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Math 521: Lecture Notes, Fall 2004
Arun Ram
University of Wisconsin-Madison
480 Lincoln Drive
Madison, WI 53706
[email protected]
1 Numbers
The positive integers is the set
Z> 0 = { 1 , 2 , 3 ,.. .} with the operation Z>^0 × (^ Zi, j>^0 )^ → 7 →^ Zi +>^0 j
The nonnegative integers is the set
Z≥ 0 = { 0 , 1 , 2 , 3 ,.. .} with the operation Z≥^0 × (^ Zi, j≥^0 )^ → 7 →^ Zi +≥^0 j
The advantage of the nonnegative integers Z≥ 0 over the positive integers Z> 0 is that Z≥ 0 contains an identity element for the operation and Z> 0 does not.
The integers is the set Z = {... , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 ,.. .}
with the operations Z × Z → Z (i, j) 7 → i + j The advantage of the integers Z over the nonnegative integers Z≥ 0 is that every element of Z has an inverse; this is not true in Z≥ 0. There is another operation on Z,
Z × Z → Z (i, j) 7 → i + (−j)
but this operation is not very well behaved: it is not associative and not commutative (though it does have an identity). There is another operation on Z
Z × Z → Z (i, j) 7 → ij
and this operation is associative, commutative and has an identity but does not have inverses.
The rationals is the set
Q =
{ (^) a b
| a, b ∈ Z, b 6 = 0
where a b
= c d
if ad = bc,
with operations defined by
a b
= ad^ +^ bc bd
and a b
c d
= ac bd
The advantage of the rationals Q over the integers Z is that the multiplication has inverses; well,
... almost has inverses–the element 0 does not have an inverse.
By long division, every rational number ab can be represented as a decimal expansion drdr− 1 · · · d 1 d 0 .d− 1 d− 2 d− 3 · · ·
where the idea is that
drdr− 1 · · · d 1 d 0 .d− 1 d− 2 d− 3 · · · =
∈Z,≤r
d10.
If a = ar... a 1 a 0 .a− 1 a− 2... is a decimal expansion let a≤n be the element of Q given by
a≤n = ar... a 1 a 0 .a− 1 a− 2... a−(n−1)a−n.
The real numbers is the set R of decimal expansions
R = { dr · · · d 1 d 0 .d− 1 d− 2 · · · | di ∈ { 0 , 1 , 2 ,... , 9 } }
with a = b for all n ∈ Z> 0 (a≤n − b≤n)≤n− 1 = 0 in Q,
and operations determined by
a + b = c if, for all n ∈ Z> 0 , (a≤n + b≤n)≤n− 1 = c≤n− 1 in Q,
and ab = c if, for all n ∈ Z> 0 , (a≤nb≤n)≤n− 1 = c≤n− 1 in Q.
An irrational number is a real number that is not a rational number.
Theorem 1.1. Q = {decimal expansions that repeat}
Theorem 1.2. Irrational numbers exist.
The complex numbers is the set
C = {a + bi | a, b ∈ R}
with operations given by
(a 1 + b 1 i) + (a 2 + b 2 i) = (a 1 + a 2 ) + (b 1 + b 2 )i and (a 1 + b 1 i)(a 2 + b 2 i) = (a 1 a 2 − b 1 b 2 ) + (a 1 b 2 + a 2 b 1 )i.
Theorem 1.3. (The fundamental theorem of algebra) If p 0 , p 1 ,... , pd ∈ C with pd 6 = 0 then there are λ 1 ,... , λd ∈ C such that
p 0 + p 1 x + p 2 x^2 + · · · + pdxd^ = (x − λ 1 )(x − λ 2 ) · · · (x − λd).
The algebraic numbers is the set
Q = {z ∈ C | there exists p(x) ∈ Q[x], p(x) 6 = 0, with p(z) = 0}.
A transcendental number is a complex number that is not an algebraic number.
