Difference between revisions of "09-240/Classnotes for Tuesday September 15"
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# <math>(-a) \cdot (-b) = a \cdot b</math> | # <math>(-a) \cdot (-b) = a \cdot b</math> | ||
# (Bonus) <math>\,\! (a + b)(a - b) = a^2 - b^2</math> | # (Bonus) <math>\,\! (a + b)(a - b) = a^2 - b^2</math> | ||
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+ | == Quotation of the Day == | ||
+ | ...... |
Revision as of 00:26, 16 September 2009
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The real numbers A set with two binary operators and two special elements s.t.
- Note: or means inclusive or in math.
Definition: A field is a set F with two binary operators : F×F → F, : F×F → F and two elements s.t.
Examples
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- is not a field because not every element has a multiplicative inverse.
- Let
- Then
- Therefore F4 fails; there is no number b in F6 s.t. a · b = 1
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Theorem: for is a field iff (if and only if) is a prime number
Tedious Theorem
- "cancellation property"
- Proof:
- By F4,
- by F2
- by choice of d
- by F3
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- Proof:
- by F3
- by adding the additive inverse of a to both sides
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- Proof:
- by F3
- by F5
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- So there is no 0−1
- (Bonus)
Quotation of the Day
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