Proving the square root of $zin mathbb{C}$ geometrically
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I hope somebody could help me to prove that the squareroot exists for all numbers in $mathbb{C}$. A complex number is Always a stretch and a Rotation at the same time. For example (3,2) Projects (1,0) to (3,2), if we trat (3,2) like we did treat (3,2), we do the same stratch and Rotation again. Visually it would look like this:
Can one derive an explicit Formula with that idea or with other words is it possible to argue that every Point on a plane can be described as the composition of two equal streteches and Rotation?
Many Thanks for your Input and time.
Here is also another idea of mine to find such a Point but not sure whether it is Right and how to prove it
complex-numbers euclidean-geometry
asked Dec 2 '18 at 12:41
RM777RM777
4029
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add a comment |
$begingroup$
I hope somebody could help me to prove that the squareroot exists for all numbers in $mathbb{C}$. A complex number is Always a stretch and a Rotation at the same time. For example (3,2) Projects (1,0) to (3,2), if we trat (3,2) like we did treat (3,2), we do the same stratch and Rotation again. Visually it would look like this:
Can one derive an explicit Formula with that idea or with other words is it possible to argue that every Point on a plane can be described as the composition of two equal streteches and Rotation?
Many Thanks for your Input and time.
Here is also another idea of mine to find such a Point but not sure whether it is Right and how to prove it
complex-numbers euclidean-geometry
asked Dec 2 '18 at 12:41
RM777RM777
4029
$endgroup$
add a comment |
$begingroup$
I hope somebody could help me to prove that the squareroot exists for all numbers in $mathbb{C}$. A complex number is Always a stretch and a Rotation at the same time. For example (3,2) Projects (1,0) to (3,2), if we trat (3,2) like we did treat (3,2), we do the same stratch and Rotation again. Visually it would look like this:
Can one derive an explicit Formula with that idea or with other words is it possible to argue that every Point on a plane can be described as the composition of two equal streteches and Rotation?
Many Thanks for your Input and time.
Here is also another idea of mine to find such a Point but not sure whether it is Right and how to prove it
complex-numbers euclidean-geometry
asked Dec 2 '18 at 12:41
RM777RM777
4029
$endgroup$
I hope somebody could help me to prove that the squareroot exists for all numbers in $mathbb{C}$. A complex number is Always a stretch and a Rotation at the same time. For example (3,2) Projects (1,0) to (3,2), if we trat (3,2) like we did treat (3,2), we do the same stratch and Rotation again. Visually it would look like this:
Can one derive an explicit Formula with that idea or with other words is it possible to argue that every Point on a plane can be described as the composition of two equal streteches and Rotation?
Many Thanks for your Input and time.
Here is also another idea of mine to find such a Point but not sure whether it is Right and how to prove it
complex-numbers euclidean-geometry
complex-numbers euclidean-geometry
asked Dec 2 '18 at 12:41
RM777RM777
4029
asked Dec 2 '18 at 12:41
RM777RM777
4029
edited Dec 2 '18 at 13:00
RM777
asked Dec 2 '18 at 12:41
RM777RM777
4029
asked Dec 2 '18 at 12:41
RM777RM777
4029
asked Dec 2 '18 at 12:41
RM777RM777
4029
4029
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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From that point of view, consider half of the rotation and the square root of the strecht. If you multiply this by itself, you will get the original number.
answered Dec 2 '18 at 12:53
José Carlos SantosJosé Carlos Santos
155k22124227
$endgroup$
$begingroup$
For example if I want to find the squareroot of (2,3) it would be $(sqrt(2),frac{3}{2})$?
$endgroup$
– RM777
Dec 2 '18 at 13:12
1
$begingroup$
The complex number $2+3i$ corresponds to a strecht of $sqrt{13}$ and a rotation of $arccosleft(frac2{sqrt{13}}right)$.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 13:18
add a comment |
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I am not sure I understood your question correctly, but I think this answer will have all you are looking for:
How do I get the square root of a complex number?
answered Dec 2 '18 at 12:53
JefferyJeffery
834
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
From that point of view, consider half of the rotation and the square root of the strecht. If you multiply this by itself, you will get the original number.
answered Dec 2 '18 at 12:53
José Carlos SantosJosé Carlos Santos
155k22124227
$endgroup$
$begingroup$
For example if I want to find the squareroot of (2,3) it would be $(sqrt(2),frac{3}{2})$?
