- Square roots of -1:
i×i=-1 (product of the imaginary unit and itself is minus 1, i.e. the real unit counterpart)
(-i)×(-i)=-1 (product of the imaginary unit counterpart and itself is also minus 1)
i×(-i)=(-i)×i=1 (product of the imaginary unit and its counterpart is 1, i.e. the real unit)
i×i×i=-i (three-time product, or cubic potence, of the imaginary unit is its counterpart)
(-i)×(-i)×(-i)=i (three-time product, or cubic potence, of the imaginary unit counterpart is the imaginary unit). - Cubic roots of 1:
(cos 120° + i×sin 120°)×(cos 120° + i×sin 120°) = cos (-120°) + i×sin (-120°) (product of the 1st complex cubic root of 1 and itself is the other complex cubic root of 1, its conjugated number)
(cos (-120°) + i×sin (-120°))×(cos (-120°) + i×sin (-120°)) = cos 120° + i×sin 120° (product of the 1st complex cubic root of 1 and itself is the other complex cubic root of 1, its conjugated number)
(cos 120° + i×sin 120°)×(cos 120° + i×sin 120°)×(cos 120° + i×sin 120°)=1
(cos (-120°) + i×sin (-120°))×(cos (-120°) + i×sin (-120°))×(cos (-120°) + i×sin (-120°))=1 (three-time product, or cubic potence, of the 1st cubic root of 1 is 1, i.e. the real unit, as well as that of the 2nd cubic root of 1).
November 29, 2009
Some identifications of complex numbers
October 28, 2009
Kinds of number | Nummerarten
There are several kinds of number which are shown below:
Es gibt verschiedene Arten vom Nummer die unten genannt werden:
- Ordinary numbers: for the order in a list. | Ordnungszahlen: um die Reihenfolge zu bezeichnen.
EN first, second, third, fourth, fifth, … , tenth, … , fifteenth, … , twentieth, … , hundredth, …
DE erst-, zweit-, dritt-, viert-, fünft-, … , zehnt-, … , fünfzehnt-, … , zwanzigst-, … , hundertst-, … - Cardinal numbers: for the quantity. | Kardinalzahlen: um die Anzahl zu nennen.
EN (zero,) one, two, three, four, five, … , ten, … , fifteen, … , twenty, … , one hundred, …
DE (null,) eins, zwei, drei, vier, fünf, …, zehn, … , fünfzehn, … , zwanzig, … , hundert, … - Fractional numbers: for a part of the unit. | Bruchzahlen , Teilzahlen: um ein Teil von eins zu bezeichnen.
EN half, third, fourth, fifth, … , tenth, … , fifteenth, … , twentieth, … , hundredth, …
DE Halbe, Drittel, Viertel, Fünftel, … , Zehntel, … , Fünfzehntel, … , Zwanzigstel, … , Hundertstel, … - Frequency numbers: the number of repetations. | Frequenzzahlen: die Anzahl der Wiederholungen.
EN once, twice, three times, four times, five times, … , ten times, … , fifteen times, … , twenty times, … , one hundred times, …
DE einmal, zweimal, dreimal, viermal, fünfmal, … , zehnmal, … , fünfzehnmal, … , zwanzigmal, … , hundertmal, … - Multiple numbers: the number of units. | Mehrfachzahlen: die Anzahl der Einheiten.
EN (simple,) double, triple, quadruple, …
DE (einfach,) doppel, dreifach, vierfach, fünffach, … , zehnfach, … - Grouping numbers | Gruppierungsnummer
EN []
DE zu zweit, zu dritt, zu viert, …
[FR, IT a 2, a 3, a 4, ...] - Grading numbers | Bestufungsnummer
EN primary, secondary, tertiary, …
DE primär, sekundär, tertiär, … - Marking numbers: arbitrary for use | Kennzeichnungsnummer: frei für Verwendung.
EN no. 1, no. 2, no. 3, …
DE Nr. 1, Nr. 2, Nr. 3, …
Except for the marking numbers, there are mathematical operations (calculations/computations) for each kind of number, which are different for this kind or that kind.
Ausgenommen die Kennzeichnungsnummer, es gibt mathematische Operationen (Berechnungen) für jede Nummerart, die verschieden sind für diese oder jene Art.
October 27, 2009
Two ways of counting
Let say we have a list, a sequence or series of things of the same kind (for example a batch of pens). Now we count these things.
- One way to count is that, we take one and count it as the 1st, then the next as the 2nd, then the 3rd and so on. The ordinary numbers here mean that these are counted from one member of the group (the 1st one).
- Another way to count is that, after one we take the next and count it as the 1st, then the 2nd next, then the 3rd next and so on. The ordinary numbers here mean that these are counted after one member of the group (‘the 0th one’).
Now we group the things into groups of each n members (for example 3, 4, 5 or 10 etc.). When we count from the 1st one, the n-th, 2×n-th, 3×n-th, … members are the last one of each group. When we count after the 0th one, the n-th, 2×n-th, 3×n-th, … members are the start one of each group, which contains except for these members the members of the next group in the counting way above.
September 25, 2009
Which function increases fastest?
Among the well-known functions the exponential function is a function which increases its value very fast. An exponential function always increases faster than a polynome.
However there is another function which increases faster than the exponential functions, that is the factorial function (i.e. n!). The factorial increases faster than any exponential function. The extrapolation of the factorial for all real numbers is the gamma function ( Γ(x) ).
