Abbildung   -  mapping; (sometimes) illustration
	aber  -  but, however, yet
	abfassen  -  compose, write up, draw up, word
	Abfassung  -  writing, formulation, wording
	Abgrenzung  -  definition
	abhangen  -  depend on
	Abkurzung  -  contraction, abbreviation, abridgement
	ableiten  -   lead out of; deduce
	Ableitung  -  derivative, derivation; deduction
	ablesen  -  pick (off), read (off), read out
	abmachen  -  settle, arrange; undo, untie
	Abmachung  -  arrangement, settlement
	abnehmen  -  decrease, take off, remove
	Abnehmen  -  reduction (see verb abnehmen)
	Absatz  -  pause, break, indentation; paragraph
	abschatzen  -  estimate, calculate, assess, appraise, etc.
	Abschatzung  -  estimate, appraisal, valuation
	Abschnitt  -  intersection; segment; cut, paragraph, part, section
	Abschnitt auf der x--Achse  -  x-intercept 
	absehen  -  foresee
	absehen von  -  disregard; refrain from
	absolut  -  absolute, unconditional, perfectly, utterly
	Abstand  -  distance
	absteigen  -  descend, come or go down
	abzahlbar  -  countable
	Abzahlbarkeit  -  countability
	abzahlen  -  count, enumerate
	achten (auf)  -  pay attention (to)
	ahnlich  -  alike, like, similar
	all  -  all, every, any
	allein  -  only, merely, solely; alone
	allerdings  -  certainly (strongest possible affirmative answer)
	allgemein  -  general, universal
	allmahlich  -  gradually, little by little
	alsdann  -  then, thereupon, whereupon, after that
	analog  -  analogous
	Anblick  -  look, view
	andere  -  other; next
	andererseits  -  on the other hand
	andern  -  alter, change, modify, amend
	Anderung  -  change, alteration, modification
	andeuten  -  indicate, suggest, hint at, allude to, imply
	Anfang  -  origin, beginning, start
	anfangen  -  start, begin, commence
	angeben  -  indicate, state
	angehoren  -  be a member (of)
	angewandt - applied
	angrenzen  -  border, adjoin
	Anhang  -  appendix, codicil
	ankommen  -  arrive (at); es kommt derauf an = it is a matter of
	Anlass  - motive, motivation; cause, occasion, incentive, reason
	Annaherung  -  approach
	Annahme  -  acceptance, adoption
	annehmen  -  accept, take, assume, suppose
	anordnen  -  arrange, order, command, direct, set in order
	Anordnung  -  arrangement
	anpassen  -  fit, adapt, adjust, accommodate
	Ansage  -  announcement
	ansagen  -  announce, declare
	anschaffen  -  procure, get
	anschliessend  -  subsequent (to), adjacent (to); next
	Anschluss  -  connection, union
	ansehen  -  look at, have a look at
	ansetzen  -  put (on), apply (to); begin
	Anspruch  -  claim
	Anstoss  -  impact, impluse
	anstossen  -  bump against, adjoin
	anstreben  -  strive for
	anwenden  -  employ, use, make use of
	Anwendung  -  application
	Anzahl  -  number, quantity
	anzusehen  -  see ansehen
	arbeiten  -  work
	Art  -  kind, sort, class
	auch  -  also, too
	auf  -  on, upon, on top of
	auf Grund  -  on the basis (of)
	aufbauen  -  construct
	auffassen  -  comprehend, grasp, understand
	Aufgabe  -  exercise, problem
	aufgehen  -  divide exactly
	Auflosung  -  solution, resolution (of)
	aufschreiben  -  write down
	Aufstellung  -  assertion; list, table
	auftragen  -  instruct, order, serve, put on
	auftreten  -  appear, occur, behave, act, proceed
	Auftritt  -  appearance, entrance
	Augenblick  -  moment, instant
	aus  -  from, out of
	ausarten  -  get out of hand, degenerate into
	Ausblick  -  view, outlook, prospect
	ausdehnen  -  stretch, expand, extend
	Ausdehnung  - stretching, expansion
	ausdruchlich  -  explicit, explictly
	Ausdruck  -  expression, phrase, term
	ausdrucken  -  express, put into (a form, words)
	ausfuhrlich  -  detailed, in detail, full
	Ausfuhrung  -  execution, construction; explanation
	ausgehen  -  proceed from
	ausgewahlt  -  selected, chosen, choice
	ausgezeichnet  -  excellent
	ausmachen  -  consist of, make up, add or amount up to, constitute
	Ausnahme  -  exception
	ausnahmslos  -  without exception
	ausreichen   -  suffice
	ausreichend  -  adequate, sufficient, enough
	Aussage  -  assertion, statement
	aussagen  -  state, express, declare
	Aussenrand  -  outer boundary
	Aussenwelt  -  environment, world, outside world
	ausser  -  out of, outside
	ausserdem  -  besides, in addition, moreover, as well as
	ausserhalb  -  outside, out of
	ausstreichen  -  cross out, delete, erase; smooth out
	Auswahl  -  choice, selection, assortment
	auswahlen  -  choose, select, pick out, fix on
	auswerten  -  evaluate
	auszahlen  -  count
	auszeichnen  -  designate (notation), mark
	Auszeichnung  -  labelling; distinction
	
	
	
