Square root - Revision history

Rational Point: /* Square roots over finite fields */ Jacobi symbols

2025-01-12T23:03:17Z

Square roots over finite fields: Jacobi symbols

← Older revision		Revision as of 23:03, 12 January 2025
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	Elements of [[finite field]]s that have square roots are known as [[quadratic residue]]s.		Elements of [[finite field]]s that have square roots are known as [[quadratic residue]]s.

	[[Legendre symbol]]s are computed by [[quadratic reciprocity]] using an [[extended Euclidean algorithm]] to determine whether or not an element of a finite field has a square root.		[[Legendre symbol]]s are computed by [[quadratic reciprocity]] with [[Jacobi symbol]]s using an [[extended Euclidean algorithm]] to determine whether or not an element of a finite field has a square root.

	Several variations on the general method exist <ref>Ebru Adiguzel-Goktas and Enver Ozdemir. “Square Root Computation In Finite Fields” arXiv preprint no. 2206.07145, 2022 https://arxiv.org/abs/2206.07145</ref>.		Several variations on the general method exist <ref>Ebru Adiguzel-Goktas and Enver Ozdemir. “Square Root Computation In Finite Fields” arXiv preprint no. 2206.07145, 2022 https://arxiv.org/abs/2206.07145</ref>.

Rational Point: /* Extracting square roots by hand */ residue

2025-01-12T22:51:24Z

Extracting square roots by hand: residue

← Older revision		Revision as of 22:51, 12 January 2025
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	# Group the digits under the radical in pairs: <math>\sqrt{6\quad 55\quad 36}</math>. You will get one digit of result for each pair of digits under the radical. That is the original reason for the vinculum or bar commonly placed over the radical, to indicate that it is a problem to be worked out in the style of a long division.		# Group the digits under the radical in pairs: <math>\sqrt{6\quad 55\quad 36}</math>. You will get one digit of result for each pair of digits under the radical. That is the original reason for the vinculum or bar commonly placed over the radical, to indicate that it is a problem to be worked out in the style of a long division.
	# Start with the largest digit whose square does not exceed the leftmost digit or pair of digits under the radical.		# Start with the largest digit whose square does not exceed the leftmost digit or pair of digits under the radical.
	# At each step, double the number you have written so far over the radical, append the next digit, the largest that will fit, multiplying by that same digit again and subtracting to calculate the ~~remainder~~, and then bring down the next pair of digits, as for long division.		# At each step, double the number you have written so far over the radical, append the next digit, the largest that will fit, multiplying by that same digit again and subtracting to calculate the residue, and then bring down the next pair of digits, as for long division.
	:<math>\begin{array}{rrr\|rrr}		:<math>\begin{array}{rrr\|rrr}
	&&& 2&5&6\\ \hline		&&& 2&5&6\\ \hline

Rational Point: /* Square roots over finite fields */ remove plurals from red links

2025-01-12T11:18:50Z

Square roots over finite fields: remove plurals from red links

← Older revision		Revision as of 11:18, 12 January 2025
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	== Square roots over finite fields ==		== Square roots over finite fields ==

	Elements of [[finite field]]s that have square roots are known as [[quadratic ~~residues~~]].		Elements of [[finite field]]s that have square roots are known as [[quadratic residue]]s.

	[[Legendre ~~symbols~~]] are computed by [[quadratic reciprocity]] using an [[extended Euclidean algorithm]] to determine whether or not an element of a finite field has a square root.		[[Legendre symbol]]s are computed by [[quadratic reciprocity]] using an [[extended Euclidean algorithm]] to determine whether or not an element of a finite field has a square root.

	Several variations on the general method exist <ref>Ebru Adiguzel-Goktas and Enver Ozdemir. “Square Root Computation In Finite Fields” arXiv preprint no. 2206.07145, 2022 https://arxiv.org/abs/2206.07145</ref>.		Several variations on the general method exist <ref>Ebru Adiguzel-Goktas and Enver Ozdemir. “Square Root Computation In Finite Fields” arXiv preprint no. 2206.07145, 2022 https://arxiv.org/abs/2206.07145</ref>.