Is there a function which increases faster than any other function? The answer is no. For every function f(x) → ∞ there is at least another function which grows faster than it. Examples can be chosen among the composed functions:
- f(P(x)) and P(f(x)): the function at a polynomial and polynomial of the function
- f(exp(x)) and exp(f(x)): the function at the exponent and exponent of the function
- f(Γ(x)) and Γ(f(x)): the function at the gamma function and the gamma function of the function in discussion
We may say that somewhat which can not be got over is a “function” which is infinity (∞) from a value (of the argument). However infinity is not a number and therefore this is not a function in the normal meaning.
Another thing is that whether these functions make sense or they are just only a catastrophe. That is also the problem with the seek for the fast increasing functions.
September 22, 2009
(Small) problem of the day
- Derivatives of a function of one variable: indicate the dependence of the function (value) on the variable.
- y = f(x) then dy/dx = f ’(x).
Second and higher derivatives: the function is derived two or more times.
- d²y/dx² ie. d/dx (dy/dx) = f ”(x)
- d³y/dx³ ie. d/dx (d/dx (dy/dx)) = f ”’(x)
- etc.
- Partial derivatives of a function of two or more variables: indicate the dependence of the function (value) on each variable or on some variables.
- u=f(x,y) then ∂u/∂x = ∂/∂x f(x,y), ∂u/∂y = ∂/∂y f(x,y). Note that for the partial derivative to one variable, the other variables are kept constant.
Second and higher derivatives: the function is derived two or more times to one variable or to an order of variables.
- ∂²u/∂x² ie. ∂/∂x (∂u/∂x) = ∂²/∂x² f(x,y)
- ∂²u/∂y² ie. ∂/∂y (∂u/∂y) = ∂²/∂y² f(x,y)
- ∂²u/∂x∂y ie. ∂/∂x (∂u/∂y) = ∂²/∂x∂y f(x,y) – The partial derivate of u to y is partially derived to x.
- ∂²u/∂y∂x ie. ∂/∂y (∂u/∂x) = ∂²/∂y∂x f(x,y) – The partial derivate of u to x is partially derived to y. If f(x,y) is a continuous function with continuous derivatives upto 2nd order then this derivative is equal to the derivative above (Theorem of Schwartz).
- Directional derivatives: for the case the variables of a function with several variables are changed in a process with one parameter the directional derivatives indicate the dependence of the function (value) to the parameter of the process.
- u = f(x,y) with x = g(t), y = h(t) then du/dt = ∂/∂x f(x,y) · g’(t) + ∂/∂y f(x,y) · h’(t) – sum of dependence over x and dependence over y. Dependence over x is the product of the partial derivative of u to x and the derivative of x to t (Rule of chain). The same for y.
Second and higher derivatives:
- d²u/dt² = d/dt (du/dt) – the function is derived 2 times in this way.
- d³u/dt³ = d/dt (d/dt (du/dt)) – the function is derived 3 times in this way.
- etc. There is a formula for the exposition of the higher derivatives using Newton binomial coefficients.
- Netto and brutto derivatives: for the case some variables of a function with several variables depend on one variable of this function. In this case one variable can not be changed while keeping other variables constant.
- u = f(x,y,z) with y = g(x), z = h(x)
The netto derivatives of u to x, y, z are the direct derivatives of u to each of them.
- ∂u/∂x = ∂/∂x f(x,y,z), ∂u/∂y = ∂/∂y f(x,y,z), ∂u/∂z = ∂/∂z f(x,y,z) – the values of the dependent variables and the independent variable are put into the partial derivatives of f(x,y,z).
The brutto derivatives of u to x is the sum of the direct dependence of u on x and the indirect dependences over y and z.
- du/dt = ∂/∂x f(x,y,z) + ∂/∂y f(x,y,z) · g’(x) + ∂/∂z f(x,y,z) · h’(x) – the chain rule is also applied here.
- This concept may also be applied for cases with several independent variables and several dependent variables among the variables of a function.
- u = f(x,y,z,t) with z = g(x,y), t = h(x,y)
Netto derivatives:
- ∂u/∂x = ∂/∂x f(x,y,z,t)
- ∂u/∂y = ∂/∂y f(x,y,z,t)
- ∂u/∂z = ∂/∂z f(x,y,z,t)
- ∂u/∂t = ∂/∂t f(x,y,z,t)
Brutto derivatives: [the symbol δ/δx or δ/δy is suggested, since neither d/dx, d/dy nor ∂/∂x, ∂/∂y can be used]
- δu/δx = ∂/∂x f(x,y,z,t) + ∂/∂z f(x,y,z,t) · ∂/∂x g(x,y) + ∂/∂t f(x,y,z,t) · ∂/∂x h(x,y) – the other independent variable y is kept constant.
- δu/δy = ∂/∂y f(x,y,z,t) + ∂/∂z f(x,y,z,t) · ∂/∂y g(x,y) + ∂/∂t f(x,y,z,t) · ∂/∂y h(x,y) – the other independent variable x is kept constant.
September 18, 2009
Hey, whats that?
- Vector function of a scalar: eg. 3D-movement, parameterisation of a curve.
- Scalar function of a vector: eg. density, temperature etc. in a 3D-body
- What if a function itself is the argument of another function/mapping? In mathematics it is not easy to point it out. In programming it has to be specially built, not just as the other variables or arrays.
