[This article is modified from a paper which originally appeared in * Technical Signs 1 & 2: Project Overview and Reading Technical Signs Diagrams* by Frank Caccamise, William Newell, Marilyn Mitchell, et alia. (Rochester, NY: National Technical Institute for the Deaf, 1993). For an exhaustive treatment of numbers, see the *Sign Media* videotape series, *ASL Numbers: Developing Your Skills (Cardinal & Ordinal Systems; Incorporating Systems; *and *Unique Systems). *]

__Cardinal numbers__

Cardinal numbers get their name from the concept of "principal, chief". As a mnemonic, consider the Catholic term "cardinal", which has the same root, and is an office second only to the pope. The cardinal numbers are the counting numbers and are the first kind that are taught to children. In ASL, the numbers 1 to 5 differ in palm orientation depending on if they are used as an adjective (see below) or as an abstract number: **FOUR"abstract" MINUS THREE"abstract", EQUAL ONE"abstract".** "Four minus three equals one" is an example of abstract numbers, that is they do not represent a quantity of things. The numerals "1" through "5" should have their palm orientation (PO) away from the signer. Extended numbers should be "spelled" out moving towards the signer's dominant side, not "backwards" so that they consumer can see them from their perspective. Deaf people are used to turning fingerspelling around to their perspective mentally.

**Quantifiers (adjectival)**

These are the numbers that do represent an answer to "how many?" and serve as adjectives, because they always modify nouns. Example:

**_____________________t
BOOK THREE"adjectival", ME HAVE.** (I have three books.)

The quantifiers "1" through "5" have their PO towards the signer and may move towards the signer a bit, even be flicked up from a closed hand for emphasis. "6" through "9" tap at their contact point. "10" through "15" are iterated 3 or more times.

**Nominal numerals**

The concept of labeling something with a number, like a room, page, road, or test, involves a special sign modulation from "1" to "15". Even if the English source message does not include the word "number", but could include it, such as in "now we will discuss quiz (number) 5", the special modulation should be used. For "1" to "9", the appropriate handshape for the numeral is repeatedly shaken slightly with the fingertip orientation (FO) away from the signer and the PO towards the non-dominant side. For "10" to "15" the numeral is repeated more than twice. Example:

**PLEASE LOOK-UP PAGE EIGHT"nominal".** (Please turn to page 8.)

**Counting (verbal)**

When people are in the act of counting things, they point up slightly with the handshape of the number that is being assigned to that object and then point directly at the object being counted, with PO towards the non-dominant side. Example:

**__________rhq
NOW #DO-DO? ME SET-UP GROUP"each" THREE"adjectival".
ONE"verbal", TWO"verbal", THREE"verbal, ONE"verbal", . . .** (Now I'll divide you into three groups. You're a "one", you're a "two", you're a "three", you're a "one", . . .)

Eye contact must be made with the thing counted. If signers are counting things they are visualizing in their head because the things are not present, they will either close their eyes or look upward.

__Ordinal numbers__

**Temporal**

Ordinal numbers get their name from the fact that they establish
the order of things. Example: **MY SON ATTEND SIXTH G-R-A-D-E.** (My son is in sixth grade.) Sixth grade is after fifth and before seventh. "1st" through "9th" are executed by taking the handshape for that number and supinating, that is, the PO is first away from the signer, and then twists until the PO is upwards. After "9th", temporal ordinals are indicated by the numeral plus T-H. This is true even for the "20s" and above that end in "1", "2" or "3". For example, "21st": **TWENTY-ONE~T-H**, "52nd": **FIFTY-TWO~T-H**, and "83rd": **EIGHTY-THREE~T-H.** The tilda (~) means the fingerspelling is blended with the number. I once noticed a poster for a Deaf event that read "June 22th".

