09-240/Classnotes for Tuesday September 15
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Yangjiay - Page 1
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: F2 is a field.
In order to prove that the associative property holds, make a table (similar to a truth table) for a, b and c.
a | b | c | |
---|---|---|---|
0 | 0 | 0 | |
0 | 0 | 1 | |
0 | 1 | 0 | |
0 | 1 | 1 | (0 + 1) + 1 =? 0 + (1 + 1) 1 + 1 =? 0 + 0 0 = 0 |
1 | 0 | 0 | |
1 | 0 | 1 | |
1 | 1 | 0 | |
1 | 1 | 1 |
Theorem: for is a field iff (if and only if) is a prime number
Proof:
Given a finite set with elements in , an element will have a multiplicative inverse iff
This can be shown using Bézout's identity:
We have shown that has a multiplicative inverse if and are relatively prime. It is therefore a natural conclusion that if is prime all elements in the set will satisfy
Multiplication is repeated addition.
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 23*23 + 23*n = 27*23. Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 23 = 8, but 32 = 9.
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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