the argument φ is unique modulo 2π; that is, if any two values of the complex argument differ by an exact integer multiple of 2π, they are considered equivalent. To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], i.e. −π < φ ≤ π. This choice of φ is sometimes called the principal value of arg(z).

[edit] Conversion from the polar form to the Cartesian form

x = r \cos \varphi
y = r \sin \varphi

[edit] Conversion from the Cartesian form to the polar form

r = |z| = \sqrt{x^2+y^2}
\varphi = \arg(z) = \operatorname{atan2}(y,x)

The value of φ can change by any multiple of 2π and still give the same angle. The function atan2 gives the principal value in the range (−π, +π]. If a non-negative value of φ in the range [0, 2π) is desired, add 2π to any negative value.

The arg function is sometimes considered as multivalued taking as possible values atan2(y, x) + 2πk, where k is any integer.

[edit] Notation of the polar form

The notation of the polar form as

z = r\,(\cos \varphi + i\sin \varphi )\,

is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ. Using Euler's formula it can also be written as

z = r\,\mathrm{e}^{i \varphi}\,

which is called exponential form.

In electronics it is common to use angle notation to represent a phasor with amplitude A and phase θ as:

A \ang \theta = A e ^ {j \theta }

Where θ may be in either radians or degrees. In electronics j is used instead of i because i is used for electric current.

[edit] Multiplication, division, exponentiation, and root extraction in the polar form

Multiplication, division, exponentiation, and root extraction have simple formulas in polar form.

Using sum and difference identities it follows that

r_1\,e^{i\varphi_1} \cdot r_2\,e^{i\varphi_2} = r_1\,r_2\,e^{i(\varphi_1 + \varphi_2)} \,

and that

\frac{r_1\,e^{i\varphi_1}}{r_2\,e^{i\varphi_2}} = \frac{r_1}{r_2}\,e^{i (\varphi_1 - \varphi_2)}. \,

Exponentiation with integer exponents; according to De Moivre's formula,

(\cos\varphi + i\sin\varphi)^n = \cos(n\varphi) + i\sin(n\varphi),\,

from which it follows that

(r(\cos\varphi + i\sin\varphi))^n = (r\,e^{i\varphi})^n = r^n\,e^{in\varphi} = r^n\,(\cos n\varphi + \mathrm{i} \sin n \varphi).\,

Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation.

Multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching, in particular multiplication by i corresponds to a counter-clockwise rotation by 90 degrees (π/2 radians). The geometric content of the equation i 2 = −1 is that a sequence of two 90 degree rotations results in a 180 degree (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

If c is a complex number and n a positive integer, then any complex number z satisfying zn = c is called an n-th root of c. If c is nonzero, there are exactly n distinct n-th roots of c, which can be found as follows. Write c=re^{i\varphi} with real numbers r > 0 and φ, then the set of n-th roots of c is

\{ \sqrt[n]r\,e^{i(\frac{\varphi+2k\pi}{n})} \mid k\in\{0,1,\ldots,n-1\} \, \},

where \sqrt[n]{r} represents the usual (positive) n-th root of the positive real number r. If c = 0, then the only n-th root of c is 0 itself, which as n-th root of 0 is considered to have multiplicity n.

[edit] Some properties

[edit] Matrix representation of complex numbers

While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form

\begin{bmatrix} a & -b \\ b & \;\; a \end{bmatrix}

where a and b are real numbers. The sum and product of two such matrices is again of this form, and the product operation on matrices of this form is commutative. Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field, isomorphic to the field of complex numbers. Every such matrix can be written as

\begin{bmatrix} a & -b \\ b & \;\; a \end{bmatrix} = a \begin{bmatrix} 1 & \;\; 0 \\ 0 & \;\; 1 \end{bmatrix} + b \begin{bmatrix} 0 & -1 \\ 1 & \;\; 0 \end{bmatrix}

which suggests that we should identify the real number 1 with the identity matrix

\begin{bmatrix} 1 & \;\; 0 \\ 0 & \;\; 1 \end{bmatrix},

and the imaginary unit i with

\begin{bmatrix} 0 & -1 \\ 1 & \;\; 0 \end{bmatrix},

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.

The square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix.

|z|^2 = \begin{vmatrix} a & -b \\ b & a \end{vmatrix} = (a^2) - ((-b)(b)) = a^2 + b^2.

