Math - Revision history

Eric Rasmusen: /* Proofs */

2024-11-27T14:34:14Z

‎Proofs

Eric Rasmusen: /* Numbers */

2024-03-02T19:19:19Z

‎Numbers

Manson Lilian: /* Line Integrals */

2022-10-24T21:46:51Z

‎Line Integrals

Eric Rasmusen: /* Equality */

2022-09-20T19:54:41Z

‎Equality

Eric Rasmusen: /* Irrational Numbers */

2022-08-02T02:29:53Z

‎Irrational Numbers

Erasmuse: /* Miscellaneous */

2022-05-22T17:35:26Z

‎Miscellaneous

Erasmuse: /* Irrational Numbers */

2022-04-18T02:48:26Z

‎Irrational Numbers

Erasmuse: /* Irrational Numbers */

2022-04-18T00:27:37Z

‎Irrational Numbers

Rasmusen p1vaim: /* Erdos Numbers */

2022-02-27T00:56:28Z

‎Erdos Numbers

Rasmusen p1vaim: /* Erdos Numbers */

2022-01-03T22:24:24Z

‎Erdos Numbers

← Older revision		Revision as of 14:34, 27 November 2024
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	*See [https://twitter.com/taz_chu/status/1763892059146948823 "A One-Sentence Proof . . .] (2024).		*See [https://twitter.com/taz_chu/status/1763892059146948823 "A One-Sentence Proof . . .] (2024).

		+	*The Huffman result, when [https://x.com/fermatslibrary/status/1861725472779878430 giving a class an unsolved problem led to a major discovery](X).
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 ==Numbers==
 *[https://en.wikipedia.org/wiki/Cistercian_numerals#:~:text=The%20medieval%20Cistercian%20numerals,%20or,were%20introduced%20to%20northwestern%20Europe Cistercian Numbers, 13th Century]:"The medieval Cistercian numerals, or "ciphers" in nineteenth-century parlance, were developed by the Cistercian monastic order in the early thirteenth century at about the time that Arabic numerals were introduced to northwestern Europe. They are more compact than Arabic or Roman numerals, with a single glyph able to indicate any integer from 1 to 9,999."
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 . You want to integrate  x^2 + y^2 over all points in the area bounded by  y = x^3 and y  = log x (or some two curves that cross).
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 ==Slide Rules==
 {{Quotation|December 6, 2021: In late 2021, Russian video appeared that showed a heavy bomber in flight with one of the crew using s slide rule, apparently to calculate course and/or fuel consumption rate. Most pilot training, especially for the crews of long-range aircraft, includes instruction on how to use special slide rules for such calculations in the event of problems with the electronic navigation and flight management instruments that do this automatically. For generations, ever since long-range flight became possible, manual tools were used for these calculations, along with a special bubble sextant to obtain the location of an aircraft. On the surface the original sextant is used for this but in the air, there is no fixed horizon to base these calculations. The bubble-sextant creates an artificial horizon that enables aerial navigation that shows position within ten kilometers or less. Surface sextant navigation is even more accurate and electronic navigation, especially using GPS, is accurate enough for landing aircraft.

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 ==Equality==
 *[https://twitter.com/SC_Griffith/status/1528140634422394880 Twitter opn what it means to be equal] (2022).
 ==Miscellaneous==

← Older revision		Revision as of 02:29, 2 August 2022
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	:If x is even, we can write it as x = 2m for some number m. But then 2y^2 = (2m)^2 = 4m^2, so y^2 = 2m^2. But then y^2 is even, so by our earlier argument, y is even too. But that means both x and y are even, which contradicts our starting assumption. So it must be that Sqrt(2) != a/b for any integers a and b. QED.		:If x is even, we can write it as x = 2m for some number m. But then 2y^2 = (2m)^2 = 4m^2, so y^2 = 2m^2. But then y^2 is even, so by our earlier argument, y is even too. But that means both x and y are even, which contradicts our starting assumption. So it must be that Sqrt(2) != a/b for any integers a and b. QED.

−		+	*The Real Numbers are Uncountable. Cantor proved this. His Diagonalization Argument is one way. The [https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument#:~:text=In%20set%20theory%2C%20Cantor's%20diagonal,cannot%20be%20put%20into%20one%2D Wikipedia article on this] is good. The basic idea is that you can convert all irrational number into a list of decimals if they're countable, and then contradict this by constructing a new decimal that is different from the first one on your list in the first decimal place, the second one in the second decimal place, the third one in the third decimal place, etc.

