Cube root - Revision history

Rational Point: gunners

2025-02-01T07:10:35Z

gunners

← Older revision		Revision as of 07:10, 1 February 2025
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	\end{array}</math>		\end{array}</math>
	An exact root has been reached when (and if) the residue is zero. Typically the number whose root is being taken is called the '''solvend''', the equations to the left of the vertical line are '''resolvent equations''', and at every step, to the right of the vertical line, the '''resolvent''' is subtracted from the previous '''resultant''' to yield a '''residue''', as each digit of the root is calculated as the greatest digit whose resolvent does not exceed the resultant. The new resultant is the residue with three more digits of the solvend brought down.		An exact root has been reached when (and if) the residue is zero. Typically the number whose root is being taken is called the '''solvend''', the equations to the left of the vertical line are '''resolvent equations''', and at every step, to the right of the vertical line, the '''resolvent''' is subtracted from the previous '''resultant''' to yield a '''residue''', as each digit of the root is calculated as the greatest digit whose resolvent does not exceed the resultant. The new resultant is the residue with three more digits of the solvend brought down.

			Knowing the value of <big>''π''</big>, and the exact density of lead, '''gunners''' would have used cube roots for calculating the precise diameter of spherical lead shot sold by weight among other matters, and '''gunnery''' was considered an important branch of applied mathematics, to calculate the trajectories of artillery shells using geometry, square and cube roots, trigonometry, calculus and so forth, long before the days of electronic calculators and computers.

Rational Point: terminology

2025-01-31T20:11:19Z

terminology

← Older revision		Revision as of 20:11, 31 January 2025
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	&&&&&&0		&&&&&&0
	\end{array}</math>		\end{array}</math>
	An exact root has been reached when (and if) the residue is zero.		An exact root has been reached when (and if) the residue is zero. Typically the number whose root is being taken is called the '''solvend''', the equations to the left of the vertical line are '''resolvent equations''', and at every step, to the right of the vertical line, the '''resolvent''' is subtracted from the previous '''resultant''' to yield a '''residue''', as each digit of the root is calculated as the greatest digit whose resolvent does not exceed the resultant. The new resultant is the residue with three more digits of the solvend brought down.

Rational Point: residue

2025-01-12T22:52:59Z

residue

← Older revision		Revision as of 22:52, 12 January 2025
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	&&&&&&0		&&&&&&0
	\end{array}</math>		\end{array}</math>
	An exact root has been reached when the ~~remainder~~ is zero.		An exact root has been reached when (and if) the residue is zero.

Rational Point: explain third step

2024-12-27T07:42:11Z

explain third step

← Older revision		Revision as of 07:42, 27 December 2024
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	:<math>(10x+y)^3=1000x^3+300x^2y+30xy^2+y^3</math>		:<math>(10x+y)^3=1000x^3+300x^2y+30xy^2+y^3</math>

	if we let <math>x=2</math>, the number written so far above the radical, and ''y'' be the next digit to calculate. The term <math>1000x^3=8000</math> has already been subtracted when we bring down the next group of three digits, so the next digit to write above the radical will be the largest ''y'' such that <math>300\times 2^2y+30\times 2y^2+y^3\le 2460</math>.		if we let <math>x=2</math>, the number written so far above the radical, and ''y'' be the next digit to calculate. The term <math>1000x^3=8000</math> has already been subtracted when we bring down the next group of three digits, so the next digit to write above the radical will be the largest ''y'' such that <math>300\times 2^2y+30\times 2y^2+y^3\le 2460</math>. Here we have found <math>y=1</math>, and for the next step set <math>x=21</math> to find the next digit of the root after that.