$endgroup$
– RM777
Dec 2 '18 at 13:12
1
$begingroup$
The complex number $2+3i$ corresponds to a strecht of $sqrt{13}$ and a rotation of $arccosleft(frac2{sqrt{13}}right)$.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 13:18
add a comment |
$begingroup$
From that point of view, consider half of the rotation and the square root of the strecht. If you multiply this by itself, you will get the original number.
answered Dec 2 '18 at 12:53
José Carlos SantosJosé Carlos Santos
155k22124227
$endgroup$
$begingroup$
For example if I want to find the squareroot of (2,3) it would be $(sqrt(2),frac{3}{2})$?
$endgroup$
– RM777
Dec 2 '18 at 13:12
1
$begingroup$
The complex number $2+3i$ corresponds to a strecht of $sqrt{13}$ and a rotation of $arccosleft(frac2{sqrt{13}}right)$.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 13:18
add a comment |
$begingroup$
From that point of view, consider half of the rotation and the square root of the strecht. If you multiply this by itself, you will get the original number.
answered Dec 2 '18 at 12:53
José Carlos SantosJosé Carlos Santos
155k22124227
$endgroup$
From that point of view, consider half of the rotation and the square root of the strecht. If you multiply this by itself, you will get the original number.
answered Dec 2 '18 at 12:53
José Carlos SantosJosé Carlos Santos
155k22124227
answered Dec 2 '18 at 12:53
José Carlos SantosJosé Carlos Santos
155k22124227
answered Dec 2 '18 at 12:53
José Carlos SantosJosé Carlos Santos
155k22124227
answered Dec 2 '18 at 12:53
José Carlos SantosJosé Carlos Santos
155k22124227
155k22124227
$begingroup$
For example if I want to find the squareroot of (2,3) it would be $(sqrt(2),frac{3}{2})$?
$endgroup$
– RM777
Dec 2 '18 at 13:12
1
$begingroup$
The complex number $2+3i$ corresponds to a strecht of $sqrt{13}$ and a rotation of $arccosleft(frac2{sqrt{13}}right)$.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 13:18
add a comment |
$begingroup$
For example if I want to find the squareroot of (2,3) it would be $(sqrt(2),frac{3}{2})$?
$endgroup$
– RM777
Dec 2 '18 at 13:12
1
$begingroup$
The complex number $2+3i$ corresponds to a strecht of $sqrt{13}$ and a rotation of $arccosleft(frac2{sqrt{13}}right)$.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 13:18
$begingroup$
For example if I want to find the squareroot of (2,3) it would be $(sqrt(2),frac{3}{2})$?
$endgroup$
– RM777
Dec 2 '18 at 13:12
$begingroup$
For example if I want to find the squareroot of (2,3) it would be $(sqrt(2),frac{3}{2})$?
$endgroup$
– RM777
Dec 2 '18 at 13:12
1
1
$begingroup$
The complex number $2+3i$ corresponds to a strecht of $sqrt{13}$ and a rotation of $arccosleft(frac2{sqrt{13}}right)$.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 13:18
$begingroup$
The complex number $2+3i$ corresponds to a strecht of $sqrt{13}$ and a rotation of $arccosleft(frac2{sqrt{13}}right)$.
$endgroup$
– José Carlos Santos
Dec 2 '18 at 13:18
add a comment |
$begingroup$
I am not sure I understood your question correctly, but I think this answer will have all you are looking for:
How do I get the square root of a complex number?
answered Dec 2 '18 at 12:53
JefferyJeffery
834
$endgroup$
add a comment |
$begingroup$
I am not sure I understood your question correctly, but I think this answer will have all you are looking for:
How do I get the square root of a complex number?
answered Dec 2 '18 at 12:53
JefferyJeffery
834
$endgroup$
add a comment |
$begingroup$
I am not sure I understood your question correctly, but I think this answer will have all you are looking for:
How do I get the square root of a complex number?
answered Dec 2 '18 at 12:53
JefferyJeffery
834
$endgroup$
I am not sure I understood your question correctly, but I think this answer will have all you are looking for:
How do I get the square root of a complex number?
answered Dec 2 '18 at 12:53
JefferyJeffery
834
answered Dec 2 '18 at 12:53
JefferyJeffery
834
answered Dec 2 '18 at 12:53
JefferyJeffery
834
answered Dec 2 '18 at 12:53
JefferyJeffery
834
834
add a comment |
add a comment |
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