	
	 Band   -  volume (as of a book)
	Bauart  - structure, build, type, model
	bauen  -  build, erect, make, manufacture
	beachten  -  note, pay attention to
	bedecken  -  cover
	bedenken  -  consider, weigh, ponder
	Bedenken  -  thought, reflection, deliberation
	bedeuten  -  mean or signify
	bedienen  -  wait on, serve
	bedingen  -  cause, determine, or condition
	Bedingung  -  condition, stipulation
	befassen  -  touch, handle
	befassen sich mit  -  concern oneself with
	befriedigen  -  satisfy
	begegnen  -  meet, come upon
	begnugen (mit)  -  be satisfied or content (with)
	begrenzt  -  bounded  (restricted, limited, circumscribed)
	Begriff  -  idea, notion, concept, term
	behandeln  -  deal with, treat, handle
	behaupten  -  declare, assert, maintain, retain
	Behauptung  -  declaration, assertion
	beherrschen  -  rule, master, control
	Behruhrungspunkt  -  accumulation point, limit point (topology)
	bei  -  at
	Beibehaltung  -  retaining
	beiden  -  two, both, either, neither
	beiderseits  -  reciprocally, mutually, on both sides
	beim  -  bei dem
	Beiwort  -  adjective
	bekannt  -  well known
	belasten  -  load, put a load on, make heavy
	belegen  -  verify (a proof)
	belegt  -  reserved, coated, husky, busy
	Beleuchtung  -  elucidation; illumination, lighting
	Belieben  -  choice, discretion, convenience, liking, pleasure
	belieben  -  wish
	beliebig(e)  -  optional, arbitrary
	bemerken  -  remark, notice, observe
	benotigen  -  need
	benutzen  -  use, make use of, employ, utilize
	bequem  -  comfortable, easy
	Berechnung  -  calculation 
	Bereich  -  region; (sometimes) open set
	bereits  -  already
	berucksichtigen  -  bear in mind, consider
	Berufung  -  call, vocation, appointment
	beruhen  -  be based (on); be due (to)
	beruhren  -  touch, mention, affect, pass through, come in contact
	besagen  -  signify, mean
	besagt  -  aforesaid
	beschaffen  -  procure, provide, supply, get, raise
	Beschaffenheit  -  quality, property
	Beschaffung  -  acquisition; procurement
	beschranken  -  limit, restrict, confine, set limits
	beschrankt  -  bounded, limited
	beschreiben  -  describe
	besitzen  -  possess, own, have
	besondere  -  particular, special
	besonders  -  especially, particularly, in particular
	bestatigen  -  confirm, corroborate, verify
	bestehen  -  exist
	Bestehen  -  existence
	bestehen auf - consists of
	bestehend  -  existing, present
	bestimmen  -  determine
	bestimmt  -  determined
	betonen  -  stress, accent, emphasize
	Betracht  -  (in) consideration
	betrachten  -  observe
	Betrachtung  -  observation, reflection, reason
	Betrag  -  value, amount; absolute value
	Betrag, absolut  -  absolute value
	betreffen  -  concern, affect, be relevant to, have to do with, (to study?)
	betrifft  -  in regards
	bewaltigen  -  conquer, overcome, master (difficulties)
	bewegen  -  move, stir
	Beweis  -  proof
	beweisen  -  prove, show, demonstrate
	bezeichnen  -  designate (by), define
	Bezeichnung  -  notation; designation
	beziehen (auf)  -  relate or refer (to); move into
	Beziehung  -  relation, connection, respect
	Bezug auf, in  -  with respect to, in reference to
	bezuglich  -  relative to, relating to
	Bezugnahme auf, unter  -  with reference to
	bieten  -  give, show, offer; present itself
	Bild - image
	bilden  -  form, shape, fashion
	binar  -  binary
	binden  -  bind, tie unite, join
	bisher  -  hitherto, up to now, so far
	bisherig(an)  -  as up to the present, former, previous; to date, so far,
	bisweilen  -  occasionally, sometimes
	bleiben  -  remain, stay, continue
	bleiben -- erhalten - remain preserved
	Blick  -  view
	bloss  -  merely, only, simply
	brauchen  -  require
	Brief - letter
	bringen - bring
	Bruch  -  fraction
	Bruch, echte - proper fraction
	Buch - book
	Bundigkeit  -  validity, conclusiveness, conciseness, brevity
	bzw.  -  respectively (beziehungsweise)
	
	
	
	 da   -  there, because
	dabei  -  by or near it or them, in addition, at the same time
	dadurch  -  thereby, by this means, in that way
	dafur  -  therefore, for it or them, in its place
	dagegen  -  on the contrary, conversely, on the other hand, nevertheless
	daher  -  therefore
	damals  -  then, in those days, at that time
	damit  -  consequently; thereby, thereupon, with it or them
	danach  -  behind or after it, afterwards, thereupon
	dann  -  then, at that time, after that
	daraus  -  out of this
	daraus folgt  -  hence it follows (that)
	dargestellt  -  described
	darin  -  in it or in them, there
	darstellen  -  represent, portray, present, describe
	Darstellung  -  representation, description
	darunter  -  beneath or under (it or them), in it, among(st) them
	dass  -  that
	dasselbe  -  (denselbe, derselbe, dieselbe, etc.) the same
	davon   -  from it, of it, away from it
	definieren  -  define
	denen  -  (to) them or those, (to) those
	denken  -  think, imagine, mean, intend
	denn  -  than, for, because, unless, except
	derart  -  in such a way or manner, to such an extent
	derartig  -  such, of that kind
	deren  -  of them, (of) that, (of) those
	derjenige  -  (adj) that; (pron) the one; he
	derselbe  -  the same
	desgleichen  -  such a thing; the likes of which; likewise
	deshalb  -  therefore, for this reason
	desselbe  -  the same
	deuten  -  interpret, explain or indicate
	d.h.  -   that is (das heisst)
	dicht  -  dense (in topology); tight, thick
	dichten  -  tighten; write
	dichtesten  -  densest
	Dichtigkeit  -  density; conciseness
	diejenige  -  (adj) that; (pron) the one; he
	diesmal  -  this time
	Ding  -  object, thing, matter
	dingen  -  hire, engage, bargain, haggle
	diskutieren  -  discuss
	doch  -  but, though, yet, however, nevertheless
	dort  -  there, over there, yonder
	dortig  -  in or of that place, there
	Dreieck  -  triangle
	drucken  -  put (into); press; squeeze
	durch  -  through
	durchblicken  -  be apparent or obvious 
	durchblicken lassen  -  make apparent
	durchdringen  -  force one's way, penetrate
	durchfuhren  -  take or lead through, accomplish
	Durchfuhrung  -  accomplishment, execution, development
	durchlaufen  -  run or pass through, traverse
	durfen  -  may, might
	Durschnitt  -  intersection (set-theoretic)
	
