1724: Proofs - Revision history

Asdf: /* Explanation */ fix transclusion to a pages to delete category

2024-09-28T02:40:48Z

‎Explanation: fix transclusion to a pages to delete category

172.69.70.228: /* Explanation */

2023-05-06T11:38:11Z

‎Explanation

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2022-05-04T22:37:52Z

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Ex Kay Cee Dee at 22:14, 4 May 2022

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Ex Kay Cee Dee at 22:13, 4 May 2022

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Ex Kay Cee Dee at 22:00, 4 May 2022

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Ex Kay Cee Dee at 21:46, 4 May 2022

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 # If the two integers have a common factor, it can be eliminated using the Euclidean algorithm.
 # Then √2 can be written as an irreducible fraction ''a''/''b'' such that ''a'' and ''b'' are coprime integers (having no common factors other than 1).
-# The equation ''a''/''b'' {{=}} √2, when multiplied by itself, gives ''a²''/''b²'' {{=}} 2, which can be rearranged as ''a²'' {{=}} 2''b²''.
+# The equation ''a''/''b'' = √2, when multiplied by itself, gives ''a²''/''b²'' = 2, which can be rearranged as ''a²'' = 2''b²''.
 # Therefore, ''a²'' is even because it is equal to 2''b²''. (2''b²'' is necessarily even because it is 2 times another whole number, and multiples of 2 are even.)
 # It follows that ''a'' must be even (as squares of odd integers are never even).
-# Because ''a'' is even, there exists an integer ''k'' that fulfills: ''a'' {{=}} 2''k''.
+# Because ''a'' is even, there exists an integer ''k'' that fulfills: ''a'' = 2''k''.
-# Substituting 2''k'' from step 7 for ''a'' in the second equation of step 4: 2''b²'' {{=}} (2''k'')''²'' is equivalent to 2''b²'' {{=}} 4''k²'', which is equivalent to ''b²'' {{=}} 2''k²''.
+# Substituting 2''k'' from step 7 for ''a'' in the second equation of step 4: 2''b²'' = (2''k'')''²'' is equivalent to 2''b²'' = 4''k²'', which is equivalent to ''b²'' = 2''k²''.
-# Because 2''k²'' is divisible by two and therefore even, and because 2''k²'' {{=}} ''b²'', it follows that ''b²'' is also even, which means that ''b'' is even.
+# Because 2''k²'' is divisible by two and therefore even, and because 2''k²'' = ''b²'', it follows that ''b²'' is also even, which means that ''b'' is even.
 # By steps 6 and 9, ''a'' and ''b'' are both even, which contradicts that ''a''/''b'' is irreducible as stated in step 3.
 ::'''''Q.E.D.'''''

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 Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to {{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.
-The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. {{w|Determinacy}} of infinite games is used as a tool in the set theory, however the deterministic process is rather a term of the {{w|stochastic process|stochastic processes theory}}, and the {{w|dynamical systems theory}}, branches of mathematics far from the abstract set theory, which makes the proof even more exotic. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]] and later in [[982: Set Theory]], another comic about a math class with a similar theme on how teachers teach their student mathematical proofs.
+The axiom of choice itself states that for every collection of disjoint nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. {{w|Determinacy}} of infinite games is used as a tool in the set theory, however the deterministic process is rather a term of the {{w|stochastic process|stochastic processes theory}}, and the {{w|dynamical systems theory}}, branches of mathematics far from the abstract set theory, which makes the proof even more exotic. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]] and later in [[982: Set Theory]], another comic about a math class with a similar theme on how teachers teach their student mathematical proofs.
 Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class. A very similar Miss Lenhart comic was later released with [[2028: Complex Numbers]].