- #1
Bruno Tolentino
- 97
- 0
I was me asking why the complex numbers are defined how z = x + i y !? Is this definition the better definition or was chosen by chance?
In mathematics, some things are defined by chance, for example: 0 is the multiplicative neutral element and your multiplicative inverse (0-) is the ∞. But, 1 is the additive neutral element and it haven't a symbol for its additive inverse (-1), the inverse additive is wrote simply how -1.
Other example: the conic equation is, actually, the vetorial form of the quadratic equation a x² + b x + c = 0. Therefore, the 'correct' form of write it is: [tex]a_{ij} : \vec{r}^2 + b_{i} \cdot \vec{r} + c = 0[/tex] In matrix form: [tex]
\begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{bmatrix}:\begin{bmatrix}
xx & xy \\
yx & yy
\end{bmatrix}
+
\begin{bmatrix}
b_1\\
b_2
\end{bmatrix}\cdot
\begin{bmatrix}
x\\
y
\end{bmatrix}
+c=0[/tex] And not this way:
https://en.wikipedia.org/wiki/Conic_section#Matrix_notation
Considering these 'mistakes' and others that I not wrote here, I me asked why the complex numbers are defined how z = x + i y.
Why i is important? Why? Why emphasize the number i ? Why? i is not important!
You known that x² - y² = (x + y) (x - y), all right!? And that x² + y² = (x + i y) (x - i y), correct!? But, you know that x³ + y³ = (x + α y) (x + β y) (x + γ y) ? No? You know that α, β and γ are the roots of ³√(-1) ?
α, β and γ appears in the cubic formula too. Write α, β and γ how the linear combination x + i y always complicates every equation! So, is necessary to define one symbol for α, β and γ too! Or, why not? Why the preconception? Why ²√(-1) has a proper symbol and ³√(-1) haven't? Why ²√(-1) is special?
I thought wrote the complex number like z = xy, because this notation no emphasize none root to the detriment of other.
But, linear combination of roots appears be a good way of write complex numbers too. In this case, 1 and i are the best base for the lienar combination?
And why the complex numbers are bidimensional numbers? If the root of the linear equation is and unidimensional number and if the roots of the quadratic equation are bidimensional numbers, so, the roots of the cubic equation are tridimensional numbers, correct or not!?
So, what you think about? I'm a little confused and unbeliever...
In mathematics, some things are defined by chance, for example: 0 is the multiplicative neutral element and your multiplicative inverse (0-) is the ∞. But, 1 is the additive neutral element and it haven't a symbol for its additive inverse (-1), the inverse additive is wrote simply how -1.
Other example: the conic equation is, actually, the vetorial form of the quadratic equation a x² + b x + c = 0. Therefore, the 'correct' form of write it is: [tex]a_{ij} : \vec{r}^2 + b_{i} \cdot \vec{r} + c = 0[/tex] In matrix form: [tex]
\begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{bmatrix}:\begin{bmatrix}
xx & xy \\
yx & yy
\end{bmatrix}
+
\begin{bmatrix}
b_1\\
b_2
\end{bmatrix}\cdot
\begin{bmatrix}
x\\
y
\end{bmatrix}
+c=0[/tex] And not this way:
https://en.wikipedia.org/wiki/Conic_section#Matrix_notation
Considering these 'mistakes' and others that I not wrote here, I me asked why the complex numbers are defined how z = x + i y.
Why i is important? Why? Why emphasize the number i ? Why? i is not important!
You known that x² - y² = (x + y) (x - y), all right!? And that x² + y² = (x + i y) (x - i y), correct!? But, you know that x³ + y³ = (x + α y) (x + β y) (x + γ y) ? No? You know that α, β and γ are the roots of ³√(-1) ?
α, β and γ appears in the cubic formula too. Write α, β and γ how the linear combination x + i y always complicates every equation! So, is necessary to define one symbol for α, β and γ too! Or, why not? Why the preconception? Why ²√(-1) has a proper symbol and ³√(-1) haven't? Why ²√(-1) is special?
I thought wrote the complex number like z = xy, because this notation no emphasize none root to the detriment of other.
But, linear combination of roots appears be a good way of write complex numbers too. In this case, 1 and i are the best base for the lienar combination?
And why the complex numbers are bidimensional numbers? If the root of the linear equation is and unidimensional number and if the roots of the quadratic equation are bidimensional numbers, so, the roots of the cubic equation are tridimensional numbers, correct or not!?
So, what you think about? I'm a little confused and unbeliever...