← Older revision		Revision as of 17:35, 22 May 2022
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		+
		+	==Equality==
		+	*[https://twitter.com/SC_Griffith/status/1528140634422394880 Twitter opn what it means to be equal] (2022).

	==Miscellaneous==		==Miscellaneous==

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 *Not using proof by contradiction is separate from not using infinity in proofs, right?
-*Theorem: The square root of 2 is irrational.
+*'''Theorem: The square root of 2 is irrational.'''
-:Proof: Suppose not.  If the square root of 2 is rational, Sqrt(2) = a/b for some integers a and b. First, note that this implies that Sqrt(2) = x/y for some integers x and y that are not both even. That's because if  Sqrt(2) = a/b  and both a and b are even, then it is also true that a and b are divisible by 2, so Sqrt(2) = .5a/.5b with both .5a and .5b being integers.  If .5a and .5b are both even, we can repeat the division by 2, and we can keep doing this till either the numerator or denominator is odd (or both).
+:'''Proof:''' Suppose not.  If the square root of 2 is rational, Sqrt(2) = a/b for some integers a and b. First, note that this implies that Sqrt(2) = x/y for some integers x and y that are not both even. That's because if  Sqrt(2) = a/b  and both a and b are even, then it is also true that a and b are divisible by 2, so Sqrt(2) = .5a/.5b with both .5a and .5b being integers.  If .5a and .5b are both even, we can repeat the division by 2, and we can keep doing this till either the numerator or denominator is odd (or both).
-So suppose Sqrt(2) = x/y, where x and y are not both even. We will show that this implies that both x and y are even, a contradiction.
+:So suppose Sqrt(2) = x/y, where x and y are not both even. We will show that this implies that both x and y are even, a contradiction.
-First, square both sides so 2 = x^2/y^2.  Then 2y^2 = x^2.   This implies that x^2 is even. But then x must be even too,  since if x is odd,  x(x-1) is even because it multiplies an odd number (x) times an even one (x-1), and  so x(x-1)+x is odd because it adds an even number (x(x-1)) to an odd number (x).
+:First, square both sides so 2 = x^2/y^2.  Then 2y^2 = x^2.   This implies that x^2 is even. But then x must be even too,  since if x is odd,  x(x-1) is even because it multiplies an odd number (x) times an even one (x-1), and  so x(x-1)+x is odd because it adds an even number (x(x-1)) to an odd number (x).
-If x is even, we can write it as x = 2m for some number m. But then  2y^2 = (2m)^2 = 4m^2, so y^2 = 2m^2. But then y^2 is even, so by our earlier argument, y is even too. But that means both x and y are even, which contradicts our starting assumption. So it must be that Sqrt(2) != a/b for any integers a and b. QED.
+:If x is even, we can write it as x = 2m for some number m. But then  2y^2 = (2m)^2 = 4m^2, so y^2 = 2m^2. But then y^2 is even, so by our earlier argument, y is even too. But that means both x and y are even, which contradicts our starting assumption. So it must be that Sqrt(2) != a/b for any integers a and b. QED.

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 ==Irrational Numbers==
 *I read that Kronecker opposed the use of irrational numbers because they can't be constructed from integers by a finite number of steps. But you can  construct pi as the ratio not only as an infinite series, but as the ratio of circumference to diameter. What's wrong with that?
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 ==Line Integrals==
 I should set up a latex file in a rasmapedia directory to explain line integrals.

← Older revision		Revision as of 00:56, 27 February 2022
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	:*Rasmusen-Ayres-Rowat-Beardon-Lehner-Erdos		:*Rasmusen-Ayres-Rowat-Beardon-Lehner-Erdos
		+
		+	----
		+	==Geometry==
		+	[https://mathcs.clarku.edu/~djoyce/java/elements/elements.html Joyce] has an amazingly good website on Euclid's Elements.

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 :*Rasmusen-Connell-Farb-Lubotzky-Alon-Erdos,
-**Rasmusen-Janssen-Sierksma-Doignon-Fishburn-Erdos
+:*Rasmusen-Janssen-Sierksma-Doignon-Fishburn-Erdos
-**Rasmusen-Ayres-Rowat-Beardon-Lehner-Erdos
+:*Rasmusen-Ayres-Rowat-Beardon-Lehner-Erdos
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