If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z.

If the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions. In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras.

It should also be noted that the two eigenvalues of the 2x2 matrix representing a complex number are the complex number itself and its conjugate.

[edit] Real vector space

C is a two-dimensional real vector space. Unlike the reals, the set of complex numbers cannot be totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. More generally, no field containing a square root of −1 can be ordered.

R-linear maps C → C have the general form

f(z)=az+b\overline{z}

with complex coefficients a and b. Only the first term is C-linear, and only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations.

The function

f(z)=az\,

corresponds to rotations combined with scaling, while the function

f(z)=b\overline{z}

corresponds to reflections combined with scaling.

[edit] Solutions of polynomial equations

A root of the polynomial p is a complex number z such that p(z) = 0. A surprising result in complex analysis is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and it shows that the complex numbers are an algebraically closed field. Indeed, the complex numbers are the algebraic closure of the real numbers, as described below.

[edit] Construction and algebraic characterization

One construction of C is as a field extension of the field R of real numbers, in which a root of x2+1 is added. To construct this extension, begin with the polynomial ring R[x] of the real numbers in the variable x. Because the polynomial x2+1 is irreducible over R, the quotient ring R[x]/(x2+1) will be a field. This extension field will contain two square roots of -1; one of them is selected and denoted i. The set {1, i} will form a basis for the extension field over the reals, which means that each element of the extension field can be written in the form a+ b·i. Equivalently, elements of the extension field can be written as ordered pairs (a,b) of real numbers.

Although only roots of x2+1 were explicitly added, the resulting complex field is actually algebraically closed – every polynomial with coefficients in C factors into linear polynomials with coefficients in C. Because each field has only one algebraic closure, up to field isomorphism, the complex numbers can be characterized as the algebraic closure of the real numbers.

The field extension does yield the well-known complex plane, but it only characterizes it algebraically. The field C is characterized up to field isomorphism by the following three properties:

* it has characteristic 0
* its transcendence degree over the prime field is the cardinality of the continuum
* it is algebraically closed

One consequence of this characterization is that C contains many proper subfields which are isomorphic to C (the same is true of R, which contains many subfields isomorphic to itself). As described below, topological considerations are needed to distinguish these subfields from the fields C and R themselves.

[edit] Characterization as a topological field

As just noted, the algebraic characterization of C fails to capture some of its most important topological properties. These properties are key for the study of complex analysis, where the complex numbers are studied as a topological field.

The following properties characterize C as a topological field:[citation needed]

* C is a field.
* C contains a subset P of nonzero elements satisfying:
o P is closed under addition, multiplication and taking inverses.
o If x and y are distinct elements of P, then either x-y or y-x is in P
o If S is any nonempty subset of P, then S+P=x+P for some x in C.
* C has a nontrivial involutive automorphism x→x*, fixing P and such that xx* is in P for any nonzero x in C.

Given a field with these properties, one can define a topology by taking the sets

* B(x,p) = \{y | p - (y-x)(y-x)^*\in P\}

as a base, where x ranges over the field and p ranges over P.

To see that these properties characterize C as a topological field, one notes that P ∪ {0} ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization.

Pontryagin has shown that the only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R by noting that the nonzero complex numbers are connected, while the nonzero real numbers are not.

//End Thread

What is the deal with quoting in different colours i missed the truck on that one

winnar ainec.

short bus imo

Playing 10/20 LIMIT Hold'em

Flop comes out A23, hand plays out.

Next hand flop comes out 456. Someone points this out and says "lol, looks like the deck is counting up! I Wonder if it will keep going?".

Next hand flop comes out 789 and the guy literally flips out. Just starts going off on how the last two flops were couting up before this one! Some guy at the table says "I'll bet you my whole stack that the next flop doesn't keep the count going". The guy says ok, but I want odds. He says, "Your $10 to my $1500" (he had about 1700 behind). He says ok.

So in suspense, they look as the next flop comes out:



Bam! Buddy immediately ships $1500 to the guy and couldn't believe it. "What are the chances?!?" He says.

That cat is pretty cool

+1

lmfao i nearly missed this from all the ****ty quoting

lol good one

is that sklansky?!?!?

I won't bother you guys with the complicating mathematical formulas, but the answer is 50/50