Comparison sign

<>
Mathematical signs
arithmetic
Plus sign	+
Minus sign	- , ./.
Mark	⋅ , ×
Divided sign	: , ÷ , /
Plus minus sign	± , ∓
Comparison sign	< , ≤ , = , ≥ , >
Root sign	√
Percent sign	%
Analysis
Sum symbol	Σ
Product mark	Π
Difference sign , Nabla	∆ , ∇
Prime	′
Partial differential	∂
Integral sign	∫
Concatenation characters	∘
Infinity symbol	∞
geometry
Angle sign	∠ , ∡ , ∢ , ∟
Vertical , parallel	⊥ , ∥
Triangle , square	△ , □
Diameter sign	⌀
Set theory
Union , cut	∪ , ∩
Difference , complement	∖ , ∁
Element character	∈
Subset , superset	⊂ , ⊆ , ⊇ , ⊃
Empty set	∅
logic
Follow arrow	⇒ , ⇔ , ⇐
Universal quantifier	∀
Existential quantifier	∃
Conjunction , disjunction	∧ , ∨
Negation sign	¬

Compare characters in the are mathematical notation usual characters to represent the proportions of two numbers or terms . The most important comparison symbols are the equal sign (=) as well as the greater than sign (>) and the less than sign (<). Comparison signs can be combined in many ways, for example with a tilde for equivalence . Crossed out variants represent the negation of the original relationship. Many of the combinations coincide with the meanings of other characters in most applications.

Typography and appearance

Angle brackets , comparison characters, and simple guillemets (half-pointed characters) in Cambria, DejaVu Serif, Andron Mega Corpus, Andika, and Everson Mono fonts

The less-than-sign and the greater-than-sign consist of a line bent in the middle with straight legs, whereby the kink angle is regularly acute (i.e. smaller than a right angle). The end points are perpendicular to each other so that the legs are of equal length; this is mostly retained in italics . The height and position of the signs regularly correspond to those of the plus sign . In most fonts, they are therefore significantly larger than Guillemets (pointed characters) and, on the other hand, significantly smaller than angle brackets . The latter also differ in that they have a much more open angle at the kink, mostly much larger than a right angle.

history

Less-than and greater-than signs with serifs in a print from 1802

The symbols "and" were introduced by the English mathematician Thomas Harriot in 1631 in his work Artis Analyticae Praxis . The sign ≥ was first used by the French mathematician Pierre Bouguer in 1734.

use

mathematics

In mathematics, comparison signs ( apart from the equal sign ) are used to form inequalities . In elementary mathematics, they denote the comparison of numbers, and they are also used as symbols for general order relations .

The less than sign (<) indicates a two-digit relation , the semantic assignment of which depends on the algebra used . It is implicitly assumed that the relation is evaluated as "true".

In the daily usage of natural numbers, it denotes the relation of a really smaller (not equally large!) Value to a really larger value. In prefix notation this means: <( a , b ) is evaluated as "true", so a is really smaller than  b .

The more common form is the infix notation a < b , if a is really smaller than  b .

example

The value of the natural number 3 is really smaller (has a lower order) than the value of the natural number 4. The order is given by the number line of the natural numbers.

One writes: ${\displaystyle 3<4}$

The following also applies to real numbers : ${\displaystyle 3{,}14<3{,}15}$

Memorabilia and donkey bridges

Donkey bridge for less than sign

Donkey bridge for greater than sign

To avoid confusion between the greater than sign and the less than sign, the comparison with a crocodile that always snaps for the bigger “bite” is sometimes seen as helpful - especially for schoolchildren: “The crocodile that always wants to eat most of it ”. In the magazine Kopf und Zahl (ZTR magazine for the treatment of arithmetic weaknesses) this motto is criticized:

“I am skeptical of such an 'explanation', as this donkey bridge is not based on mathematical logic, but is based solely on the desire for a child-friendly image. What if the reptile is just a little hungry? (...) Instead, I prefer an explanation that refers to the origin of the symbol: 'On the side on which the symbol is larger, there is also the larger number.' In this way you also get an elegant transition to 'is the same': this symbol is equally open on both sides. "

- Journal Heads and Tails , 8th edition, 2007

Alternatively, the less than sign can be stylized with a vertical line to a k (or K) and the greater than sign with a semicircle to a G.

Markup languages

In some markup languages such as HTML or XML , less-than and greater-than characters are used to identify the start and end of all (main) elements ( tags ) in their own language. In order to still be able to display such markups in HTML, the elements named <and (also abbreviated in English) can be used as a substitute >- for example, for the paragraph -start and end characters <p>and </p>(compare also paragraph characters and see generally under masking characters ).

linguistics

In linguistics , the greater-than sign is used to mean that the grammatical or phonetic form on the right can be derived from the form on the left. Conversely, the less than sign means that the shape on the left is or can be a derivation from the shape on the right. So here both signs are to be understood as arrowheads.

