Difference between revisions of "09240/Classnotes for Tuesday September 15"
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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 27n=23 23*23 + 23*n = 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.  
Revision as of 17:46, 17 September 2009

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

 is not a field because not every element has a multiplicative inverse.
 Let
 Then
 Therefore F4 fails; there is no number b in F_{6} s.t. a · b = 1

 


Theorem: F_{2} 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 27n=23 23*23 + 23*n = 27*23. Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 2^{3} = 8, but 3^{2} = 9.
Tedious Theorem
 "cancellation property"
 Proof:
 By F4,
 by F2
 by choice of d
 by F3

 Proof:
 by F3
 by adding the additive inverse of a to both sides


 Proof:
 by F3
 by F5

 So there is no 0^{−1}
 (Bonus)
Quotation of the Day
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