diff --git a/airpower/aircraft.py b/airpower/aircraft.py
index 67fb9111..144f7476 100644
--- a/airpower/aircraft.py
+++ b/airpower/aircraft.py
@@ -10,6 +10,8 @@ class Aircraft:
 
   def __init__(self, name, x, y, azimuth, altitude):
 
+    apaltitude._checkaltitude(altitude)
+
     self._turn          = 0
     self._name          = name
     self._x             = x
@@ -128,22 +130,11 @@ def _L(self, facingchange):
         self._y += 0.25
     self._facing = (self._facing + facingchange) % 360
 
-
-  def _adjustaltitude(self, altitudechange):
-    # Here we do altitude arithmetic, ensuring that _altitude stays as an
-    # integer and keeping track of fractions in _altitudecarry.
-    self._altitude = self._altitude + self._altitudecarry + altitudechange
-    if altitudechange >= 0:
-      self._altitudecarry = self._altitude - math.floor(self._altitude)
-    else:
-      self._altitudecarry = self._altitude - math.ceil(self._altitude)
-    self._altitude = self._altitude - self._altitudecarry
-
   def _D(self, altitudechange):
-    self._adjustaltitude(-altitudechange)
+    self._altitude, self._altitudecarry = apaltitude._adjustaltitude(self._altitude, self._altitudecarry, -altitudechange)
 
   def _C(self, altitudechange):
-    self._adjustaltitude(+altitudechange)
+    self._altitude, self._altitudecarry = apaltitude._adjustaltitude(self._altitude, self._altitudecarry, +altitudechange)
 
   def _report(self, s):
     print("%s: turn %d: %s" % (self._name, self._turn, s))
@@ -168,8 +159,8 @@ def start(self, turn, nfp, s):
     self._report("--- start of turn ---")
     self._report("%d FPs available." % self._nfp)
     self._report("initial azimuth        = %s" % apazimuth.toazimuth(self._facing))
-    self._report("initial altitude       = %2.0f %s" % (self._altitude, apaltitude.altitudeband(self._altitude)))
-    self._report("initial altitude carry = %+.02f" % (self._altitudecarry))
+    self._report("initial altitude       = %d %s" % (self._altitude, apaltitude._altitudeband(self._altitude)))
+    self._report("initial altitude carry = %s" % apaltitude._formataltitudecarry(self._altitudecarry))
 
     if s != "":
       self.next(s)
@@ -270,10 +261,12 @@ def next(self, s):
       self._report("all FPs used.")
 
       self._report("final azimuth        = %s" % apazimuth.toazimuth(self._facing))
-      self._report("final altitude       = %2.0f %s" % (self._altitude, apaltitude.altitudeband(self._altitude)))
-      self._report("final altitude carry = %+.02f" % (self._altitudecarry))
-      if apaltitude.altitudeband(self._initialaltitude) != apaltitude.altitudeband(self._altitude):
-        self._report("altitude band changed from %s to %s." % (apaltitude.altitudeband(self._initialaltitude), apaltitude.altitudeband(self._altitude)))
+      self._report("final altitude       = %2.0f %s" % (self._altitude, apaltitude._altitudeband(self._altitude)))
+      self._report("final altitude carry = %s" % apaltitude._formataltitudecarry(self._altitudecarry))
+
+      if apaltitude._altitudeband(self._initialaltitude) != apaltitude._altitudeband(self._altitude):
+        self._report("altitude band changed from %s to %s." % (apaltitude._altitudeband(self._initialaltitude), apaltitude._altitudeband(self._altitude)))
+
       self._report("--- end of turn ---")
 
       self._save(self._turn)
diff --git a/airpower/altitude.py b/airpower/altitude.py
index 98cc153f..20e061d3 100644
--- a/airpower/altitude.py
+++ b/airpower/altitude.py
@@ -1,6 +1,55 @@
+import math
+
 print("airpower.altitude")
 
-def altitudeband(altitude):
+# _altitudequantum must be 1 over an integral power of 2.
+_altitudequantum = 1/8
+
+def _checkaltitude(altitude):
+  if not isinstance(altitude, (int, float)) or altitude % 1 != 0 or altitude < 1:
+    raise ValueError("invalid altitude %s." % altitude)
+
+def _adjustaltitude(altitude, altitudecarry, altitudechange):
+
+    # Here we do altitude arithmetic, ensuring that _altitude stays as an
+    # integer and keeping track of fractions in _altitudecarry. We require 
+    # altitude changes to be multiples of _altitudequantum.
+
+    assert altitude % 1 == 0 and altitude > 0
+    assert altitudecarry % _altitudequantum == 0
+    assert altitudechange % _altitudequantum == 0
+
+    altitude = altitude + altitudecarry + altitudechange
+    if altitudechange >= 0:
+      altitudecarry = altitude - math.floor(altitude)
+    else:
+      altitudecarry = altitude - math.ceil(altitude)
+    altitude = altitude - altitudecarry
+
+    assert altitude % 1 == 0
+    assert altitudecarry % _altitudequantum == 0
+
+    return altitude, altitudecarry
+
+def _formataltitudecarry(altitudecarry):
+
+  assert altitudecarry % _altitudequantum == 0
+
+  n = altitudecarry / _altitudequantum
+  m = 1 / _altitudequantum
+
+  if n == 0:
+    return "0"
+
+  while n % 2 == 0:
+    n = n / 2
+    m = m / 2
+  return "%+d/%d" % (n, m)
+  
+def _altitudeband(altitude):
+
+  assert altitude % 1 == 0 and altitude > 0
+
   if altitude <= 7:
     return "LO"
   elif altitude <= 16: