UEBC and Numbers
July 2002

Q. I understand that UEBC has adopted the traditional "upper" numbers, as found in English literary braille, for all purposes. Did you do this just because they are traditional?

A. Our decision-making involved balancing many issues, including tradition, but no one issue dominated all the others. In the case of numbers, we considered the matter from the beginning, weighing the merits of all three systems that have been used in various braille codes for different purposes. We kept in mind that UEBC is to be used for general reading, including math and science, but not only for those subject areas. In the end, we reached the same conclusion that Louis Braille himself reached in his original design, namely that upper numbers serve general reading purposes best.

Q. When you speak of decision-making, who made the decision on numbers?

A. All the members of the working committee that made the original recommendation were knowledgeable in technical subjects including math, psychology and computer notation. And at all times, a majority of those members were blind persons and users of braille. Following an open international evaluation, their recommendation was later ratified by unanimous vote of the countries participating in the General Assembly of the International Council on English Braille. In general, the entire UEBC development process has been open and democratic, in the spirit of blind persons controlling their own destiny--and it remains so.

Q. But don't lower numbers work best for math and science?

A. There are some limited contexts in which the letters a through j may immediately follow digits with higher frequency than in general literature, and in those circumstances lower numbers appear to have an advantage because they do not require an indicator to signal the switch to letters. Algebraic expressions in which those letters appear as variables or coefficients, and hexadecimal numbers where those letters are themselves used as digits, are commonly given as examples. However, after carefully evaluating the real frequency of such cases, even in works that are mathematical or scientific in nature, the committee found that it is far more common for punctuation marks such as the period, comma or colon to follow digits--by a margin of seven or more to one. Wherever that happens, an indicator--or dual representation--is required with lower numbers but not with upper numbers.

Q. Have others validated those findings? And what do they mean in concrete cases?

A. Any reader can easily verify those statistics by sampling random pages from books and magazines, including highly technical ones. Or, you can simply reflect how often you really encounter equations and hexadecimal numbers in real life (not counting examples prepared for a UEBC discussion) as compared with dates, times and other cases where punctuation marks touch numbers. Then imagine reading examples such as

  At 12:00 noon on July 4, 2002 ...

  with an indicator or dual representation at the punctuation marks, again and again ...

Q. What do you mean by "dual representation" as an alternative to an indicator?

A. A group that advocates lower numbers for all purposes also advocates using dots 16 for a comma and dots 12456 for period or decimal following digits. This saves the extra cell that would be required for an indicator in those cases, but at the cost of having two braille forms of those punctuation marks--an example of what can happen when "save cells" dominates all other design considerations.

Q. Is the problem with punctuation marks the only reason for not choosing lower numbers?

A. No, another very important reason derives from the fundamental design of braille. If you examine the familiar "seven line" table in which Louis Braille presented the 63 braille signs, and reflect upon the way in which those signs are used, it becomes obvious that the first four lines, all of which are upper signs, are the ones that are used for "primary" symbols that need to be able to stand by themselves, such as the letters. By contrast, the signs in lines five through seven are mainly used for "auxiliary" symbols, such as punctuation marks and indicators, that are normally found in close proximity to primary symbols. Following this principle, it is natural that the numbers, no less than the letters, deserve to be treated as primary and hence upper symbols.

Q. What does this design principle affect?

A. It greatly affects the "look and feel" of braille as we know it. For example, words are mostly in the upper dots while punctuation marks are "out of the way" in the lower dots. That in turn affects not only aesthetics but the way that braille is most naturally learned and understood. For example, two dots, one just above the other, are naturally understood to be dots 12 unless cells in close proximity force one of the other possibilities, such as dots 23. Interestingly, that effect also makes lower numbers on their own, without a supporting upper-cell indicator, unsuitable for use in hexadecimal numbers. For example, with lower numbers you couldn't tell the difference between fd and 64 unless some unambiguous sign happened to be close enough so that you could tell whether the dots were upper or lower.