	:<math>\begin{array}{rrr\|rrrr}		:<math>\begin{array}{rrr\|rrrr}
	&&&2&1\\ \hline		&&&2&1&\_\\ \hline
	&&\sqrt[3]{}&10&460&353&203\\		&&\sqrt[3]{}&10&460&353&203\\
	\underline2^3&=&8&-8&\downarrow\\ \hline		\underline2^3&=&8&-8&\downarrow\\ \hline

Rational Point: let URL link automatically

2024-12-27T03:09:15Z

let URL link automatically

← Older revision		Revision as of 03:09, 27 December 2024
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	\end{array}</math>		\end{array}</math>

	The next step is based on the identity <ref>WikiHow: How to Calculate Cube Root by Hand https://www.wikihow.com/Calculate-Cube-Root-by-Hand</ref>		The next step is based on the identity <ref>WikiHow: How to Calculate Cube Root by Hand https://www.wikihow.com/Calculate-Cube-Root-by-Hand</ref><ref>Paul E. Black, “cube root” in ''Dictionary of Algorithms and Data Structures'', [online], ed. 6 May 2019. https://www.nist.gov/dads/HTML/cubeRoot.html</ref>

	:<math>(10x+y)^3=1000x^3+300x^2y+30xy^2+y^3</math>		:<math>(10x+y)^3=1000x^3+300x^2y+30xy^2+y^3</math>

Rational Point: misprint

2024-12-27T01:59:10Z

misprint

← Older revision		Revision as of 01:59, 27 December 2024
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	{}+30\times 21 \times\underline8^2+\underline8^3&=&1099232&-1&099&232&\downarrow\\ \hline		{}+30\times 21 \times\underline8^2+\underline8^3&=&1099232&-1&099&232&\downarrow\\ \hline
	300\times 218^2\times \underline7&&&&100&121&203\\		300\times 218^2\times \underline7&&&&100&121&203\\
	{}+30\times 218\times\underline7^2~~\times~~\underline7^3&=&100121203&&-100&121&203\\ \hline		{}+30\times 218\times\underline7^2+\underline7^3&=&100121203&&-100&121&203\\ \hline
	&&&&&&0		&&&&&&0
	\end{array}</math>		\end{array}</math>
	An exact root has been reached when the remainder is zero.		An exact root has been reached when the remainder is zero.

Rational Point: exact root

2024-12-27T01:57:49Z

exact root

← Older revision		Revision as of 01:57, 27 December 2024
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	\end{array}</math>		\end{array}</math>
	:<math>\begin{array}{rrr\|rrrr}		:<math>\begin{array}{rrr\|rrrr}
	&&&2&1&8\\ \hline		&&&2&1&8&7\\ \hline
	&&\sqrt[3]{}&10&460&353&203\\		&&\sqrt[3]{}&10&460&353&203\\
	\underline2^3&=&8&-8&\downarrow\\ \hline		\underline2^3&=&8&-8&\downarrow\\ \hline
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	&&&&&&0		&&&&&&0
	\end{array}</math>		\end{array}</math>
			An exact root has been reached when the remainder is zero.
	...

Rational Point: completed root

2024-12-27T01:55:31Z

completed root

← Older revision		Revision as of 01:55, 27 December 2024
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	== Extracting cube roots by hand ==		== Extracting cube roots by hand ==

	Start by grouping the digits under the radical into groups of three as you normally would. This seems natural to do, even if it is a little more		Start by grouping the digits under the radical into groups of three as you normally would. This seems natural to do, even if it is a little more complicated than the [[square root]].

	:<math>\sqrt[3]{10,\!460,\!353,\!203}</math>		:<math>\sqrt[3]{10, 460, 353, 203}</math>

	The first step here is to find the largest digit whose cube does not exceed the first group of digits under the radical, cube it, subtract to find the remainder, and bring down the next three digits.		The first step here is to find the largest digit whose cube does not exceed the first group of digits under the radical, cube it, subtract to find the remainder, and bring down the next three digits.
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	\end{array}</math>		\end{array}</math>

	The next step is based on the identity		The next step is based on the identity <ref>WikiHow: How to Calculate Cube Root by Hand https://www.wikihow.com/Calculate-Cube-Root-by-Hand</ref>

