Nemeth Code: How Math Is Written in Braille
The Nemeth Code is the braille code for mathematics and science used in the United States. It was created by Abraham Nemeth, a blind mathematician, and its full name is The Nemeth Braille Code for Mathematics and Science Notation. It writes numbers, fractions, exponents and symbols in a compact way that does not depend on the rules of ordinary braille text: in Nemeth, x² is ⠭⠘⠆ and ½ is ⠹⠂⠌⠆⠼.
Since 2016, the United States has used Unified English Braille (UEB) for ordinary text, and UEB has its own way of writing math. So American students read math in one of two ways: UEB Math or UEB with Nemeth. This article explains how Nemeth works, how it compares with UEB, and how the choice is made.
Key points
- Abraham Nemeth (1918–2013) was a blind American mathematician and a professor at the University of Detroit. His code was first written up in 1952 and revised in 1956, 1965 and 1972.
- The current edition is BANA's 2022 revision, updated so Nemeth can be used inside UEB text.
- Nemeth digits sit in the lower part of the cell: they are the letters a to j moved down one row, so a number never looks like a letter.
- Fractions have opening and closing indicators, exponents a superscript indicator, and the equals sign a space on each side.
- When the United States adopted UEB, Nemeth "was maintained as an official code". BANA says the choice between UEB math and Nemeth "should be made based on braille readers' individual needs".
- Inside a UEB book, Nemeth is switched on with ⠸⠩ and off with ⠸⠱.
Who was Abraham Nemeth
Abraham Nemeth (October 16, 1918 – October 2, 2013) was a blind American mathematician. He was a professor of mathematics at the University of Detroit, now the University of Detroit Mercy, and he developed his code to read and write the mathematics he studied and taught.
The code was first written up in 1952 and revised in 1956, 1965 and 1972. The 1972 revision was the standard for fifty years, produced, as its foreword says, "under the able tutelage of Dr. Abraham Nemeth". In 2012 the Braille Authority of North America (BANA) adopted UEB, which meant Nemeth had to be updated to work inside UEB text. That project began in 2016 and was adopted in 2022: The Nemeth Braille Code for Mathematics and Science Notation, 2022.
Braille itself has a longer history with music than with math: Louis Braille was an organist, and music was in the title of his first book.
Why math needs its own braille code
Ordinary braille is enough for 2 + 2 = 4 in a sentence. It is not enough for a textbook.
Printed math is spatial. An exponent sits higher than the line, a fraction stacks one expression over another, a square root covers what is under it. The position of each symbol is information. A line of braille, on the other hand, is a single row of identical cells: there is no "higher" and no "under". So every math code has to turn geometry into characters, making explicit with signs everything print says through position.
A second problem is collisions. Braille has only 63 dot patterns, and ordinary text already uses most of them for letters, contractions and punctuation. In UEB, the lower cells double as punctuation and contractions (see braille punctuation), and numbers are written with letters after a numeric indicator (see braille numbers). A math code has to avoid all of that, or live with it.
Nemeth's answer was to build a separate system with its own rules. UEB's answer was to extend the literary code with technical symbols. Both work; they make different trade-offs.
How Nemeth writes math
These are the core Nemeth symbols. Every cell is transcribed from BANA's 2022 edition.
| Symbol | Braille | Dots | Use |
|---|---|---|---|
| Numeric indicator | ⠼ | 3-4-5-6 | Before a number at the start of a line or after a space |
| Plus | ⠬ | 3-4-6 | Sign of operation |
| Minus | ⠤ | 3-6 | Sign of operation |
| Multiplication cross | ⠈⠡ | 4, 1-6 | Sign of operation |
| Equals | ⠨⠅ | 4-6, 1-3 | Sign of comparison, with a space on each side |
| Decimal point | ⠨ | 4-6 | Inside a number |
| Opening simple fraction | ⠹ | 1-4-5-6 | Starts a fraction |
| Fraction line | ⠌ | 3-4 | Between numerator and denominator |
| Closing simple fraction | ⠼ | 3-4-5-6 | Ends a fraction |
| Superscript | ⠘ | 4-5 | Moves up a level, as in exponents |
| Baseline | ⠐ | 5 | Returns to the main line |
| Opening Nemeth indicator | ⠸⠩ | 4-5-6, 1-4-6 | Switches from UEB text into Nemeth |
| Nemeth terminator | ⠸⠱ | 4-5-6, 1-5-6 | Switches back to UEB |
Numbers live in the lower cells
This is the most distinctive idea in Nemeth. In UEB, the digits are the letters a to j after a numeric indicator. In Nemeth, they are the same patterns moved down one row:
| Digit | Nemeth | Dots | UEB letter it comes from |
|---|---|---|---|
| 1 | ⠂ | 2 | a ⠁ |
| 2 | ⠆ | 2-3 | b ⠃ |
| 3 | ⠒ | 2-5 | c ⠉ |
| 4 | ⠲ | 2-5-6 | d ⠙ |
| 5 | ⠢ | 2-6 | e ⠑ |
| 6 | ⠖ | 2-3-5 | f ⠋ |
| 7 | ⠶ | 2-3-5-6 | g ⠛ |
| 8 | ⠦ | 2-3-6 | h ⠓ |
| 9 | ⠔ | 3-5 | i ⠊ |
| 0 | ⠴ | 3-5-6 | j ⠚ |
Because a Nemeth digit never uses dot 1 or dot 4, it cannot be mistaken for a letter. That is why Nemeth needs the numeric indicator mainly at the start of a line or after a space: inside an expression such as 2+2, the digits identify themselves.
Fractions, exponents and comparison signs
A simple fraction is enclosed by an opening indicator, ⠹, and a closing indicator, ⠼, with ⠌ as the fraction line. So ½ is ⠹⠂⠌⠆⠼: open, 1, fraction line, 2, close.
An exponent is introduced by the superscript indicator, ⠘, which moves the reader up a level. x² is ⠭⠘⠆: x, up, 2. The baseline indicator, ⠐, brings the reader back down when more follows.
Comparison signs such as equals, ⠨⠅, always have a space on each side. That space is a reading aid: it splits an equation into its sides at a glance.
Nemeth and UEB math, side by side
The same five expressions in both codes. The Nemeth versions are BANA's own examples; the UEB versions come from liblouis, the open-source braille translator. Cell counts leave out spaces.
| Nemeth | Cells | UEB | Cells | ||
|---|---|---|---|---|---|
| Addition | 2 + 2 = 4 | ⠼⠆⠬⠆ ⠨⠅ ⠼⠲ | 8 | ⠼⠃⠐⠖⠼⠃⠐⠶⠼⠙ | 10 |
| Multiplication | 2 × 4 | ⠼⠆⠈⠡⠲ | 5 | ⠼⠃⠐⠦⠼⠙ | 6 |
| Decimal | 3.14 | ⠼⠒⠨⠂⠲ | 5 | ⠼⠉⠲⠁⠙ | 5 |
| Fraction | ½ | ⠹⠂⠌⠆⠼ | 5 | ⠼⠁⠌⠃ | 4 |
| Superscript | x² | ⠭⠘⠆ | 3 | ⠭⠰⠔⠼⠃ | 5 |
The table shows the real trade-off, without the slogans. Nemeth is shorter where numbers and levels pile up: in 2 + 2 = 4 it needs a numeric indicator only after a space, while UEB puts one before every number, and an exponent takes one indicator instead of two. UEB is shorter for a simple numeric fraction such as ½, because it does not enclose it. And for a decimal like 3.14 they tie.
The deeper difference is not length. In UEB, the same cell can be a letter, a digit or a contraction depending on context, and math shares the rules of ordinary text. In Nemeth, math has its own rules, so a reader always knows they are in math.
