Difference between revisions of "09-240/Classnotes for Tuesday September 15"

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'''Theorem''': <math>\,\! F_p </math> for <math>p > 1</math> is a field ''iff'' <small>([http://en.wikipedia.org/wiki/If_and_only_if if and only if])</small> <math>p</math> is a prime number
 
'''Theorem''': <math>\,\! F_p </math> for <math>p > 1</math> is a field ''iff'' <small>([http://en.wikipedia.org/wiki/If_and_only_if if and only if])</small> <math>p</math> is a prime number
  
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Multiplication is repeated addition.
 
Multiplication is repeated addition.

Revision as of 01:46, 16 September 2009

The real numbers A set \mathbb R with two binary operators and two special elements 0, 1 \in \mathbb R s.t.

F1.\quad \forall a, b \in \mathbb R, a + b = b + a \mbox{ and } a \cdot b = b \cdot a
F2.\quad \forall a, b, c, (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)
\mbox{(So for any real numbers } a_1, a_2, ..., a_n, \mbox{ one can sum them in any order and achieve the same result.}
F3.\quad \forall a, a + 0 = a \mbox{ and } a \cdot 0 = 0 \mbox{ and } a \cdot 1 = a
F4.\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1
\mbox{So } a + (-a) = 0 \mbox{ and } a \cdot a^{-1} = 1
\mbox{(So } (a + b) \cdot (a - b) = a^2 - b^2)
\forall a, \exists x, x \cdot x = a \mbox{ or } a + x \cdot x = 0
Note: or means inclusive or in math.
F5.\quad (a + b) \cdot c = a \cdot c + b \cdot c

Definition: A field is a set F with two binary operators \,\!+: F×FF, \times\,\!: F×FF and two elements 0, 1 \in \mathbb R s.t.

F1\quad \mbox{Commutativity } a + b = b + a \mbox{ and } a \cdot b = b \cdot a \forall a, b \in F
F2\quad \mbox{Associativity } (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)
F3\quad a + 0 = a, a \cdot 1 = a
F4\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1
F5\quad \mbox{Distributivity } (a + b) \cdot c = a \cdot c + b \cdot c

Examples

  1. F = \mathbb R
  2. F = \mathbb Q
  3. \mathbb C = \{ a + bi : a, b \in \mathbb R \}
    i = \sqrt{-1}
    \,\!(a + bi) + (c + di) = (a + c) + (b + d)i
    \,\!0 = 0 + 0i, 1 = 1 + 0i
  4. \,\!F_2 = \{ 0, 1 \}
  5. \,\!F_7 = \{ 0, 1,2,3,4,5,6 \}
  6. \,\!F_6 = \{ 0, 1,2,3,4,5 \} is not a field because not every element has a multiplicative inverse.
    Let a = 2.
    Then a \cdot 0 = 0, a \cdot 1 = 2, a \cdot 3 = 0, a \cdot 4 = 2, a \cdot 5 = 4
    Therefore F4 fails; there is no number b in F6 s.t. a · b = 1
Ex. 4
+ 0 1
0 0 1
1 1 0
Ex. 4
× 0 1
0 0 0
1 0 1
Ex. 5
+ 0 1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 1 2 3 4 5 6 0
2 2 3 4 5 6 0 1
3 3 4 5 6 0 1 2
4 4 5 6 0 1 2 3
5 5 6 0 1 2 3 4
6 6 0 1 2 3 4 5
Ex. 5
× 0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 1 3 5
3 0 3 6 2 5 1 4
4 0 4 1 5 2 6 3
5 0 5 3 1 6 4 2
6 0 6 5 4 3 2 1

Theorem: F2 is a field.

In order to prove that the associative property holds, make 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: \,\! F_p for p > 1 is a field iff (if and only if) p is a prime number


Multiplication is repeated addition.

23 \times 27 = \begin{matrix} 27 \\ \overbrace{23 + 23 + 23 + \cdots + 23} \end{matrix} = 621

27 \times 23 = \begin{matrix} 23 \\ \overbrace{27 + 27 + 27 + \cdots + 27} \end{matrix} = 621

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; 23 = 8, but 32 = 9.

Tedious Theorem

  1. a + b = c + d \Rightarrow a = c "cancellation property"
    Proof:
    By F4, \exists d \mbox{ s.t. } b + d = 0
    \,\! (a + b) + d = (c + b) + d
    \Rightarrow a + (b + d) = c + (b + d) by F2
    \Rightarrow a + 0 = c + 0 by choice of d
    \Rightarrow a = c by F3
  2.  a \cdot b = c \cdot b , (b \ne 0) \Rightarrow a = c
  3. a + O' = a \Rightarrow O' = 0
    Proof:
    \,\! a + O' = a
    \Rightarrow a + O' = a + 0 by F3
    \Rightarrow O' = 0 by adding the additive inverse of a to both sides
  4. a \cdot l' = a, a \ne 0 \Rightarrow l' = 1
  5. a + b = 0 = a + b' \Rightarrow b = b'
  6. a \cdot b = 1 = a \cdot b' \Rightarrow b = b' = a^{-1}
    \,\! \mbox{Aside: } a - b = a + (-b)
    \frac ab = a \cdot b^{-1}
  7. \,\! -(-a) = a, (a^{-1})^{-1}
  8. a \cdot 0 = 0
    Proof:
    a \cdot 0 = a(0 + 0) by F3
    = a \cdot 0 + a \cdot 0 by F5
    = 0 = a \cdot 0
  9. \forall b, 0 \cdot b \ne 1
    So there is no 0−1
  10. (-a) \cdot b = a \cdot (-b) = -(a \cdot b)
  11. (-a) \cdot (-b) = a \cdot b
  12. (Bonus) \,\! (a + b)(a - b) = a^2 - b^2

Quotation of the Day

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