**Rank**

If the order is one of importance, and not of time, there are two possibilities. (1) For things other than rank in sports, from "1st" to "5th", the thumb of the non-dominant hand is struck with the bottom finger of the appropriate numeral handshape, which is twisted slightly. From "6th" to "9th", the contact point of the handshape is twisted at the thumb of the non-dominant hand. (2a) For sports, the FO is towards the non-dominant side, held high and pulled quickly toward the dominant side. Alternately the FO may be away from the signer and the hand moved away from the signer. (2b) In rapid discourse the FO is away from the signer and brought towards the signer in ballistic movements.

**Repetition**

If something is being repeated, then for the first three iterations, **the number is held with the DH PO towards, and the FT is rubbed upwards against the NDH [B], PO > DS.** For "fourth time" and so on, one may sign **FOURTH"temporal" TIME.** After nine, one adds **~T-H** as in **TWENTY-FOUR~T-H TIME.**

**Listing**

If the interpreter can sense that a list is being generated, then the list should be counted on the fingers. If it is known how many the total will be, say three, then the list should be labeled as **ONE-ON-LIST-OF-THREE, TWO-ON-LIST-OF-THREE, THREE-ON-LIST-OF-THREE.** Otherwise it could be listed as **ONE-ON-LIST-OF-ONE, TWO-ON-LIST-OF-TWO, THREE-ON-LIST-OF-THREE,** etc.

__Special Applications for Math and Science__

**Fractions and decimals**

For fractions where the numerator and denominator are pure numbers and not extended expressions (see below), represent the numerator, then move the hand down to a lower level and represent the denominator. Bearing in mind again that numbers should be "spelled" out moving towards the signer's dominant side, not "backwards", put a **PERIOD** for the decimal point. Every number after the decimal point is a separate number, that is, even though we say "four point twenty two" for "4.22", it should be signed **FOUR PERIOD TWO TWO**, not **FOUR PERIOD TWENTY-TWO.**

**Subscripts**

Subscripts are numbers (or letters) that label something by being written immediately after and slightly below the line of the thing they label. These are represented by fingerspelling the number slightly below the level of the item signed. Example:

**"X" ONE'subscripted' MINUS "X" TWO'subscripted' EQUAL FOUR.**
(x_{1} - x_{2} = 4.)

The sentence is read "x sub one minus x sub two equals four."

**Superscripts**

Superscripts are the inverse of subscripts and are handled in a similar way. They are generally used to express exponents, see below.

**Exponents**

Exponents are special superscripts that show how many times a number is multiplied by itself. Example: **SIXTEEN SECOND'superscripted'** (16^{2}). The sentence is read "sixteen squared." For "three" it is read "cubed" and from then on it is read "to the nth power", where "nth" is "fourth", "fifth", "sixth", etc. For the exponents from "2" to "9" superscript the temporal ordinals **SECOND** through **NINTH**. For "10" and above, simply superscript the number itself.

**Roots**

Roots are the inverse operation of exponents and are represented by the number of the root (no number shown means it is a two) followed by the symbol " ". The "sign" for this is to draw this symbol with the index finger. The "second" root is called the "square root", the third, the "cube root", and from then on the "nth root". For the roots from "2nd" to "9th" superscript the temporal ordinals **SECOND** through **NINTH.** Example: for "the fourth root of x" sign **FOURTH'superscripted' 1-CL'draws a root sign' "X".** For "10" and above, simply superscript the number itself.

**Operations on "quantities"**

When an extended expression is operated on, such as when "x^{2}-4" is itself squared, it is written (x^{2}-4)^{2} and is read "x squared minus four (the) quantity squared." It is represented by signing **"X" SECOND'superscripted' MINUS FOUR, then (2h)L:-CL is held so that it looks like parentheses and encloses the space that the "X" SECOND'superscripted' MINUS FOUR occupied and is followed by SECOND'superscripted'.**

**Representing equations**

In general, it is counterproductive to fingerspell equations or sign what is being written on the board, unless you want to use the opportunity to introduce the student to how you will represent concepts in "signs" that are being put on the board for the first time, the presenter's handwriting is difficult to read, or abbreviations are used that have not been previously introduced. It is easier for the student to read it off the board then off your fingers, especially if it is a lengthy equation.

