diff --git a/chapter-03/binseq/main.py b/chapter-03/binseq/main.py
index c39ee1b..4819a15 100644
--- a/chapter-03/binseq/main.py
+++ b/chapter-03/binseq/main.py
@@ -2,7 +2,10 @@ from binseq import REAL
 from math import isqrt  # integer sqrt
 
 # Theorem 3.1 (2)
+# Es gibt eine berechenbare Folge (qn)n∈N rationaler Zahlen, die schnell gegen x konvergiert.
+# d.h. |x − qi| < 2^(−i) für alle i ∈ N.
 
+# bin_seq(1/3) = 0.010101...
 def one_third(n):
     if n == 1: return "."
     if n % 2 == 0: return "0"
diff --git a/chapter-03/decidable/__pycache__/decidable.cpython-313.pyc b/chapter-03/decidable/__pycache__/decidable.cpython-313.pyc
deleted file mode 100644
index d03a1f4..0000000
Binary files a/chapter-03/decidable/__pycache__/decidable.cpython-313.pyc and /dev/null differ
diff --git a/chapter-03/decidable/main.py b/chapter-03/decidable/main.py
index 3ee7393..9309044 100644
--- a/chapter-03/decidable/main.py
+++ b/chapter-03/decidable/main.py
@@ -2,6 +2,7 @@ from decidable import REAL
 from gmpy2 import mpq
 
 # Theorem 3.1 (5)
+# Es gibt eine ganze Zahl z ∈ Z und eine entscheidbare Menge A ⊆ N mit x = z + xA, wobei xA := SUMi∈A 2^(−i−1)
 
 print(REAL.from_rational(mpq(1,3)).as_str(10))
 print(REAL.from_rational(mpq(41,5)).as_str(10))
diff --git a/chapter-03/dedekint/__pycache__/dedekint.cpython-313.pyc b/chapter-03/dedekint/__pycache__/dedekint.cpython-313.pyc
deleted file mode 100644
index 4ab8701..0000000
Binary files a/chapter-03/dedekint/__pycache__/dedekint.cpython-313.pyc and /dev/null differ
diff --git a/chapter-03/dedekint/main.py b/chapter-03/dedekint/main.py
index 15fd838..4bf32b8 100644
--- a/chapter-03/dedekint/main.py
+++ b/chapter-03/dedekint/main.py
@@ -3,6 +3,7 @@ from dedekint import REAL
 from gmpy2 import mpq
 