	
	
	
	 Ebene   -  plane
	ebenfalls  -  likewise, also, too
	ebenso  -  likewise, in the same manner
	echte  -  genuine, real, true
	Eck  -  corner
	Eichkorper  -  standard ball?  unit ball?
	Eigenschaft  -  property, quality
	eigentlich  -  actual; actually, really
	einander  -  one another, each other
	eindeutig  -  unique; unambiguous, clear, plain
	Eindeutigkeit  -  uniqueness
	eineindeutig  -  one to one
	einerseits  - on the one hand
	einfach  -  simply; simple, easy
	Einfluss  -  influx, influence
	einfuhren  -  introduce
	Einfuhrung  -  introduction, presentation
	Eingang(en)  -  entries
	eingehen  -  enter into
	eingreifen  -  catch, hold; intervene or interfere in
	Einheit  -  unit
	einheiten  -  unimodular
	einig  -  united, at one
	einige  -  (pron) some, several
	einigen  -  unite, unify; agree
	einklammern  -  bracket, put into parentheses
	Einlagerung  -  deposit, intercalation, storage
	einleiten  -   lead into; introduce
	Einleitung  -  introduction, prelude
	einmal  -  once, at once
	einschalten  -  insert; intervene
	einschliessen  -  include
	Einschrankung  -  restriction
	einsehen  -  inspect, consult, realize
	einsetzen  -  insert, put or place in, substitute
	Einsetzung  -  putting in; substitution?
	einstellen  -  put or place (in the proper place); appear, show up
	einteilen  -  divide, distribute, classify
	Einteilung  -  classification
	einzelnen  -  individual
	einzig  -  only (darstellen einzig = be unique)
	endlich  -  finite; (final, ultimate)
	entdecken  -  discover, detect, find out
	entfernen  -  move away
	entfernt - distant
	Entfernung  -  distance, range
	enthalten  -  hold, contain, comprise, include
	Entscheidung  -  decision
	entsprechen  -  correspond (to)
	entsprechend  -  correspondingly, accordingly
	entstehen  -  originate, arise from
	entweder  -  either
	Entwicklung  -  development (sp. power series)
	erfahren  -  experienced
	erfahren  -  find out, experience
	erfordern  -  require, need, necessitate, render necessary
	erfullen  -  fill, fulfill, perform, carry out
	erganzen  -  complete, complement, restore
	erganzend  -  complementary
	erganzt - completed
	ergeben  -   produce, yield, show, establish, prove
	ergeben sich (aus)  -  result (from)
	Ergebnis  -  result, solution, answer
	ergiebig  -  productive, rich, abundant
	erhalten  -  obtain, get, receive
	erhalten - preserved
	erinnern  -  remind
	erkennen  -  make out, recognize, detect, realize
	erklaren  -  explain
	erlangen  -  reach, get 
	erlauben  -  allow
	erlautern  -  explain, elucidate, comment on
	Erlauterung  -  explanation, illustration
	erleichtern  -   facilitate
	ermitteln  -  ascertain, find out, discover
	Ermittelung  -  determination
	ernennen  -  name, select
	erneuern  -  renew, renovate
	eroffnen  -  open, unseal, disclose, reveal
	erortern  -  discuss
	erreichbar  -  attainable, accessible, available, within reach or distance
	erreichen  -  reach, attain, obtain, achieve
	Ersatz  -  replacement, substitution
	erscheinen  -  appear, show itself
	Erscheinung  -  appearance; phenomenon
	erschopft  -  spent, exhausted, at an end
	ersehen  -  gather (from)
	ersetzen  -  replace, compensate for
	Ersetzung  -  replacement, substitution
	ersichtlich  -  obvious, evident, apparent
	ersinnen  -  think out, devise, think up
	erst  -  first, at first, just, only
	erstrecken  -  extend, stretch, run, reach
	erwahnen  -  mention, make mention of, refer to
	erwahnt  -  mentioned
	Erweis  -  proof
	erweisen  -  prove, establish, demonstrate
	erweitern  -  widen, extend, broaden (one's knowledge)
	erweiterte  -  extended
	erzeugen  -  produce, make, generate
	erzeugend  -  productive, generative
	erzielen  -  obtain, attain, achieve, reach
	es gibt  -  there is
	es gilt  -  it is true
	es moge hinreichen zu  -  suffice it to say (let it suffice to say)
	es mogen  -  let...
	etwa  -  around, about, approximately; perhaps, by (any) chance
	etwas  -  something, some, a little, anything, somewhat
	eventuell  -  eventual; possibly
	
	
	
	 fahig   -  capable, liable, apt
	fahig, ist  -  is permitted
	Fahigkeit  -  ability, capability
	Fall  -  (in this) case, instance
	falls  -  in (the) case
	Faltung  -  fold, convolution
	Faser  -  fiber
	Faserraum  -  fiber bundle
	fassen  -  grasp, seize, take hold of
	fast  -  almost, about
	Fehlen  -  absence
	fehlen  -  do wrong, fail, be short of; be missing, be absent
	fern  -  far, distant, remote, far off
	ferner  -  further, farther, furthermore, moreover
	fest  -  firm, stable, fixed, stationary
	festhalten  -  hold on to
	festsetzen  -  arrange, settle, establish, fix, stipulate, prescribe
	Festsetzung  -  requirement, stipulation
	feststehen  -  stand firm or steady, be certain, settled, established as a fact
	feststellen  -  ascertain
	Flache  -  surface
	Folge  -  consequence
	Forderung  -  claim
	Forderung (w/ o umlaut)  -  encouragement
	Form  -  form, figure, shape
	formal  -  formal
	formig  -  (comb.) -shaped, -formed
	Formulierung  -  formulation, wording, definition
	fortsetzen  -  continue (sp. analytic continuation); see also setzen
	Fortsetzung  -  continuation (sp. analytic)
	fortwahrend  -  lasting, continual; always
	Frage  -  question
	fragen - ask
	freilich  -  indeed, certainly
	fruher  -  earlier, previous, prior, former
	fugen  -  join; be fitted or proper
	fuhren  -  lead
	
