09-240/Classnotes for Tuesday September 15 - Revision history

Kjgry: Update file caption

2009-11-27T02:44:33Z

Update file caption

← Older revision		Revision as of 22:44, 26 November 2009
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	Image:09-240 Classnotes for Tuesday September 15 2009 page 4.jpg\|Page 4		Image:09-240 Classnotes for Tuesday September 15 2009 page 4.jpg\|Page 4
	Image:09-240 Classnotes for Tuesday September 15 2009 page 5.jpg\|Page 5		Image:09-240 Classnotes for Tuesday September 15 2009 page 5.jpg\|Page 5
	Image:ALA240-2009_-_September_15th.pdf		Image:ALA240-2009_-_September_15th.pdf\|A complete copy of notes for the lecture given on September 15th by Professor Natan (in PDF format)
	</gallery>		</gallery>
	(In the above gallery, there is a complete set of notes for the lecture given by Professor Natan on September 15th in PDF form.)		(In the above gallery, there is a complete set of notes for the lecture given by Professor Natan on September 15th in PDF form.)

2009-11-06T01:15:38Z

← Older revision		Revision as of 21:15, 5 November 2009
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	Image:ALA240-2009_-_September_15th.pdf		Image:ALA240-2009_-_September_15th.pdf
	</gallery>		</gallery>
			(In the above gallery, there is a complete set of notes for the lecture given by Professor Natan on September 15th in PDF form.)

	The real numbers A set <math>\mathbb R</math> with two binary operators and two special elements <math>0, 1 \in \mathbb R</math> s.t.		The real numbers A set <math>\mathbb R</math> with two binary operators and two special elements <math>0, 1 \in \mathbb R</math> s.t.

2009-11-05T23:35:41Z

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			Image:ALA240-2009_-_September_15th.pdf
	</gallery>		</gallery>

2009-10-22T02:42:15Z

2009-09-19T22:04:55Z

← Older revision		Revision as of 18:04, 19 September 2009
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	<u>Proof</u>:		<u>Proof</u>:

	<math>a \in Z</math> has a multiplicative inverse modulo <math>m</math> if and only if a and m are relatively prime.		<math>a \in \mathbb Z</math> has a multiplicative inverse modulo <math>m</math> if and only if a and m are relatively prime.

	This can be shown using [http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity]:		This can be shown using [http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity]:

2009-09-19T22:04:29Z

← Older revision		Revision as of 18:04, 19 September 2009
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	<u>Proof</u>:		<u>Proof</u>:

	~~Given a finite set with~~ <math>~~m</math> elements~~ in ~~<math>\mathbb~~ Z</math>, an ~~element~~ <math>a</math> ~~will~~ ~~have~~ a ~~multiplicative~~ ~~inverse~~ ~~'''iff'''~~ ~~<math>gcd(a,~~m) = ~~1</math>~~		<math>a \in Z</math> has a multiplicative inverse modulo <math>m</math> if and only if a and m are relatively prime.

	This can be shown using [http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity]:		This can be shown using [http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity]:
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	: <math>x = a^{-1}</math>		: <math>x = a^{-1}</math>

	We have shown that <math>a</math> has a multiplicative inverse if <math>a</math> and <math>m</math> are relatively prime. It is therefore a natural conclusion that if <math>m</math> is prime all elements in the set will ~~satisfy~~ ~~<math>gcd(a,~~ m) = ~~1</math>~~		We have shown that <math>a</math> has a multiplicative inverse modulo m if <math>a</math> and <math>m</math> are relatively prime. It is therefore a natural conclusion that if <math>m</math> is a prime number all elements in the set will be relatively prime to m.

	----		----

2009-09-19T19:19:14Z

Tedious Theorem

← Older revision		Revision as of 15:19, 19 September 2009
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	== Tedious Theorem ==		== Tedious Theorem ==
	# <math>a + b = c + d \Rightarrow a = c </math> "cancellation property"		# <math>a + b = c + b \Rightarrow a = c </math> "cancellation property"
	#: Proof:		#: Proof:
	#: By F4, <math>\exists d \mbox{ s.t. } b + d = 0</math>		#: By F4, <math>\exists d \mbox{ s.t. } b + d = 0</math>

2009-09-17T23:02:42Z

27-n = 27*23? Where does the subtraction and variable come from?

← Older revision		Revision as of 19:02, 17 September 2009
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	One may interpret this as counting the units in a 23×27 rectangle; one may choose to count along either 23 rows or 27 columns, but both ways lead to the same answer.		One may interpret this as counting the units in a 23×27 rectangle; one may choose to count along either 23 rows or 27 columns, but both ways lead to the same answer.


	You may also think of it as 27-n=23 2323 + 23n = 27*23.
	Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 2<sup>3</sup> = 8, but 3<sup>2</sup> = 9.		Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 2<sup>3</sup> = 8, but 3<sup>2</sup> = 9.

2009-09-17T21:46:45Z

Examples

← Older revision		Revision as of 17:46, 17 September 2009
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	One may interpret this as counting the units in a 23×27 rectangle; one may choose to count along either 23 rows or 27 columns, but both ways lead to the same answer.		One may interpret this as counting the units in a 23×27 rectangle; one may choose to count along either 23 rows or 27 columns, but both ways lead to the same answer.

			You may also think of it as 27-n=23 2323 + 23n = 27*23.

	Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 2<sup>3</sup> = 8, but 3<sup>2</sup> = 9.		Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 2<sup>3</sup> = 8, but 3<sup>2</sup> = 9.

2009-09-17T15:47:54Z