	:<math>(10x+y)^3=1000x^3+300x^2y+30xy^2+y^3</math>		:<math>(10x+y)^3=1000x^3+300x^2y+30xy^2+y^3</math>
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	{}+30\times 21 \times\_^2+\_^3&=&		{}+30\times 21 \times\_^2+\_^3&=&
	\end{array}</math>		\end{array}</math>
	~~<blockquote>~~<math ~~display="block"~~>\begin{array}{rrr\|rrrr}		:<math>\begin{array}{rrr\|rrrr}
	&&&2&1&8\\ \hline		&&&2&1&8\\ \hline
	&&\sqrt[3]{}&10&460&353&203\\		&&\sqrt[3]{}&10&460&353&203\\
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	300\times 2^2\times\underline1&&&2&460\\		300\times 2^2\times\underline1&&&2&460\\
	{}+30\times 2\times\underline1^2+\underline1^3&=&1261&-1&261&\downarrow\\ \hline		{}+30\times 2\times\underline1^2+\underline1^3&=&1261&-1&261&\downarrow\\ \hline
	300\times 21^2 \times 8&&&1&199&353\\		300\times 21^2 \times \underline8&&&1&199&353\\
	{}+30\times 21 \~~times8~~^2+8^3&=&1099232&-1&099&232\\ \hline		{}+30\times 21 \times\underline8^2+\underline8^3&=&1099232&-1&099&232&\downarrow\\ \hline
	&&&&100&121		300\times 218^2\times \underline7&&&&100&121&203\\
	\end{array}</math>~~</blockquote>To be continued <ref>WikiHow: How to Calculate Cube Root by Hand https://www.wikihow.com/Calculate-Cube-Root-by-Hand</ref>~~ ...		{}+30\times 218\times\underline7^2\times\underline7^3&=&100121203&&-100&121&203\\ \hline
			&&&&&&0
			\end{array}</math>

			...

Rational Point: next step

2024-12-27T01:38:04Z

next step

← Older revision		Revision as of 01:38, 27 December 2024
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	{}+30\times 21 \times\_^2+\_^3&=&		{}+30\times 21 \times\_^2+\_^3&=&
	\end{array}</math>		\end{array}</math>
			<blockquote><math display="block">\begin{array}{rrr\|rrrr}
	To be continued <ref>WikiHow: How to Calculate Cube Root by Hand https://www.wikihow.com/Calculate-Cube-Root-by-Hand</ref> ...		&&&2&1&8\\ \hline
			&&\sqrt[3]{}&10&460&353&203\\
			\underline2^3&=&8&-8&\downarrow\\ \hline
			300\times 2^2\times\underline1&&&2&460\\
			{}+30\times 2\times\underline1^2+\underline1^3&=&1261&-1&261&\downarrow\\ \hline
			300\times 21^2 \times 8&&&1&199&353\\
			{}+30\times 21 \times8^2+8^3&=&1099232&-1&099&232\\ \hline
			&&&&100&121
			\end{array}</math></blockquote>To be continued <ref>WikiHow: How to Calculate Cube Root by Hand https://www.wikihow.com/Calculate-Cube-Root-by-Hand</ref> ...

Rational Point: x = no. so far written

2024-12-27T00:03:29Z

x = no. so far written

← Older revision		Revision as of 00:03, 27 December 2024
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	300\times 2^2\times\underline1&&&2&460\\		300\times 2^2\times\underline1&&&2&460\\
	{}+30\times 2\times\underline1^2+\underline1^3&=&1261&-1&261&\downarrow\\ \hline		{}+30\times 2\times\underline1^2+\underline1^3&=&1261&-1&261&\downarrow\\ \hline
	300\times 1^2 \times \_&&&1&199&353\\		300\times 21^2 \times \_&&&1&199&353\\
	{}+30\~~times1~~ \times\_^2+\_^3&=&		{}+30\times 21 \times\_^2+\_^3&=&
	\end{array}</math>		\end{array}</math>

	To be continued <ref>WikiHow: How to Calculate Cube Root by Hand https://www.wikihow.com/Calculate-Cube-Root-by-Hand</ref> ...		To be continued <ref>WikiHow: How to Calculate Cube Root by Hand https://www.wikihow.com/Calculate-Cube-Root-by-Hand</ref> ...