	
	
	 gab   -  is given
	ganz  -  integer; whole; entire (sp. complex functions)
	ganzzahlige  -  integral (eg., linear combination), integer-valued
	gar  -  entirely, absolutely, fully, very, quite
	Garbe  -  sheaf
	geben  -  give
	Gebiet  -  area, region, domain; (sometimes) open connected set
	gebildet  -  educated, cultured, refined, shaped, fashioned
	Gebrauch  - use
	gebrauchen  -  use, employ
	gebrochen  -  broken, bent, deflected
	gebunden  -  bound, latent, tied up (with), controlled
	geeignete  -  suitable, appropriate, desired, wanted
	Gefallen  -  favor
	gefallen  -  please
	gefunden  -  found (pp of finden)
	gegeben  -  given, state, accepted
	gegen  -  against; wrt. convergance, to (konvergieren gegen)
	Gegenbeispiel  -  counterexample
	Gegensatz  -  converse theorem
	Gegensatz, in  -  in contrast
	Gegenstand  -  object, thing, subject matter
	gegenuber  -  opposite to; with respect to; compared with
	gehen  -  go
	gehoren  -  belong (to, with); be proper, be right
	gehorig  -  belonging to
	gehort  -  belonging (to, with); proper
	gelangen  -  attain, reach
	gelegen  -  located; convenient, opportune
	gelernt  -  practiced, experienced (at)
	gelochte  -  punctured
	gelten  -  hold true
	geltend  -  effective, applicable, valid
	gemach  -  slowly, by degrees
	gemachlich  -  leisurely, comfortably
	gemass  -  according to
	gemein  -  common
	gemeinsam  -  common, mutual, combined
	gemeinschaftlich  -  common, joint, mutual
	gemischt  -  mixed, combined, diffused
	Genanigkeit  -  precision
	genannt  -  above mentioned, called, surnamed
	genau  -  exact, accurate, precise, detailed
	genau dann  -  then and only then (ie, if and only if)
	genugen  -  be sufficient or enough, suffice; satisfy (a condition)
	genugt  -  suffices (inf. genugen); "es genugt": it suffices
	geordnet  -  orderly, disciplined, systematic
	gerade  -  even (numbers); just, right, now
	Gerade  -  line
	geradlinig  -  straight-lined, rectilinear, lineal
	gerichtlich  -  legal, judicial, court
	gern  -  willingly
	gesamt  -  whole, entire, complete, common, general
	geschehen  -  take place, happen, occur
	geschictlich  -  historical
	Geschlecht  -  genus (of a surface)
	geschlossen  -  closed, enclosed
	geschult  -  schooled, trained
	gesetzmassig  -  legal, legitimate
	gesetzt  -  fixed
	Gestalt  -   form, figure, shape
	gestatten  -  permit, allow, grant
	gestern  -  yesterday
	gesucht  -  desired, sought after, choice
	getan  -  no sooner said than done
	getrennt  -  separated
	gewahlt  -  chosen; selected, choice, select
	Gewicht  -  weight
	gewinnen  -  win, obtain, acquire
	gewiss  -  sure, certain, assured, true, positive
	gewisse Seiten  -  certain sides
	Gewissen  -  conscience
	gewohnlich  -  usual, customary, normal
	gezeichnet  -  drawn, marked
	gezeigt  -  shown
	gibt  -  gives (inf. geben); "es gibt": there is
	gilt  -  is true (inf. gelten)
	Gitter  -  lattice
	gitterformigen  -  lattice-shaped
	Gittermasche  -  fundamental paralleltope of a lattice
	glatt  -  smooth, even, glossy, flat
	gleich  -  equal
	gleichbedeutend  -  equivalent (to)
	gleichen  -  equal
	gleichfalls  -  likewise, as well, too, in the same way
	Gleichheit  -  equality
	Gleichheitzeichen  -  equality sign (=)
	gleichmassig  -  uniform (continuity); proportional, constant, symmetric
	Gleichung  -  equality
	Glied  -  term; link
	Grad  -  degree, order
	greifen  -  grab, seize; choose
	Grenze  -  limit
	grenzen  -  border, adjoin
	Grenzen, uber all  -  over all bounds
	Grenzwert  -  limit, limiting value
	gross  -  large
	Grosse  -  quantity, size
	grunden  -  establish, found; be based on
	Grundgedanke  -  basic idea, motivation, fundamental idea
	Grundlage  -  foundation, basis, groundwork
	grundlegend  -  basic, fundamental
	Gruppe  -  group
	Gultigkeit  -  validity, legality, availability
	Gute  -  goodness
	
	
	 haben   -  have
	haften  -  cling, adhere, stick
	handeln  -  act, take action, proceed
	handelt  -  acts (inf. handeln); "es handelt sich um": it is a matter of
	hangen   -  hang, suspend, attach, fasten, fix
	hatten  -  approach
	haufig  -  frequent
	Haufung  -  accumnnulation, multiplication, piling up
	Haufungspunkt  -  accumulation point
	Haupt  -  head, chief, principal, main
	heben  -  lift, raise; increase; enhance
	heissen  -  to be called
	heisst  -  "das heisst": that is, in other words, i.e.
	heisst - called
	herausgreifen  -  choose, select, pick, single out
	hergeleiten  -  derived, deduced
	herleiten  -  derive, deduce
	Herleitung  -  derivation
	herruhren  -  originate, derive, spring, proceed
	herstellbar   -  producible
	herstellen  -  produce, prepare, make
	herum  -  around, about
	hervor  -  forth, out, out of, from under
	hervorgehen  -  come (from); emerge (from)
	heute  -  today
	hierin  -  herein, in this
	hiermit  -  herewith, (along) with this, saying or doing this
	Hilfe  -  help, aid, support
	Hilfesatz  -  lemma
	Hilfsmittel  -  device, tool, aid
	hindurch  -  through, throughout, during
	hinfort  -  in the future
	hinlanglich  -  sufficient, adequate
	hinreichend  -  sufficient
	hinweisen  -  point to or towards, indicate
	hinzu  - to, towards, near; in addition, moreover
	hinzufugen  -  append, attach
	hoch  -  high, tall, lofty
	hochstens  -  at the most, not exceeding
	Hohe  -  height, altitude
	hohe  -  high, tall, lofty
	Hulle, abgeschlossene  -  closure
	