Q. What about in algebra?

A. It is even more clear in algebra than in other subject areas that numbers and letters are parallel in our thinking. Letters stand for numbers, and consequently occupy the same places in equations in relationship to the signs of operation and comparison. Notice how, for example, "two plus two equals 4":

#b "6 #b "7 #d

has the same rhythm as "x plus y equals z":

;x "6 ;y "7 ;z

Q. But I've heard that the indicators required by upper numbers lead to problems with alignment in spatial math, and in general make math expressions so long and cumbersome that it is difficult for people who use upper numbers to become proficient in math. Is this true?

A. No. Spatial math can be aligned with upper numbers, just as with other number forms. Some math expressions are expanded in UEBC, some are actually reduced in length, as compared with other codes. The speculative notion that learning and use of math is inherently hampered by the use of upper numbers is completely disproved by the long experience of braille users in the many other countries where upper numbers have always been used for most purposes in math and science. There is no evidence that blind people in those countries are less likely to achieve proficiency in technical areas just because they use upper numbers.

Q. Did Louis Braille himself consider lower numbers?

A. Yes, he did consider them--and in the end, as we have noted, chose the upper number system that is still used for most purposes in most codes throughout the world.

Q. Has any other kind of number system been considered?

A. Yes, there is a third type of system that is used for technical notation in some European codes. It uses the same upper dot patterns as the upper numbers, but with dot 6 added--except for zero, which would clash with the letter w. These "dot-6" digits, using dots 346 for zero, look like this:

* < % ? : $ ] \ [ +

Q. What are the advantages of dot-6 numbers?

A. First, they do not clash with letters nor with punctuation marks. While they do clash with some English contractions, that would not be a problem within a passage marked as in grade 1, which UEBC provides for. Consequently, in such passages, digits would have their own unique identity and there would be no need for any associated indicators. Second, they are upper forms and therefore suitable for use as primary symbols. In fact, they are "strong" symbols whose braille dot patterns cannot be confused no matter how isolated they might be from other symbols on the page.

Q. Why, then, did UEBC not adopt dot-6 numbers?

A. While there was stronger support for them than for lower numbers, in the end the dot-6 numbers were judged to be too dense--that is, there would be too many dots per cell over longer numbers for optimum readability.

Q. Is readability, then, the main strength of UEBC?

A. You could say that, because "readability" pretty well sums up the point of most of the goals listed for UEBC, as well as being a goal in its own right.

Q. But what about writeability, and in particular the need for experts working in technical areas to write and manipulate notation quickly, even at the expense of some ambiguity?

A. We are aware of this need, and while we feel that unambiguous readability is more important for the basic code and the general reader and so needs to be developed first, we also believe that UEBC can eventually be extended to include a "rapid writing mode" that would meet the needs of experts. In fact, the ability to fall back to a "shorthand" of this kind has been considered in the UEBC design process right along.

Q. Would such a mode amount to a separate code, such as we now have?

A. No. Being based upon a well-established UEBC, the symbols in a rapid-writing mode would be closely similar--e.g. the same root, but with the prefix dropped.

Q. What would such a mode be like?

A. An obvious possibility would be to drop the dot 5 prefix from many 2-cell operators, so that for example the plus sign would be just dots 235. Now that would create an ambiguity with the exclamation mark, which does occur in math expressions as the factorial operator--though far less frequently than the plus sign. The user of the rapid-writing mode would have the option of using a grade 1 symbol indicator so that factorials were clear, or simply assuming that, as the writer, he or she would be able to figure out which is which when reading back. That illustrates another feature of rapid-writing mode as it would likely be used in practice--the dropping of indicators that the user can safely assume to be implicit when reading back.

Q. Is this like anything we do in braille now?

A. Yes. Grade 3 braille is an extension of grade 2 that well serves those expert users who feel a need for very rapid reading and writing, while regular grade 2 is the formal code that serves for general reading and is used in publishing. It would be the same with this projected rapid-writing mode and regular UEBC.

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