If comments are made about the equation it will be necessary to interpret these, of course, and if the comments are interwoven with the equation writing, it may be necessary to include the equation in your interpretation so that it makes sense. Do represent it, as in fingerspelling words or numbers, as an equation that moves toward your dominant side. Don't fingerspell it backwards in an attempt to help the client "see it." If the client is ASL, they can reverse it as part of their semantic abilities. If the person is not ASL, you might as well follow ASL rules, since reversing equations will eventually lead to conflict with regular fingerspelling and your own visualization.

**Equations with extended numerators and denominators**

If a factor (item) in an equation has an extended numerator or denominator, such as

It may be necessary to use your non-dominant forearm (as in **TABLE**) as the line between numerator and denominator. Or alternately draw a line with the index finger.

**Roots of lengthy expressions**

If an extended expression is under a root sign, such as

the extended expression should be signed first, and then a root sign drawn over the entire expression.

**Representing points in space**

Points in two- or three-dimensional space are represented as
(2,-4) or (3, -17, -6). Everything within the parentheses should be represented first, then the parentheses should be "drawn" around the entire expression. For example, the three-dimensional point shown above would be represented **THREE COMMA NEGATIVE SEVENTEEN COMMA NEGATIVE SIX, then (2h)L:-CL is held so that it looks like parentheses and encloses the space that the expression occupied.**

**Examples for practice**

- "Equation 4", "Quiz 13", "Test 9", "page 403".
- "four equations", "three scientists", "two books", "nine problems"
- Numerals (nominal): 4; 522; 69; 42,603.
- "four-fifths", "fifty seven three-hundred-sixty-seconds", 1/2, 24/97.
- Count windows, count the buttons on someone's blouse/shirt, count chairs, count people.
- "third edition", "eighty-third precinct", "8th commandment", "78th floor".
- Rank ordinal: "eighth in importance", "secondary", "primary", "sixth in importance".
- (E
_{3}- E_{5})^{2}, (4, -33, 27/2), 3.1415, "four point sixty-eight".

Bienvenu, MJ & Colonomos, B. ASL Numbers: Developing your skills (Cardinal & ordinal systems; Incorporating systems; and Unique systems). Sign Media Inc. If you can have only one resources, this is it! Lots of interactive practice.

HandSpeak.com. A paid site but worthwhile. Click on "Signs & categories" and then "Numbers (ASL)" to see the numbers from 1-100 and beyond.

MacDougall, C.

(1997). Number signs for everyone: Numbering in American Sign Language. This deals with numbering for money, measurements, age, sports, and scientific numbers. This is the videotape version. Dawn Sign Press.

(1998). Number signs for everyone: Numbering in American Sign Language. This deals with numbering for money, measurements, age, sports, and scientific numbers. It has over 1,000 illustrations. This is a book, so the images are static; I prefer the videotape version. Dawn Sign Press.

Palatine, Inc. Interactive Sign Language fingerspelling and numbers. This computer program allows you to master fingerspelling at your own speed and pace. You can create custom vocabularies, such as medical and legal terms, as well as common word ending and letter groups. When the word is shown in the signing window, you can choose the correct English word. This program also generates the numbers from 1-100.

Penilla, A. R. & A. L. Taylor. (June 2003). Counting on numbers in Sign Language. ISBN: 0-7645-5436-0.

Signing Online: Numbers. In the trial, you can view numbers 0 thru 10. Enroll or Login for the complete glossary content.

Tuccelli, M. Receptive number practice with Dr. Sign. Fractions, decimals, phone numbers, social security numbers, addresses, etc. 60 minutes.

- All number images are from HandSpeak.