There were two different number systems in use in Middle Earth; the duodecimal system (base 12) and the decimal system that we use today. Interestingly Tolkien tells us that although "in Common Eldarin the multiples of three, especially six and twelve, were considered especially important" the decimal system developed first - "and eventually beside the decimal numeration a complete duodecimal system was devised for calculations".

Although he goes on to say that "the special words for 12 (dozen), 18 and 144 (gross) were in general use" we don't have any record of what that special number for 18 might have been in any language. Tolkien further noted that "for general purposes the numeral names were decimal in origin". This is why the numbers for 20, 30 and so on mean 'two tens', 'three tens'.

Cardinals (1-19)

The numbers 13-19 are reconstructed from Quenya.

Sindarin	English
Mîn	1
Tâd (Tad- in compounds)	2
Neledh (Nel- in compounds)	3
Canad (Can- in compounds)	4
Leben (Leb- in compounds)	5
Eneg	6
Odog	7
Tolodh	8
Neder	9
Pae (-paen in compounds)	10
Minib	11
Ýneg	12
Neleb	13
Canab	14
Lebem	15
Eneph	16
Odoph	17
Toloph	18
Nederph	19

The "in compounds" part should make sense when you now look at the numbers from 20 - 90 and see how they are formed. Instead of Neledhchaen for thirty (a long not aesthetically pleasing word) we have the shorter Nelchaen.

Cardinals (20 onwards)

Originally the word for 'ten' was Cae (-caen in compounds), which is why in the King's Letter Aragorn writes
"a Pherhael ar am Meril suilad uin aran o Minas Tirith nelchaenen ned Echuir"
"to Samwise and to Rose greetings from the King of Minas Tirith on the 30th of Spring"
However later (c. 1967 - 1969, long after the Lord of the Rings was published) Tolkien wrote an essay called The Rivers and Beacon-Hills of Gondor (found in Vinyar Tengwar #42) which includes a long section on numbers. In this he clearly gave the Sindarin for 'ten' as pae (-paen in compounds).

These then, are the deduced numbers 20 - 90 using Tolkien's later material. I suggest that these are for Sindarin as spoken by the Elves, and the original forms as written in the King's Letter are Gondorian Sindarin (see Lesson 39).

Sindarin	English
Taphaen	20
Nelphaen	30
Cambaen	40
Lephaen	50
Enephaen	60
Odophaen	70
Tolophaen	80
Nederphaen	90
*Haran	100
Host	144 (a gross, 12*12)
Meneg	1000 (or a great gross of 1728)

*This word is reconstructed.

Numbers like 33, 67, 82 etc. can be formed like this:
Nelphaen a neledh = Thirty and three (33) or Neledh a nelphaen = Three and thirty
Enephaen a odog = Sixty and seven (67) or Odog a enephaen = Seven and sixty
Tolophaen a dâd = Eighty and two (82) or Tâd a dolophaen = Two and eighty

Ordinals

For any number after 12 you just need to put -ui on the end to form the ordinal, e.g. nederphaenui = 90th

Sindarin	English
Minui	1st
Tadui	2nd
Nelui	3rd
Canthui	4th
Lefnui	5th
Enchui	6th
Othui	7th
Tollui	8th
Nedrui	9th
Paenui	10th
Minibui	11th
Ýnegui	12th
Taphaenui	20th
Haranui	100th
Menegui	1000th

Using numbers

When we use cardinal numbers in a sentence they come before the noun, like so:
Mîn gaud = One machine
Mîn hadhod = One dwarf

If there's more than one of something, we pluralise the noun as we would normally:
Eneg goed = Six machines
Neder hedhyd = Nine dwarves

If we're using ordinals (1st, 2nd etc.) then they are treated like any other adjective (placed after the noun, with soft mutation):
I 'aud vinui = The first machine
I chadhod dollui = The eighth dwarf

Fractions and shapes; a musing on Elven mathematics

Fractions
The attested words we have are Perin = Half and Canath = Quarter
We also have some reconstructions for some of the other fractions:
Nelest = Third
Lebent = Fifth
Enaith = Sixth
Odost = Seventh
Tolost = Eighth
Nederth = Ninth
Paerth = Tenth

Addition
This is not attested anywhere, but I imagine it to be as simple as using the word 'and' in the same way that we can say an addition in English:
Mîn a eneg anna odog = One and six gives seven

Multiplication
In Parma Eldalamberon #14, we have the following in Qenya:
Hue yullume i hualqe = Twice nine is eighteen, where -lume is a suffix meaning 'times'. Qenya is the forerunner of Quenya; if we look at the word Lume in Quenya, it means 'time', which led me to the following idea from a reconstructed word of mine - minlû = once (literally, one time).
Perhaps therefore, in Sindarin lû = time (pl. lui = times) could be used as a suffix in the same way:
Tâd tadlui anna canad = Two, two times gives four

Shapes
The attested word we have is Nelthil = Triangle, which literally means 'three points', Nel + Till
Using this as a basis, we can reconstruct the following:
Canthil = Quadrilateral
Lebthil = Pentagon
Enethil = Hexagon
Odothil = Heptagon
Tolothil = Octagon
Nederthil = Nonagon
Paethil = Decagon