	
	 identisch   -  identical
	ihnen - (to) them
	im voraus  -  in advance
	immer  -  always
	in Anbetracht  - in consideration of
	in der Tat  -  in fact
	indem  -  while, as; by
	Inhalt  -  area, volume; contents, subject matter
	Inhaltverzeichnis  -  table of contents
	innere  -  interior, internal, inner
	irgend  -  any
	irgendein  -  someone (or other), anyone
	
	
	 je   -  ever, at any time; always, at all times
	je nachdem  -  depending on how;  as the case may be
	jede  -  every
	jedoch  -  however
	jetzt  -  now, at present
	jeweils  -  at times, from time to time, at any given time; respectively
	
	
	 kaufen   -  buy
	kein  -  none, not any
	kennen  -   know, become acquainted with
	Kenntnis  -  knowledge
	Kennzeichen  -  distinguishing mark, criterion, sign, characteristic
	Kern  -  kernel, core, heart
	Kette  -  chain, range
	Kettenkomplex  -  chain complex
	Klammer  -  clamp, clip
	Klammer, eckige  -  bracket
	Klammer, runde  -  parenthesis
	klammern  -  clamp, clasp
	klar  -  clear, transparent
	Klasseneinteilung  -  classification
	klein  -  little, small, tiny, minute
	knupfen  -  tie, join, attach, fasten
	kommen  -  come, happen
	komponieren  -  compose
	kontrahierbar  -  contractible
	konvergieren  -  converge
	kraftig  -  with force
	Kreisring  -  annulus
	Kreisscheibe  -  closed disk
	Kriterium  -  criterion
	Kugel  -  ball, globe, sphere
	
	
	 Lange   -  length
	langsam  -  slow
	lassen  -  let; leave, part with
	lassen sich  -  let itself; can be (eg, es lasst sich zeigen = it can be shown)
	lauten  -  sound; read, say (a word)
	lauten (a umlaut)  -  ring or toll (a bell)
	lauter  -  clear, pure, unmixed
	Leben  -  life, existence
	leben  -  live
	lediglich  -  merely, only
	leer  -  empty, vacant, void
	legen  -  put, place, set
	Lehre  -  theory, teaching, instruction, lesson
	Lehrsatz  -  thesis, proposition, theorem, precept
	leicht  -   light, easy, lightly, easily
	leiden   -  suffer, bear, tolerate
	leiten  -   lead
	lenken  -  guide, direct
	letzen  -  gratify, refresh, comfort
	letzt  -  last, final
	licht  -  light
	liefern  -  supply, provide, yield
	liegen  -  lie, be located or situated
	links  -  left (side)
	Lochrand  -  hole (= Loch) boundary (= rand)
	losbar  -  soluble, solvable, resolvable
	losen  -  settle, resolve, answer
	Losung  -  solution
	Lot  -  perpendicular (line)
	
	
	 Madchen   -  girl
	mag  -  wants (inf. mogen)
	Mangel  -  want, need, deficiency, scarcity
	Mannigfaltigkeit  -  manifold
	Mass  -  measure
	mehrere  -  several
	mehrfach  -  repeated, multiple, over and over again
	meiden  -  avoid, shun, flee
	meist  -  most
	Menge  -  set
	mindestens  -  at least, at the very least, not less than
	mithin  -  consequently
	mitsamt  -  together with
	Mittel  -  means, measures, medium, instrument, device, method
	mittels  -  by means of
	mittlerer  -  interior
	modificirt  -  see modifizieren
	modifizieren  -  modify, qualify
	mogen  -  want, wish, desire 
	Moglichkeit  -  possibility
	muss  -  see mussen
	mussen  -  must
	
	
	 nach   -  after, following, behind, to, toward, according to
	nach dem  -  relative to
	nachdem  -  afterwards
	nachprufen  -  verify
	nachst  -  next, following
	nachtraglich  -  supplementary, subsequent
	Nachweis  -  proof, evidence
	nachweisen  -  prove, establish, authenticate
	nahe  -  near, close, forthcoming
	naheliegend  -  obvious, nearby, near lying
	naher  -  nearer, closer, shorter, more precise, specific, detailed
	nahern  -  approach, close in (on), bring or place near
	Naherung  -  approximation
	namlich  -  the same; namely, that is to say; of course
	neben   -  beside, in addition to, near
	nehmen  -  take, receive, accept
	nehmen, in Anspruch  -  demand, require, claim
	nennen  -  name, call, designate, term
	neu  -  new
	nichtleere  -  nonempty
	nie  -  never
	niedrig  -  low, humble, base, mean
	niemals  -  never
	nirgends  -  nowhere (at all), not anywhere
	noch   -  still, in addition to, even, nor
	nochmal  -  once more
	normieren  -  standardize, normalize
	normieren  -  standardize, normalize
	Normierung  -  normalizing
	notigen  -  urge, force
	notwendig  -  necessary
	Nullpunkt  -  origin
	nullte  -  0-th
	nun  -  now
	nunmehr  -  now, at present, by this time, henceforth, then, since then
	nur  -  only, but
	nutzlich  -  useful
	
	
	 ob   -  over, above, on, upon;  if, whether
	oben  -  above, at the top
	oben erwahnten  -  mentioned above, aforesaid
	obig  -  above, above mentioned
	obliegen  -  apply, be devoted to, prevail over, be incumbent upon
	obwohl  -  though, although
	offen  -  open
	offenbar  -  apparent, obvious
	offenbaren  -  show, disclose
	offensichlich  -  obviously
	oft  -  often
	ohne  -  without
	ohne weiteres  - without further ado
	ordnen  -  arrange or order
	ort - local
	