 # Theorem 3.1 (4)
+# Die Menge {q ∈ Q | q < x} ist eine entscheidbare Menge rationaler Zahlen (Dedekind’scher Schnitt)
 
 def sqrt_two(q):
     if q <= 0 or q * q < 2: return True
diff --git a/chapter-03/interval/__pycache__/interval.cpython-313.pyc b/chapter-03/interval/__pycache__/interval.cpython-313.pyc
new file mode 100644
index 0000000..63dadc4
Binary files /dev/null and b/chapter-03/interval/__pycache__/interval.cpython-313.pyc differ
diff --git a/chapter-03/interval/interval.py b/chapter-03/interval/interval.py
new file mode 100644
index 0000000..17aef63
--- /dev/null
+++ b/chapter-03/interval/interval.py
@@ -0,0 +1,25 @@
+class REAL:  # nested.py
+    def __init__(self, l, u):
+        self.lower = l
+        self.upper = u
+
+    def asString(self, n):
+        l = self.lower(n)
+        u = self.upper(n)
+        return ("[ " + l.numerator.digits() + "/" + l.denominator.digits() +
+                ", " + u.numerator.digits() + "/" + u.denominator.digits() + "]")
+
+    def __add__(self, y):
+        return REAL_add(self, y)
+
+
+class REAL_add(REAL):
+    def __init__(self, x, y):
+        self.x = x
+        self.y = y
+
+    def lower(self, n):
+        return self.x.lower(n) + self.y.lower(n)
+
+    def upper(self, n):
+        return self.x.upper(n) + self.y.upper(n)
diff --git a/chapter-03/interval/main.py b/chapter-03/interval/main.py
new file mode 100644
index 0000000..80cfb61
--- /dev/null
+++ b/chapter-03/interval/main.py
@@ -0,0 +1,17 @@
+from interval import REAL
+from gmpy2 import mpq # rationale Zahlen
+
+
+def lower_one(n):
+    return mpq(1,1)-mpq(1,n)
+def upper_one(n):
+    return mpq(1,1)+mpq(1,n)
+
+x = REAL(lower_one,upper_one); y = x + x
+
+print ("x: "+x.asString(10))
+print ("y: "+y.asString(10))
+
+
+print ("x: "+x.asString(10000))
+print ("y: "+y.asString(10000))
\ No newline at end of file
diff --git a/chapter-03/ratseq/binseq.py b/chapter-03/ratseq/binseq.py
new file mode 100644
index 0000000..dea2445
--- /dev/null
+++ b/chapter-03/ratseq/binseq.py
@@ -0,0 +1,11 @@
+from collections.abc import Callable
+
+class REAL:
+    def __init__(self, f: Callable[[int], str]) -> None:
+        self.binseq = f
+
+    def as_string(self, w: int) -> str:
+        s = ""
+        for n in range(w):
+            s += self.binseq(n)
+        return s
\ No newline at end of file
diff --git a/chapter-03/ratseq/main.py b/chapter-03/ratseq/main.py
new file mode 100644
index 0000000..e9aa7f3
--- /dev/null
+++ b/chapter-03/ratseq/main.py
@@ -0,0 +1,66 @@
+from gmpy2 import mpq  # rationale Zahlen
+from binseq import REAL as REAL_bs
+from ratseq import REAL as REAL_ra
+
+
+def ra_oneThird(p):
+    return mpq(1, 3)
+
+
+x = REAL_ra(ra_oneThird)
+print("x:  " + x.asString(-50))
+
+# Theorem 3.1 (3)
+# Es gibt eine berechenbare Intervallschachtelung von Intervallen mit rationalen Endpunkten, die gegen x konvergiert
+# d.h. es gibt berechen-
+# bare Folgen (an)n∈N und (bn)n∈N
+
+class REAL_ra_from_bs(REAL_ra):  # Subklasse
+    def __init__(self, x):
+        self.x = x
+
+    def approx(self, p):
+        n = 0
+        q = mpq(0, 1)
+        v = mpq(1, 1)
+        while True:
+            c = self.x.binseq(n)
+            if c == "-": v = mpq(-1, 1)
+            if c == ".": break
+            q = 2 * q
+            if c == "1":  q += v
+            n = n + 1
+        k = 1
+        while k <= -p:
+            v = v / 2
+            c = self.x.binseq(n + k)
+            if c == "1": q += v
+            k = k + 1
+        return q
+
+
+def bs_one_third(n):
+    if n == 1: return "."
+    if n % 2 == 0: return "0"
+    return "1"
+
+
+def bs_almost_one(n):
+    if n == 0: return "1"
+    if n == 1: return "."
+    if n < 40: return "0"
+    return "1"
+
+
+# Create binseq
+y1 = REAL_bs(bs_almost_one)
+# Create rational approx of the created binseq
+z1 = REAL_ra_from_bs(y1)
+
+# Create binseq
+y2 = REAL_bs(bs_one_third)
+# Create rational approx of the created binseq
+z2 = REAL_ra_from_bs(y2)
+
+print("y1: " + y1.as_string(50) + "\nz1: " + z1.asString(-50))
+print("y2: " + y2.as_string(50) + "\nz2: " + z2.asString(-50))
\ No newline at end of file
diff --git a/chapter-03/ratseq/ratseq.py b/chapter-03/ratseq/ratseq.py
new file mode 100644
index 0000000..470c135
--- /dev/null
+++ b/chapter-03/ratseq/ratseq.py
@@ -0,0 +1,6 @@
+class REAL:
+    def __init__(self, f): self.approx = f
+
+    def asString(self, p):
+        q = self.approx(p)
+        return q.numerator.digits() + "/" + q.denominator.digits()
\ No newline at end of file