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(R,+)
GR
0∈G, ∀(x, y)∈G2:x+y∈G, ∀x∈G:−x∈G
x∈G q ∈Zng ∈G
x−x
A B RA B
∀b∈B, ∀ε > 0:]b−ε, b +ε[∩A6=∅
G(R,+) G α
G∩]0, α[= ∅
R
G
G G
∀x∈R,∀α > 0, G ∩[x, x +α[6=∅
G0
α G ∩]0, α[ = ∅
Iα
2G∩I
G
G∩R∗
+m
G={km, k ∈Z}Zm
x y
X=Zx={kx, k ∈Z}, Y =Zy={ky, k ∈Z}, S ={mx +ny, (m, n)∈Z2}
X Y S (R,+) X
x Y y S
x y
Sx
y∈Q
x
yA B
A={kx, k ∈Z∗}, B ={ky, k ∈Z∗}
A∩B=∅
inf{|a−b|,(a, b)∈A×B}= 0
π
{cos n, n ∈Z}[−1,1]

(R,+)
G
∀α > 0, G ∩]0, α[6=∅
G x α > 0
g∈G∩]0, α[nx
g
ng ≤x < ng +g
ng +g < ng +α≤x+α
(n+ 1)g∈G∩]x, x +α[G
x y I
|x−y| ≤ α
2< α
x y G ∩I
x−y y −x G ∩]0, α[
α
2
G G
G g G −g∈G
g > 0
G∩]0, g]
g
m
m= min G∩R∗
+
m∈G∩]0, g]⊂R∗
+
k∈G∩R∗
+
k≤g k ∈G∩]0, g]m≤h
g < k m < h m ≤g
m G ∩]0, g]
Zm⊂G
g∈G∩R∗
+kg
m
k≤g
m< k + 1 ⇒km ≤g < (k+ 1)m⇒0≤g−km < m
G−km ∈G g −km ∈G∩[0, m[
m g −km ∈G∩R∗
+
g−km = 0 g=km g ∈Zm
S m > 0S=Zm
x= 1 ×x+ 0 ×y∈S p ∈Zx=pm
y= 0 ×1+1×y∈S q ∈Z∗y=qm
m=x
p=y q⇒x y=p q∈Q
x
yp q
x
y=p q∈Q⇒x p=y q
m x =pm y =qm x y Zm
i j
ix +jy = (ip +jq)m∈Zm
S⊂Zm S ∩]0, m[S
A∩B p q px =qy
x
y=p q∈Q
S α > 0
m n mx +ny ∈]0, α[
α < min(x, y)m n
a=mx b =−ny a ∈A b ∈B
∀α < min(x, y),inf{|a−b|,(a, b)∈A×B} ≤ α

[u, v] [−1,1]
ncos n∈[u, v]
α= arccos v β = arccos u[α, β]R
2πZ+ 2πZ1 2pi
Rm n m + 2πn ∈[α, β]
cos m∈[u, v]