Rational Point: finite fields

2025-01-11T20:46:05Z

finite fields

← Older revision		Revision as of 20:46, 11 January 2025
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			== Square roots over finite fields ==

			Elements of [[finite field]]s that have square roots are known as [[quadratic residues]].

			[[Legendre symbols]] are computed by [[quadratic reciprocity]] using an [[extended Euclidean algorithm]] to determine whether or not an element of a finite field has a square root.

			Several variations on the general method exist <ref>Ebru Adiguzel-Goktas and Enver Ozdemir. “Square Root Computation In Finite Fields” arXiv preprint no. 2206.07145, 2022 https://arxiv.org/abs/2206.07145</ref>.

	== Extracting square roots by hand ==		== Extracting square roots by hand ==

			The grade-school arithmetic lesson long forgotten by most adults!

	# Group the digits under the radical in pairs: <math>\sqrt{6\quad 55\quad 36}</math>. You will get one digit of result for each pair of digits under the radical. That is the original reason for the vinculum or bar commonly placed over the radical, to indicate that it is a problem to be worked out in the style of a long division.		# Group the digits under the radical in pairs: <math>\sqrt{6\quad 55\quad 36}</math>. You will get one digit of result for each pair of digits under the radical. That is the original reason for the vinculum or bar commonly placed over the radical, to indicate that it is a problem to be worked out in the style of a long division.
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	&&& 2&5&6\\ \hline		&&& 2&5&6\\ \hline
	&&\sqrt{} & 6 & 55 & 36 \\		&&\sqrt{} & 6 & 55 & 36 \\
	\underline 2\times \underline 2&=&4&-4 &~~\downarrow~~& \\ \hline		\underline 2\times \underline 2&=&4&-4 & & \\ \hline
	2\times 2 &=&4~~ & 2 & 55 & \\		2\times 2 &=&4~~ & 2 & 55 & \\
	4\underline 5\times\underline 5 &=& 225 &-2 & 25 &~~\downarrow~~ \\ \hline		4\underline 5\times\underline 5 &=& 225 &-2 & 25 & \\ \hline
	25\times 2&=&50~~&& 30 & 36 \\		25\times 2&=&50~~&& 30 & 36 \\
	50\underline6\times\underline6 &=& 3036&&-30&36\\ \hline		50\underline6\times\underline6 &=& 3036&&-30&36\\ \hline

Rational Point: too many arrows

2025-01-07T08:48:55Z

too many arrows

← Older revision		Revision as of 08:48, 7 January 2025
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	&&& 2&5&6\\ \hline		&&& 2&5&6\\ \hline
	&&\sqrt{} & 6 & 55 & 36 \\		&&\sqrt{} & 6 & 55 & 36 \\
	\underline 2\times \underline 2&=&4&-4 &\downarrow&~~\downarrow~~ \\ \hline		\underline 2\times \underline 2&=&4&-4 &\downarrow& \\ \hline
	2\times 2 &=&4~~ & 2 & 55 &~~\downarrow~~\\		2\times 2 &=&4~~ & 2 & 55 & \\
	4\underline 5\times\underline 5 &=& 225 &-2 & 25 &\downarrow \\ \hline		4\underline 5\times\underline 5 &=& 225 &-2 & 25 &\downarrow \\ \hline
	25\times 2&=&50~~&& 30 & 36 \\		25\times 2&=&50~~&& 30 & 36 \\

Rational Point: another link

2024-12-26T21:52:10Z

another link

← Older revision		Revision as of 21:52, 26 December 2024
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	:<math>\sqrt{65536}=256</math>		:<math>\sqrt{65536}=256</math>

	The method <ref>Wikihow: How to Calculate a Square Root by Hand: Method 2: Finding Square Roots Manually https://www.wikihow.com/Calculate-a-Square-Root-by-Hand#Finding-Square-Roots-Manually</ref> is based on the identity		The method <ref>Wikihow: How to Calculate a Square Root by Hand: Method 2: Finding Square Roots Manually. https://www.wikihow.com/Calculate-a-Square-Root-by-Hand#Finding-Square-Roots-Manually</ref><ref>Indeed: Career development: How To Calculate Square Root by Hand. https://www.indeed.com/career-advice/career-development/how-to-calculate-square-root</ref> is based on the identity

	:<math>(10x+y)^2=100x^2 + 20xy + y^2</math>.		:<math>(10x+y)^2=100x^2 + 20xy + y^2</math>.