	
	 Paar   -  pair
	passend  -  appropriate, fitting, suited
	Pflanze  -  plant
	pflegen  -  tend, care for; be used to, be in the habit of
	Pfund  -  pound
	Potenzemenge  -  power set
	Potenzereihe  -  power series
	Primzahl  -  prime number
	Prinzip  -  principle
	Punkt  -  point, dot, spot
	punktfremde  -  disjoint
	
	
	 Quadrat   -  square
	quadratisch  -  square
	Quadratwurzel  -  square root
	Quadratzahlen  -  squares (integers that are squares)
	quadrieren  -  square
	
	
	 Rand  -  boundary
	Randpunkte  -  boundary point
	rasch  -  rapid
	Raum  -  space (eg, metric, vector, etc.)
	Rechenregel  -  calculation-rule
	rechnen  -  count, calculate, reckon
	Rechnung  -  calculation
	rechten Seite  -  right side
	rechtfertigen  -  justify, vindicate
	rechtwinlige  -  orthogonal (sp. coordinates)
	Regel  -  rule, standard, regulation
	Regel, in der  -  as a rule
	rein  -  clean, neat, tidy, pure
	reisen  -  travel
	Rest  -  remainder (rest, residue, remains, difference)
	retten  -  save, rescue, escape, deliver
	richtig  -  right, correct, true, accurate
	Richtung  -  direction, course, trend
	rotieren  -  revolve, rotate
	Rucksicht  -  respect, regard, consideration, notice
	rund  -  round
	
	
	 sagen   -  say, tell
	samtlich(en)  -  complete, whole, entire, altogether
	Satz  -  proposition, theorem
	Schar  -  group, bunch, crowd
	scheinen  -  seem, appear, look
	schliessen  -  shut, close, finish, conclude, terminate
	schliesslich  -  last, final, ultimate, eventual; finally, at last
	schliesslich doch  -  after all, when all is said and done
	Schluss  -  closing, end, termination, conclusion, inference, deduction, result
	schneiden  -  intersect; cut
	schnell  -  fast
	Schnelligkeit  -  speediness
	Schnitt  -  intersection, cut, incision, slice, cross-section
	Schnittpunkte  -  point of intersection, vertex
	schon  -  already
	Schranke  -  boundary, bound
	Schranke, obere  -  upper bound
	Schranke, untere  -  lower bound
	Schritt  -  step, measure
	schwer  -  heavy, difficult
	schwierig  -  difficult, hard, arduous
	sehr  -  very
	sein  -   to be
	seinerseits  - at one time, formerly, in due course, in its proper time
	Seite  -  side
	Seitensimplexe  -  faces (of a simplex)
	selbst  -  self-
	Selbstabbildung  -  self-mapping
	selbstverstandlich  -  obvious, self-evident
	selten  -  rare
	senkrecht  -  perpendicular
	setzen  -  set (esp. a value); put or place
	setzen das Verfahren fort - continue/repeat the procedure
	setzt man  -  if one sets
	sich  -  oneself, itself
	sind  -  are
	Singularitat  -  singularity
	Sinn  -  sense (in the sense of); faculty, mind
	sinnen  -  think, brood, meditate, reflect, ponder, speculate
	Sinnen  -  thinking, planning, thoughts, aspirations
	sinnend  -  musing, pensive, contemplative
	sinnvoll  -  meaningful, sensible
	skizzieren  -  sketch
	so  -  so, in such a way or manner
	so um wie  -  as was done (in)?
	so... wie  -  as... as
	sobald  -  as soon as
	sofern  -  provided, if, inasmuch as, (in) so far as
	sofort  -  immediately, right away
	sogar  -  even
	sogleich  -  immediately, right away
	solcher  -  such
	Solide  -  solidity
	sollen  -  should be, follows from
	somit  -  consequently
	sondern  -  but
	sondern  -  separate, sever, part, sort out, classify
	sonst  -  otherwise, besides, formerly, moreover
	soweit  -  as far as, so far
	sowie  -  just as, as soon as, the moment when, as well as
	sowohl...als auch  -  both...and
	Spalten  -  columns
	spalten  -  split (sp. exact sequences)
	sparen  -  save
	spat  -  late
	spater  -  later
	spaterhin  -  later on
	spatern  -  future
	speziell  -  specifically
	spielen  -  play
	stammen  -  originate
	standig  -  permanent, steady, constant
	Standpunkt(e)  -  standpoint, point of view
	statt  -  instead of, in the place of
	stattfinden  -  take place, happen, occur
	stehen  -  be, occur, exist
	Stelle  -  place, spot, position, job, agency
	stellen  -  put, place
	stetig  -  continuous (function); continual, steady
	Stetigkeit  -  continuity
	stets   -  always, regularly, continually, constantly
	stillschweigen  -  be silent, acquiesce in
	stillschweigend  -  still, silent, tacit
	stimmen  -  be true or correct
	Stoff  -  subject, material
	Storung  -  disturbance, trouble, breakdown, interruption
	streng  -  strict, rigorous, severe
	suchen  -  seek, desire, want
	suss  -  sweet
	
	
	 Tat   -  deed, action, feat, accomplishment
	Tatsache  -  fact
	tatsachlich  -  real, actual, factual, matter of fact
	Teil  -  part
	teilbar   -  divisible, separable
	Teilerfolge  -  partial success
	Teilmenge  -  subset
	Teilraum  -  subspace
	ternar  - ternary
	Tier  -  animal
	tragen  -  carry
	Trager  -  support (of a function)
	Treffen  -  encounter, meeting
	treffen  -  meet; touch; affect
	trennen  -  separate, divide, part, sever, detach
	Trennungseigenschaft  -  separation property
	treten - step, walk, tread, stride
	Tritt  -  step
	trotzdem  -  in spite of, nevertheless, although
	tun  -  do, perform, execute, make, act
	
