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Directly Under/Directly Over

When an arrow or other symbol, number, or letter is placed on top of another symbol or under another symbol, the change in level must be identified.

There are two ways to identify this level change:

1) Common under and over accents have their own braille symbol, and

2) Other less common modifiers use the directly under or directly over indicators to identify the level change.

Accents

The following are common accents that appear over or under a letter, symbol, or expression. Just as in print, they have multiple meanings depending on the math expression. Each has its own braille symbol. These symbols apply to the previous item only. Any exception to the "item" rule requires braille grouping symbols to identify what is affected by the accent. The bar and hat accents have a contracted meaning, so a grade 1 indicator is required unless the expression is already in uncontracted mode.

Accent	Directly Over	Directly Under
Arc	⠨⠸⠱
Right pointing arrow	⠘⠱	⠠⠘⠱
Bar	⠱	⠠⠱
Tilde	⠸⠱	⠠⠸⠱
Dot	⠘⠲	⠠⠘⠲
Hat	⠐⠱	⠠⠐⠱

Reminder: The following symbols are considered an "item" or a "group" and the accent applies to the entire group or item:

• An entire number

• An expression enclosed in grouping symbols (fractions indicators, radical indicators, parenthesis, square brackets, curly braces, or braille grouping indicators)

• An arrow or arbitrary shape

Examples:

$$\overrightarrow{x}$$ ⠭⠘⠱ $$\widetilde{x}$$

⠭⠸⠱

$$ṃ$$ ⠍⠠⠘⠲ $$A\overline{B}C$$

⠰⠰⠠⠁⠠⠃⠱⠠⠉

Note: The bar over applies only to the B.

$$\widehat{a}\ \perp \ \widehat{b}$$ ⠁⠰⠐⠱⠀⠼⠤⠀⠃⠰⠐⠱Note: The grade 1 indicator in needed so that "hat directly over" it is not read as the contraction "where." $\overline{x}$ is the mean of x

⠭⠰⠱⠀⠊⠎⠀⠮⠀⠍⠂⠝⠀⠷⠀⠰⠭

Note: The bar directly over requires a grade 1 indicator so that it is not read as "xwh."

$$\underline{x} = \{ x_{1},\ x_{2},\ldots,x_{n}\}\$$

⠰⠰⠰⠭⠠⠱⠀⠐⠶⠀⠸⠣⠭⠢⠼⠁⠂⠀⠭⠢⠼⠃⠂⠀⠲⠲⠲⠂⠀⠭⠢⠝⠸⠜⠰⠄

Note: The grade 1 passage indicator places this expression in uncontracted mode, so a grade 1 indicator is not needed before the bar directly under symbol.

More Examples:

The following examples use grouping symbols to identify what is included in the level change.

$$\underline{a + b}$$ ⠰⠰⠣⠁⠐⠖⠃⠜⠠⠱$$\overrightarrow{AB}$$

⠰⠰⠣⠠⠠⠁⠃⠜⠘⠱

$$Arc\ \widehat{GH}$$ ⠠⠜⠉⠀⠰⠰⠣⠠⠠⠛⠓⠜⠨⠸⠱⧋ ⠰⠰⠫⠼⠉⠱⠠⠱$$\overset{yields}{\rightarrow}$$

⠰⠰⠣⠽⠊⠑⠇⠙⠎⠜⠠⠘⠱

$$\frac{1}{3} = 0.\overline{3}$$ ⠼⠁⠌⠉⠀⠐⠶⠀⠼⠚⠲⠣⠼⠉⠜⠱Note: The number 3 is the only digit that has the bar over. Braille grouping signs are needed or the bar would apply to the entire number as an "item."

Directly Under/Directly Over Level Change Indicators

Other less common modifiers that appear under or over another symbol or expression use directly under or directly over indicators to identify the level change. These indicators follow the same rules for grouping and grade 1 indicator placement as subscript and superscript level changes.

Directly Below indicator	⠨⠢
Directly Above indicator	⠨⠔

$$line\ \ \overset{\overleftrightarrow{}}{FG}\ is\ the\ diameter\ of\ ◯A\$$

⠇⠔⠑⠀⠰⠰⠣⠠⠠⠋⠛⠜⠨⠔⠳⠺⠗⠕⠀⠊⠎⠀⠮⠀⠙⠊⠁⠍⠑⠞⠻⠀⠷⠀⠰⠫⠿⠀⠠⠁

$$\overset{⃑}{a}\ \ \perp \ \ \overset{⃑}{b}\ \$$ ⠁⠨⠔⠰⠳⠈⠗⠕⠀⠼⠤⠀⠰⠃⠨⠔⠰⠳⠈⠗⠕$\underset{x\to 0}{lim}f\left(x\right)$

⠰⠰⠇⠊⠍⠨⠢⠣⠭⠳⠕⠼⠚⠜⠀⠋⠐⠣⠭⠐⠜

FYI: Although the arrow is considered a sign of comparison, it is unspaced when it is not on the base line.

$$\overset{\mathrm{\Delta}}{\rightarrow}$$ ⠰⠰⠳⠺⠗⠕⠨⠔⠠⠨⠙Note: That is capital Delta, not a triangle. Notice the Greek letter indicator in front of the letter d.

Summation Notation

Summation notation requires both subscript and superscript and directly under and directly over level change indicators. Just like subscript and superscript, the directly under level change is listed first, followed by directly over. The equal sign is unspaced when it is not on the base line.

Example:

$$\sum _{i=1}^n{x}_1$$

⠰⠰⠠⠨⠎⠨⠢⠣⠊⠐⠶⠼⠁⠜⠨⠔⠝⠭⠢⠼⠁