Rational Point: verbal description

2024-12-26T21:33:46Z

verbal description

← Older revision		Revision as of 21:33, 26 December 2024
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			== Extracting square roots by hand ==

			# Group the digits under the radical in pairs: <math>\sqrt{6\quad 55\quad 36}</math>. You will get one digit of result for each pair of digits under the radical. That is the original reason for the vinculum or bar commonly placed over the radical, to indicate that it is a problem to be worked out in the style of a long division.
			# Start with the largest digit whose square does not exceed the leftmost digit or pair of digits under the radical.
			# At each step, double the number you have written so far over the radical, append the next digit, the largest that will fit, multiplying by that same digit again and subtracting to calculate the remainder, and then bring down the next pair of digits, as for long division.
	:<math>\begin{array}{rrr\|rrr}		:<math>\begin{array}{rrr\|rrr}
	&&& 2&5&6\\ \hline		&&& 2&5&6\\ \hline

Rational Point: square root by hand

2024-12-25T06:01:08Z

square root by hand

← Older revision		Revision as of 06:01, 25 December 2024
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	<math>\begin{array}{r\|rrr}		:<math>\begin{array}{rrr\|rrr}
	& 2&5&\\ \hline		&&& 2&5&6\\ \hline
	\sqrt{} & 6 & 55 & 36 \\		&&\sqrt{} & 6 & 55 & 36 \\
	\underline 2\times \underline 2=4&-4 &\downarrow& \\ \hline		\underline 2\times \underline 2&=&4&-4 &\downarrow&\downarrow \\ \hline
	& 2 & 55 & \\		2\times 2 &=&4~~ & 2 & 55 &\downarrow\\
	4\underline 5\times\underline 5 = 225 & -2 & 25 &\downarrow \\ \hline		4\underline 5\times\underline 5 &=& 225 &-2 & 25 &\downarrow \\ \hline
	& & 30 & 36		25\times 2&=&50~~&& 30 & 36 \\
			50\underline6\times\underline6 &=& 3036&&-30&36\\ \hline
			&&&&&0
	\end{array}		\end{array}
	</math>		</math>

	~~to be continued~~ <ref>Wikihow: How to Calculate a Square Root by Hand: Method 2: Finding Square Roots Manually https://www.wikihow.com/Calculate-a-Square-Root-by-Hand#Finding-Square-Roots-Manually</ref>		:<math>\sqrt{65536}=256</math>

			The method <ref>Wikihow: How to Calculate a Square Root by Hand: Method 2: Finding Square Roots Manually https://www.wikihow.com/Calculate-a-Square-Root-by-Hand#Finding-Square-Roots-Manually</ref> is based on the identity

			:<math>(10x+y)^2=100x^2 + 20xy + y^2</math>.

Rational Point: how-to

2024-12-25T04:47:47Z

how-to

New page

<math>\begin{array}{r|rrr}
& 2&5&\\ \hline
\sqrt{} & 6 & 55 & 36 \\
\underline 2\times \underline 2=4&-4 &\downarrow& \\ \hline
& 2 & 55 & \\
4\underline 5\times\underline 5 = 225 & -2 & 25 &\downarrow \\ \hline
& & 30 & 36
\end{array}
</math>

to be continued <ref>Wikihow: How to Calculate a Square Root by Hand: Method 2: Finding Square Roots Manually https://www.wikihow.com/Calculate-a-Square-Root-by-Hand#Finding-Square-Roots-Manually</ref>