	
	 Ubelstand(en)   -  defect, disadvantage, drawback
	uberall  -  everywhere, all over
	Uberaufgaben  -  practice exercises
	uberblicken  -  survey, view, prospect
	uberdecken  -  cover (as in open covering)
	uberdies  -  besides, moreover, what is more
	ubereinstimmen  -  be in agreement, concur
	ubereinstimmend  -  agreeing, harmonious, corresponding
	uberfuhren  -  conduct or lead across, transport, transform, bring over
	Ubergang  -  transformation
	ubergehen  -  go over, carry over; omit, overlook; run across, go over
	Uberlegung  -  reflection, thought, consideration, deliberation, investigation
	uberschneiden  -  intersect, cross, overlap
	ubertragbar  -  transferable
	ubertragen  -  carry over
	Ubertragung  -  carrying over, assignment, transfer, translation
	uberzahlig  -  surplus, spare, left over
	ublich  -  customary, usual, common
	ubrig  -  remaining, residual, superfluous
	ubriglassen  -  leave over, leave  a remainder
	Umfang  -  circumference; range
	umfassend  -  far reaching, sweeping, drastic, complete, full
	umformen  -  reshape, convert
	umgeben  -  surround
	Umgebung  -  neighborhood
	umgekehrt  -  conversely
	Umgrenzung  -  boundary, enclosure, limitation
	umkehrbar  -  invertible
	umkehren  -  invert
	Umkehrung  -  converse
	Umstand  -  circumstance, factor, fact; details
	umwandeln  -  change
	unabhangig  -  independent
	unausfuhrbar  -  impracticable, not feasble, impossible
	unbekannt  -  unknown
	unendlich  -  infinite, endless
	ungeandert  -  unchanged
	ungefahr  -  approximately
	ungelost  -  unsolved
	ungerade  -  odd (number)
	Ungleichheit  -  inequality
	Ungleichung - inequality
	unmittelbar  -  immediate, direct; directly
	untergeschieden  -  subdivided
	unterhalb  -  below
	unterliegen  -  be dominated (as a series); be beaten (by), lose (to)
	unterscheiden  -  distinguish
	Unterschied  -  difference, distinction
	untersuchen  -  examine, investigate
	Untersuchung  -  investigation, examination, scrutiny
	Unterteilung  -  subdivision
	unterwerfen  -  subdue, subjugate, submit, yield
	unveranderlich  -  constant, unchangeable, invariant
	unzerlegbar  -  indivisible, non-decomposable
	Urbild  -  preimage
	Urbild - preimage
	Ursprung  -  origin, source, beginning, cause
	ursprunglich  -  original, primitive, primary, initial
	u.s.f.  -  and so on (und so folgen?)
	
	
	 verallgemeinen   -  generalize
	Verallgemeinerung  -  generalization
	veranderlich  -  changing, changeable, variable
	Veranderlich  -  variable
	verandern  -  to change
	veranschaulichen  -  make clear, illustrate
	verbessern  -  improve
	verbinden  -  unite, join, combine
	Verbindung  -  binding, joining, connecting, relationship
	verbunden  -  united, joined, combined
	Vereinfachung  -  simplification, reduction
	Vereinigung  -  union (set-theoretic)
	vereinzeln  -  isolate
	vereinzelt  -  sporadic
	verfahren  -  bungle, err; act, behave, proceed
	Verfahren  -  procedure, method, process
	Vergleich  -  comparison
	vergleichen  -  to compare
	vergrobern  -  coarsen, make (an estimate) less sharp
	verhalten  -  hold back, hold, suppress
	Verhaltnis  -  proportion, ratio
	Verhaltnisse  -  circumstances, conditions
	verhandeln  -  discuss, negotiate, argue
	verkaufen  -  sell
	verknupfen  -  tie together, connect, combine, relate
	verknupft  -  related, associated
	verkorpern  -  embody, personify
	verlangen  -  require
	Verlauf  -  course (in the course of things)
	verletzen  -  violate
	Verminderung  -  lessening, diminution, reduction
	vermitteln  -  meditate, intervene, adjust, arrange, settle
	vermittels  -  by means of, through
	vermoge  -  by virtue of
	verneinen  -  deny, answer in the negative
	versagen  -  deny, refuse; fail, fail to work
	verschaffen  -  get,  obtain
	verscharfen  -  sharpen (an inequality); intensify
	verscharfte  -  sharpened (an inequality); intensified
	verschieben  -  translate (as an isometry or affine transformation)
	verschieden  -  distinct, different
	verschobenen  -  translated (as an isometry or affine transformation)
	verschwinden  -  vanish, disappear
	versehen  -  hold (a position), perform (a job)
	versehen mit  -  provide with
	versehentlich  -  by mistake, erroneously, inadvertently
	Versharfung  -  sharpening (of an estimate)
	verstandlich  -  understandable
	verstehen  -  understand, see, make out, realize
	Versuch  -  attempt, test, try, trial
	versuchen  -  try, tempt, taste
	vertauschbar  -  interchangeable; (algebra) commutable
	vertauschen (gegen)  -  exchange (for), interchange
	Vertauschung  -  exchange, substitution
	vertrauten  -  familiar
	verwandeln  -  change or transform (into)
	verweisen  -  refer (to); banish, expel
	verwenden  -  use, employ, apply (to)
	Verwendung  -  application, use, employment
	Verwirklichung  -  realization
	Verzeichnis  -  index (of a book); list, catalogue
	vgl.  -  compare (vergleichen)
	viel  -  much, many
	Vielfaches  -  multiple
	vielmehr  -  rather, on the contrary
	viertens   -  fourthly, in the fourth place
	vollenden  -  bring to a close; finish, terminate, achieve
	vollig  -  full, total, entire, complete
	vollkommen  -  perfect, consummate, complete
	vollstandig  -  complete (sp. metric spaces), total, integral
	vollziehen  -  effect, carry out; take place
	Vollziehung  -  execution, discharge, accomplishment, fulfilment
	von  -  from, off of, of, by, at
	vorangehend  -  preceding, foregoing
	voraus  -  in front, ahead (of)
	vorausgesetzt - assumed
	voraussetzen  -  suppose, assume
	Voraussetzung   -  assumption, supposition, (induction) hypothesis
	Vorbehalt  -  reservation, proviso
	Vorbemerkung  -  preliminary remark, preamble
	vorbereitend  -  preparatory
	Vorbereitung  -  preparation, introductory material
	Vorgabe  -  points, handicap
	vorhanden  -  present, at hand, available, existing
	vorigen  -  former, preceding, previous
	vorkommen  -  happen, take place, be found
	vorlaufig  -  for the present, for the time being, preliminary, tentative
	Vorlesung  -  lecture, reading, recital
	vorliegen  -  be under consideration, be present or there, exist
	vorliegend  -  present, existing, at issue, in hand or in question
	vornehmen  -  take up, undertake
	vornherein  - from the first
	vorstehenden  -  preceding?
	vorweg  -  before, beforehand, to begin with
	Vorzeichen  -  sign (of a number)
	Vorzug  -  preference, advantage, superiority, priority, excellence
	
