# Nemeth Braille

The Nemeth Braille Code for Mathematics and Science Notation is a Braille code for encoding mathematical and scientific notation linearly using standard six-dot Braille cells for tactile reading by the visually impaired. The code was developed by Abraham Nemeth. The Nemeth Code was first written up in 1952. It was revised in 1956, 1965, and 1972. It is an example of a compact human-readable markup language.
Nemeth Braille is just one code used to write mathematics in braille. There are many systems in use around the world.

## Principles of the Nemeth Code

The Nemeth Code Book (1972) opens with the following words:
   This Braille Code for Mathematics and Science Notation has been prepared to provide a system of symbols which will allow technical literature to be presented and read in braille. The Code is intended to convey as accurate an impression as is possible to the braille reader of the corresponding printed text, and this is one of its principal features. When the braille reader has a clear conception of the corresponding printed text, the area of communication between himself and his teacher, his colleagues, his associates, and the world at large is greatly broadened. A test of the accuracy with which the Code conveys information from the print to the braille text is to effect a transcription in the reverse direction. The amount of agreement between the original printed text and one transcribed from the braille is a measure of the Code's accuracy.
One consequence is that the braille transcriber does not need to know the underlying mathematics. The braille transcriber needs to identify the inkprint symbols and know how to render them in Nemeth Code braille. For example, if the same math symbol might have two different meanings, this would not matter; both instances would be brailled the same. This is in contrast to the International Braille Music Code, where the braille depends on the meaning of the inkprint music. Thus a knowledge of music is required to produce braille music.

## Table of Nemeth braille codes
### General signs

| Symbol | * | , | ... |
| Braille | | | |

### Number signs

| Symbol | Number prefix | Decimal point | Thousands separator | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Braille | | | | | | | | | | | | | |

### Operators

| Symbol | + | - | × | · | ÷ | . (decimal) | √ |
| Braille | | | | | | | |
| Symbol | inner √ (1st) | inner √ (2nd) | end 1st √ | end 2nd √ | long division | / | ! |
| Braille | | | | | | | |

### Parentheses and brackets

| Symbol | ( | ) | [ | ] | { | } |
| Braille | | | | | | |

### Fractions

| Symbol | simple | simple | simple | complex | complex | complex | hypercomplex | hypercomplex | hypercomplex | fraction in mixed number | fraction in mixed number | fraction in mixed number |
| Symbol | open | line | close | open | line | close | open | line | close | open | line | close |
| Braille | | | | | | | | | | | | |

### Other modifiers

| Symbol | ∵ | ∴ | : | ∷ | ′ | ″ |
| Braille | | | | | | |
| Symbol | ∞ | ∝ | ° | % | ✓ |
| Braille | | | | | |

### Braille indicators

| Symbol | Punctuation | Superscript | Subscript | Baseline | Omission | Cancel (close) | Cancel (open) | Run Over | Directly Over | Directly Under | Beginning | Termination |
| Braille | | | | | | | | | | | | |

### Comparison signs

| Symbol | > | < | ≥ | ≤ | ≧ | ≦ |
| Braille | | | | | | |
| Symbol | ⋝ | ⋜ | ⪚ | ⪙ |
| Braille | | | | |
| Symbol | >< | <> | >=< | <=> |
| Braille | | | | |
| Symbol | = | ≠ | ≯ | ≮ | ~ | ≈ | ≅ | ± | ∓ |
| Braille | | | | | | | | | |

### Geometry

| Symbol | Parallel | Perpendicular | Angle | Right angle |
| Braille | | | | |
| Symbol | Arc down | Arc up | Trapezoid | Square | Rectangle | Parallelogram |
| Braille | | | | | | |
| Symbol | Rhombus | Isosceles trapezoid | Triangle | Right triangle | Circle | Ellipse |
| Braille | | | | | | |

### Arrows

| Symbol | ⟶ | ⟵ | ⟷ |
| Braille | | | |
| Symbol | → | ⇌ | ⇒ |
| Braille | | | |

### Trig functions

| Function | Sine | Cosine | Tangent |
| Braille | | | |
| Function | Secant | Cosecant | Cotangent | Logarithm |
| Braille | | | | |

### Set theory

| Symbol | ∈ | ∋ | ∪ | ∩ | ∅ | { | } |
| Braille | | | | | | | |
| Symbol | ⊂ | ⊃ | ⊆ | ⊇ |
| Braille | | | | |

### Misc. signs

| Symbol | ⊕ | ⊗ | ⊡ |
| Braille | | | |
| Symbol | Intersecting lines | @ | Ditto | Tally mark | +- | -+ |
| Braille | | | | | | |
| Symbol | | | ≃ | ≡ | — | | | (modulus) | . dot |
| Braille | | | | | | |

### Polygons

| Symbol | Pentagon | Hexagon | Heptagon | Octagon | Nonagon | $n$-agon |
| Braille | | | | | | + (Nemeth Braille of $n$) |

### Triangles

| Symbol | Acute | Right | Isosceles | Obtuse | Scalene |
| Braille | | | | | |

### Other signs

| Symbol | Acute angle | Right angle | Obtuse angle | Hexagon star | ¶ | § |
| Braille | | | | | | |
| Symbol | ref. indicator | <i> | <b> | letter sign | composite function | |det| matrix | ∫ | ∂ |
| Braille | | | | | | | | |

### Greek letters and Latin letters

Greek and Latin letters are based on the assignments of International Greek Braille.
| Symbol | small Latin letters | capital Latin letters | small Greek letters | capital Greek letters |
| Braille | (Braille of the letter) | + (Braille of the letter) | + (Braille of the letter) | + (Braille of the letter) |