09240/Classnotes for Tuesday September 15

Yangjiay  Page 1
(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 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:
has a multiplicative inverse modulo if and only if a and m are relatively prime.
This can be shown using Bézout's identity:
We have shown that has a multiplicative inverse modulo m if and are relatively prime. It is therefore a natural conclusion that if is a prime number all elements in the set will be relatively prime to m.
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.
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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