Nemeth inside a UEB book
Since UEB is the code for text in the United States, Nemeth appears inside UEB books, switched on and off with indicators. This sentence is BANA's Example 4-31, What is 2 + 2 = 4?:
⠠⠱⠁⠞ ⠊⠎ ⠸⠩ ⠼⠆⠬⠆ ⠨⠅ ⠼⠲ ⠸⠱⠦
| Braille | What it is |
|---|---|
| ⠠⠱⠁⠞ | What (capital sign, wh contraction, a, t), in UEB |
| ⠊⠎ | is, in UEB |
| ⠸⠩ | Opening Nemeth indicator |
| ⠼⠆⠬⠆ | 2 + 2 in Nemeth: numeric indicator, 2, plus, 2 |
| ⠨⠅ | Equals, spaced |
| ⠼⠲ | 4 in Nemeth |
| ⠸⠱ | Nemeth terminator |
| ⠦ | Question mark, back in UEB |
Everything before ⠸⠩ is ordinary UEB, with its contractions: What starts with the one-cell wh. Everything between the indicators is Nemeth. And the question mark after ⠸⠱ is UEB again. For a single word of text inside math, Nemeth also has a single-word switch indicator, ⠠⠄.
UEB Math or Nemeth: who decides
When BANA adopted UEB in 2012, it did not replace Nemeth. In BANA's words, "The Nemeth Code for Mathematics & Science Notation was maintained as an official code for use in the United States." It recommends calling technical materials either "UEB Math/Science" or "UEB with Nemeth", so everyone knows which one is meant.
And on the choice itself: "The decision to use UEB or the Nemeth Code within UEB context for technical materials should be made based on braille readers' individual needs." In practice, that decision is made for each student by the people who plan their education.
The arguments on each side are predictable. For UEB math: one code for everything, and alignment with the other English-speaking countries that use UEB. For Nemeth: compactness in complex math, a long track record, and the large body of existing Nemeth materials and teachers who know it.
Math on a computer
Today much math reaches blind readers digitally rather than on paper. The usual path is MathML, the W3C format for math on the web, which a math engine turns into speech or braille on the reader's braille display. MathCAT, the engine used by the NVDA screen reader, produces Nemeth, UEB technical and several other national codes, so the same equation can reach each reader in the code they learned.
What we could not confirm
- We do not cover the full code. Nemeth has rules for radicals, matrices, shapes, modifiers and much more. We publish only symbols and examples we could transcribe directly from BANA's 2022 edition.
- We left radicals out of the comparison. The UEB side of that example would need the full UEB technical construction, and the translator we use does not produce it from plain text, so we would have had to write it by hand.
- How many students use each code. We found no reliable national figures for how many American students read UEB math versus Nemeth.
Frequently asked questions
What is the Nemeth Code?
The Nemeth Braille Code for Mathematics and Science Notation is the braille code for math and science developed by Abraham Nemeth, a blind American mathematician. It was first written up in 1952, and its current edition is BANA's 2022 revision, used inside Unified English Braille text.
Is Nemeth still used?
Yes. When the United States adopted UEB, BANA kept Nemeth as an official code. Technical materials are produced either in UEB Math or in "UEB with Nemeth", depending on the reader's needs.
How are numbers written in Nemeth?
With the digits in the lower part of the cell: the letters a to j moved down one row. The numeric indicator, ⠼, is used mainly at the start of a line or after a space. So 2 + 2 is ⠼⠆⠬⠆.
How do you write a fraction in Nemeth?
A simple fraction is enclosed between an opening indicator, ⠹, and a closing indicator, ⠼, with ⠌ as the fraction line. One half is ⠹⠂⠌⠆⠼.
How do you write an exponent in Nemeth?
With the superscript indicator, ⠘, before the exponent. x² is ⠭⠘⠆. The baseline indicator, ⠐, returns to the main line when more follows.
What is the difference between Nemeth and UEB math?
Nemeth is a separate code with its own rules for math; UEB math extends the literary code. Nemeth digits sit in the lower cells and need fewer indicators; UEB digits are letters after a numeric indicator. Nemeth is often shorter in complex expressions, UEB sometimes shorter in simple ones.
Can I write math with the Braillito translator?
Only simple arithmetic. Our translator writes UEB, so it handles 2 + 2 = 4 in a sentence, but not Nemeth or full technical notation. For that you need a specialized tool or a transcriber.