	
	 waagrecht   -  horizontal
	Wachsen  -  growth, development, increase, expansion
	wachsen  -  increase
	wachsend  -  growing, increasing
	wachst  -  grown, increased
	Wahl  -  choice, option, seletion
	wahlen  -  choose
	wahrend  -  during, while, whereas
	Wahrheit  -  truth, reality, fact
	waren - would be, would have
	Wechsel  -  change, shift
	weder  -  neither
	weder...noch  -  neither...nor
	wegen  -  because of, on account of
	wegfallen  -  be omitted or suppressed, become void, come to nothing
	weggeben  -  give away, dispose of
	weghaben  -  have received, got the knack of
	weglassen  -  leave out, omit, let go
	weil  -  because
	Weise  -  manner, way, method, mode
	weiss  -  white
	weit  -  far, distant, extensive, wide, broad
	weiten  -  widen, broaden
	weiter  -  further, farther, additional
	weiterhin  -  furthermore, again
	welcher  -  which
	wenden  -  turn, turn over, turn up, change
	Wendung  -  turn; change
	wenig  -  little, few
	weniger  -  less, minus
	wenigstens  -  at least, at all events
	werden  -  become
	Wert -  value, price, worth
	Wesen  -  essence, substance, entity
	wesentlich(e)  -  essential, fundamental, substantial
	wichtiges  -  weighty, important, essential
	Widerspruch  - contradiction (fuhren zu Widerspruch = leads to a c.)
	wie  -  as, like, such as
	wieder  -  again, afresh, once more
	wiederentdecken  -  rediscover
	wiederholen  -  repeat
	wiederum  -  again; on the other hand
	wiegen  -  weigh
	willkurlich  -  arbitrary, random
	Winkel  -  angle
	wird - becomes (pres. of werden)
	wirklich  -  actual, real, substantial
	Wirklichkeit  -  (actual) fact, reality
	wissen - know
	Wissenschaft - knowledge     
	wobei  -  whereby, whereat, in the course of which
	wodurch  -  by or through which, whereby ... How? By what means?
	wofern  -  provided that, so far as, in case
	wohin  -  where (to)
	wohl  -  probably
	wohlbekannt  -  well known, familiar
	wollen  -  wish, desire
	worauf  -  on which, upon or after which, whereupon
	woraus  -  from which
	Wort  -  word
	wortlich  -  word-for-word, verbatim
	woruber  -  upon, over, or about which
	wunschen  -  wish, desire, want, request
	wurde - could, (could) become (pret. of werden)
	Wurfel  -  cube
	
	
	 z. B.   -  for example (zum Bespiele)
	Zahl - number
	Zeichen  -  symbol
	zeigen  - show, display, exhibit
	Zeile  -  row
	zerfallen  -  can be separated; disconnect; fall apart, decompose
	zerfallen sein mit  -  to be at variance with
	zerlegen  -  resolve, cut up, decompose
	Zerlegung  -  partition, decomposition; dismantling, dissection
	Zerlegung der Einheit - parition of unity
	zerspalten  -  split
	zerstorbar  -  destroyable
	zerstoren  -  destroy, demolish, ruin
	zerstreuten - scattered
	zgl.  -  resp. (zugleich)
	ziehen  -  extract; pull, pull out
	Ziel  -  aim, mark, goal
	ziemlich oft   -  not often
	Ziffer  -  digit, numeral
	zitieren  -  quote
	zudem  -  moreover, besides
	zuerst  -  first
	zufolge  -  in consequence of, as a result of
	zuganglich  -  accessible
	zugehorig  -  belonging to
	zugelassen  -  allowed, admitted
	zugleich  -  at the same time
	zugleich mit  -  together with
	zulassen  -  allow, admit
	zulassig  -  permissible
	zunachst  -  for the time being, in the first place
	zuordnen  -  adjoin, coordinate, associate
	zurechnen  -  add, number, reckon
	Zurechnung  -  addition, inclusion, attribution
	zuruck  -  back, backwards, behind, points down
	zuruckkehren  -  return (to)
	zusammenfallen mit  -  coincide with
	zusammenfassen  -  condense, summarize, combine, unite
	Zusammenhang  -  connectedness (topolgical); connection
	Zustand - state, condition
	zustreben  -  strive for
	zuweilen  -  sometimes
	zwar  -  indeed, certainly
	Zweck  -  aim, purpose, objective
	zwischen  -  between, betwixt, among