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Unit 15 Lines, Arrows, and Modifiers

Lines and Arrows

Mathematical expressions are commonly modified with graphics of arrows, bars, or lines appearing within, directly over, or directly under a character(s). Nemeth Code uses the shape symbol and other modifiers to express these graphics.

Right Pointing Arrow →	⠫⠒⠒⠕
Right Arrow (Contracted) →	⠫⠕
Left Pointing Arrow ←	⠫⠪⠒⠒
Two-way Pointing Arrow ↔︎	⠫⠪⠒⠒⠕
Parallel to ∥	⠫⠇
Perpendicular to ⟂	⠫⠏

Examples:

q → r

⠟⠀⠫⠕⠀⠗

a' ∥ b

⠁⠄⠀⠫⠇⠀⠃

s ↔︎ t

⠎⠀⠫⠪⠒⠒⠕⠀⠞

* Notice that these three examples use the shape as a sign of comparison and, therefore, a space is left between the sign of comparison and the numbers or letters.

Nemeth Code Switch Indicators

All lines, arrows, and modifications are written within Nemeth Code switch indicators.

Examples:

Is DE ∥ FG?

⠠⠊⠎⠀⠸⠩⠀⠠⠙⠠⠑⠀⠫⠇⠀⠠⠋⠠⠛⠀⠸⠱⠦

line ae ⟂ line bc

⠸⠩⠀⠇⠊⠝⠑⠀⠁⠑⠀⠫⠏⠀⠇⠊⠝⠑⠀⠃⠉⠀⠸⠱

Note: Examples in this lesson that are not embedded in UEB have omitted the Nemeth Code switch indicators.

Modification Indicators

When these shapes are used as modifiers written directly over or directly under a number(s), letter(s), or an expression, modification indicators are used to identify the character(s) that are affected by the modification.

Multipurpose Indicator	⠐
Directly Over Indicator	⠣
Directly Under Indicator	⠩
Terminator	⠻

Formula for Modification for Directly Over or Under

Multipurpose Indicator	Expression to be modified	Directly over/under Indicator	Modifier	Termination Indicator
⠐		⠣ or ⠩		⠻

The (1) multipurpose indicator signals that a modification will be made to the (2) expression. The (3) directly over/under indicator plus the (4) modifier is placed directly after the character(s) it modifies. The (5) termination indicator signals an end to the modification.

Examples for Directly Over:

$\overrightarrow{EF}$

1	2	3	4	5
⠐	⠠⠑⠠⠋	⠣	⠫⠕	⠻

⠐⠠⠑⠠⠋⠣⠫⠕⠻

(1) The multipurpose indicator signals the beginning of the modified expression

(2) The expression to be modified is EF

(3) Modification is to be Directly Over

(4) Modifier is (Right Arrow [Contracted] →)

(5) Termination indicator signals the end

More Examples:

$\overleftarrow{n}$

⠐⠝⠣⠫⠪⠒⠒⠻

* The letter "n" is modified with the left pointing arrow.

$\overleftrightarrow{CD}║\overleftrightarrow{AB}$

⠐⠠⠉⠠⠙⠣⠫⠪⠒⠒⠕⠻⠀⠫⠇⠀⠐⠠⠁⠠⠃⠣⠫⠪⠒⠒⠕⠻

Other Common Modifiers:

Arc ⌒	⠫⠁
Dot ˙	⠡
Hollow Dot ˚	⠨⠡
Delta ∆	⠨⠠⠙
Horizontal Bar ―	⠱
Math Caret (hat) ^	⠸⠣

Examples:

ẋ(t)

⠐⠭⠣⠡⠻⠷⠞⠾

$\overline{x + y}$

⠐⠭⠬⠽⠣⠱⠻

$\underrightarrow{\mathrm{\Delta }}$

⠐⠫⠕⠣⠨⠠⠙⠻

$\stackrel{\mathrm{⌒}}{XY}$

⠐⠠⠭⠠⠽⠣⠫⠁⠻

$\widehat{y}=d$

⠐⠽⠣⠸⠣⠻⠀⠨⠅⠀⠙

Examples for Directly Under:

ṃ

⠐⠍⠩⠡⠻

589

⠼⠢⠐⠦⠔⠩⠱⠻

* Note: There is no typeform for underline. The directly under modification is used instead.

589

⠐⠢⠦⠩⠱⠻⠔

* Notice: Because the number 5 is not preceded by a space, a numeric indicator is not used. Also, notice that the modification ends before the number 9 is written.

Horizontal Bar Exception for Directly Over:

When a horizontal bar appears directly over a single digit or letter, it is expressed by the horizontal bar symbol (⠱) placed unspaced after the character it modifies. No other indicators are used.

Examples:

$\stackrel{\_}{x}$

⠭⠱

$\stackrel{\_}{x}y$

⠭⠱⠽

$\mathrm{e}\stackrel{\_}{\mathrm{f}}\mathrm{g}$

⠑⠋⠱⠛

${\displaystyle \frac{1}{3}}=0.333333\text{…}=\stackrel{\phantom{0.}\_}{0.3}$

⠹⠂⠌⠒⠼⠀⠨⠅⠀⠼⠴⠨⠒⠒⠒⠒⠒⠒⠀⠄⠄⠄⠀⠨⠅⠀⠼⠴⠨⠒⠱

$\stackrel{\_}{x}+\stackrel{\_}{y}$

⠭⠱⠬⠽⠱

Horizontal Bar Exception for Directly Under

When a horizontal bar appears directly under a single digit or letter, it is expressed by the "directly under" indicator (⠩) followed by the horizontal bar symbol (⠱) placed unspaced after the character it modifies. No other indicators are used.

Examples:

589

⠼⠢⠦⠩⠱⠔

xyz

⠭⠽⠵⠩⠱

Ṉ₂

⠠⠝⠩⠱⠆