Radical - Revision history

Lfox: /* Notes */

2024-04-18T02:18:56Z

Notes

← Older revision		Revision as of 02:18, 18 April 2024
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	The '''square root''' of an area <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length a square with area <math>x</math>.		The '''square root''' of an area <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length a square with area <math>x</math>.

	Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation.<ref group="note">Proof: ~~[TODO]~~ </ref> Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.		Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation.<ref group="note">Proof by contradiction: Suppose <math>(p/q)^2 = 2</math>, where <math>p, q</math> are integers sharing no common factors. Then <math>p^2 = 2 q^2</math>, so 2 divides <math>p^2</math>. It follows that 2 divides <math>p</math>, so write <math>p= 2k</math>. Then <math>4k^2 = 2q^2 </math>, so <math>2k^2 = q^2</math>. But that means that 2 divides <math>q </math> as well. We have reached a contradiction. </ref> Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.

	Objective Mathematics does not require an infinite amount of precision in its definition of square root. The amount of precision required is determined by one's context. This means that if one desires to convert a square root into a fraction, then which fraction one chooses depends on the level of precision at which one wishes to measure the real, physical ''square'' that is under consideration. [TODO I'm not completely sure about this.]		Objective Mathematics does not require an infinite amount of precision in its definition of square root. The amount of precision required is determined by one's context. This means that if one desires to convert a square root into a fraction, then which fraction one chooses depends on the level of precision at which one wishes to measure the real, physical ''square'' that is under consideration. [TODO I'm not completely sure about this.]

Lfox at 20:31, 17 April 2024

2024-04-17T20:31:53Z

← Older revision		Revision as of 20:31, 17 April 2024
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	I am not yet sure how to define radicals generally.		I am not yet sure how to define radicals generally. The natural concept to define here in standard mathematics would be algebraic numbers, but it's not clear what those mean in real life.

	== Square roots ==		== Square roots ==
	The '''square root''' of ~~a number~~ <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length ~~of the diagonal of~~ a square with area <math>x</math>.		The '''square root''' of an area <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length a square with area <math>x</math>.

	Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation.<ref group="note">Proof: [TODO] </ref> Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.		Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation.<ref group="note">Proof: [TODO] </ref> Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.
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	== Cube roots ==		== Cube roots ==
	The '''cube root''' of a number <math>x</math>, denoted <math>\sqrt[3]{x}</math>, is the side length of a cube with volume <math>x</math>.		The '''cube root''' of a number <math>x</math>, denoted <math>\sqrt[3]{x}</math>, is the side length of a cube with volume <math>x</math>.

			== Imaginary numbers ==
			We can extend the concept of square root to ''signed'' areas, beyond just areas. A signed area is a signed difference of areas (see the article [[integers]] for a detailed explanation), meaning that it [[reduces]] to some ordered pair of areas.

			A complex integer reduces to an ordered pair of integers (which reduces to an ordered 4-tuple of natural numbers). These things satisfy some special multiplication law. The point of the multiplication law is that, if <math>x</math> is ''any'' signed area, then we want the multiplication law to be defined in such a way that <math>\sqrt{x} \cdot \sqrt{x} = x </math>, just like was the case for the square root.

	== Notes ==		== Notes ==
	<references group="note" />		<references group="note" />

Lfox: /* Square roots */

2024-02-04T00:03:26Z

Square roots

← Older revision		Revision as of 00:03, 4 February 2024
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	== Square roots ==		== Square roots ==
	The '''square root''' of a number <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length of a square with area <math>x</math>.		The '''square root''' of a number <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length of the diagonal of a square with area <math>x</math>.

	Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation.<ref group="note">Proof: [TODO] </ref> Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.		Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation.<ref group="note">Proof: [TODO] </ref> Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.

Lfox: /* Square roots */

2024-01-27T23:54:22Z

Square roots

← Older revision		Revision as of 23:54, 27 January 2024
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	The '''square root''' of a number <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length of a square with area <math>x</math>.		The '''square root''' of a number <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length of a square with area <math>x</math>.

	Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation.<ref group="note">Proof: </ref> Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.		Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation.<ref group="note">Proof: [TODO] </ref> Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.

	Objective Mathematics does not require an infinite amount of precision in its definition of square root. The amount of precision required is determined by one's context. This means that if one desires to convert a square root into a fraction, then which fraction one chooses depends on the level of precision at which one wishes to measure the real, physical ''square'' that is under consideration. [TODO I'm not completely sure about this.]		Objective Mathematics does not require an infinite amount of precision in its definition of square root. The amount of precision required is determined by one's context. This means that if one desires to convert a square root into a fraction, then which fraction one chooses depends on the level of precision at which one wishes to measure the real, physical ''square'' that is under consideration. [TODO I'm not completely sure about this.]

Lfox at 23:54, 27 January 2024

2024-01-27T23:54:02Z

← Older revision		Revision as of 23:54, 27 January 2024
Line 4:		Line 4:
	The '''square root''' of a number <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length of a square with area <math>x</math>.		The '''square root''' of a number <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length of a square with area <math>x</math>.

	Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation. Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.		Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions\|rational number]] satisfies that equation.<ref group="note">Proof: </ref> Solutions to that equation can at best be approximated by a [[Sequences\|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.

	Objective Mathematics does not require an infinite amount of precision in its definition of square root. The amount of precision required is determined by one's context. This means that if one desires to convert a square root into a fraction, then which fraction one chooses depends on the level of precision at which one wishes to measure the real, physical ''square'' that is under consideration. [TODO I'm not completely sure about this.]		Objective Mathematics does not require an infinite amount of precision in its definition of square root. The amount of precision required is determined by one's context. This means that if one desires to convert a square root into a fraction, then which fraction one chooses depends on the level of precision at which one wishes to measure the real, physical ''square'' that is under consideration. [TODO I'm not completely sure about this.]
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	== Cube roots ==		== Cube roots ==
	The '''cube root''' of a number <math>x</math>, denoted <math>\sqrt[3]{x}</math>, is the side length of a cube with volume <math>x</math>.		The '''cube root''' of a number <math>x</math>, denoted <math>\sqrt[3]{x}</math>, is the side length of a cube with volume <math>x</math>.

			== Notes ==
			<references group="note" />

Lfox: Created page with "I am not yet sure how to define radicals generally. == Square roots == The '''square root''' of a number $x$ , denoted $\sqrt{x}$ , is the side length of a square with area $x$ . Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number $r$ such that $r^2 = 2$ . It is well known that no rational number satisfies that equation. Solutions to that equation can at..."

2024-01-21T01:59:29Z

Created page with "I am not yet sure how to define radicals generally. == Square roots == The '''square root''' of a number <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length of a square with area <math>x</math>. Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no rational number satisfies that equation. Solutions to that equation can at..."

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I am not yet sure how to define radicals generally.

== Square roots ==
The '''square root''' of a number <math>x</math>, denoted <math>\sqrt{x}</math>, is the side length of a square with area <math>x</math>.

Loosely speaking, standard mathematics defines the square root of, say, 2, to be the positive number <math>r</math> such that <math>r^2 = 2 </math>. It is well known that no [[Fractions|rational number]] satisfies that equation. Solutions to that equation can at best be approximated by a [[Sequences|sequence]] of fractions. This means that standard math's definition of <math>\sqrt{2}</math> comes hand in hand with a requirement for an infinite amount of precision.

Objective Mathematics does not require an infinite amount of precision in its definition of square root. The amount of precision required is determined by one's context. This means that if one desires to convert a square root into a fraction, then which fraction one chooses depends on the level of precision at which one wishes to measure the real, physical ''square'' that is under consideration. [TODO I'm not completely sure about this.]

== Cube roots ==
The '''cube root''' of a number <math>x</math>, denoted <math>\sqrt[3]{x}</math>, is the side length of a cube with volume <math>x</math>.