An example: " Greek alphabet " -ancient Greek ἑλληνικός ἀλφάβητος  > modern Greek ελληνικό αλφάβητο or in the opposite direction ελληνικό αλφάβητο  < ἑλληνικός ἀλφάβητος.

Representation in computer systems

Keyboard input

On standard German keyboards, the less than sign and the greater than sign are entered using the key to the right of the left shift key .

On German standard keyboards with the assignment T2 according to DIN 2137 : 2012-06, the less than or equal sign is entered with the key combination AltGr+ a, the greater than or equal sign with the key combination AltGr+ s.

In macOS , the less than or equal to sign is entered with the key combination Alt+ <, the greater than or equal to sign with the key combination Alt+ ⇧+ >.

List of comparison characters

Mathematical comparison symbols

character		Unicode		meaning		character		Unicode		meaning
=	≠	U+003D	U+2260	equal / unequal		≈	≉	U+2248	U+2249	almost the same / not almost the same
<	>	U+003C	U+003E	smaller / larger than		≺	≻	U+227A	U+227B	previous / next
≤	≥	U+2264	U+2265	less than / greater than or equal to		≼	≽	U+227C	U+227D	previous / next or same
≮	≯	U+226E	U+226F	not smaller / larger than		⊀	⊁	U+2280	U+2281	not preceding / following
≰	≱	U+2270	U+2271	neither less than / greater than nor equal		⋠	⋡	U+22E0	U+22E1	neither preceding / following nor equal
≲	≳	U+2272	U+2273	smaller / larger than or equivalent		≾	≿	U+227E	U+227F	preceding / following or equivalent
⋜	⋝	U+22DC	U+22DD	equal to or less / greater than		⋞	⋟	U+22DE	U+22DF	same or previous / next
⋦	⋧	U+22E6	U+22E7	smaller / larger than, but not equivalent		⋨	⋩	U+22E8	U+22E9	preceding / following, but not equivalent
≴	≵	U+2274	U+2275	neither smaller / larger than nor equivalent		⊰	⊱	U+22B0	U+22B1	previous / next in relation
≦	≧	U+2266	U+2267	smaller / larger than about equal to		≨	≩	U+2268	U+2269	less than / greater than, but not equal to
≪	≫	U+226A	U+226B	much smaller / larger than		⋘	⋙	U+22D8	U+22D9	much smaller / larger than
≶	≷	U+2276	U+2277	smaller / larger or larger / smaller than		≸	≹	U+2278	U+2279	neither smaller / larger nor larger / smaller than
⋚	⋛	U+22DA	U+22DB	less / greater than, equal or greater / less than		⋖	⋗	U+22D6	U+22D7	smaller / larger than with point

The ASCII set includes the less than sign (code 0x3C), the equal sign (code 0x3D), and the greater than sign (code 0x3E).

Typographic variants

Depending on the tradition of the formula set , slightly different variants are used for the less than or equal to and greater than or equal to sign:

character		Unicode		Latex		HTML
≤	≥	U+2264	U+2265	\leq	\geq	≤	≥
≦	≧	U+2266	U+2267	\leqq	\geqq	≦	≧
⩽	⩾	U+2A7D	U+2A7E	\leqslant	\geqslant	⩽	⩾

In DIN 1302 “General mathematical symbols and terms” , the variants of the first line are specified for the less than or equal to and greater than or equal to symbols. It is also these characters that can be entered with the German standard keyboard ( assignment E1 ) in accordance with DIN 2137-01: 2018-12 and assignment T2 in accordance with the previous standard DIN 2137-01: 2012-06.

See also

• Comparison operator

Individual evidence

1. ↑ Johann Friedrich Ludwig Häseler : Beginnings of arithmetic . Meyersche Buchhandlung, Lemgo 1802, part 1, p. 89.
2. ^ Clifford A. Pickover: A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality . John Wiley & Sons, 2005, ISBN 978-0-471-69098-6 , pp. 22 . ( wordpress.com ( Memento of the original from March 4, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this note. (PDF); October 2015) 
3. ↑ heads and tails. (PDF; 621 KB) In: JOURNAL of the Association for Learning Therapy and Dyscalculia e. V. in cooperation with the Mathematical Institutes for the Treatment of Mathematical Weaknesses (ZTR), 8th edition, 2007. Association for Learning and Dyscalculia Therapy, November 6th, 2007, p. 8 , accessed on September 1st, 2018 . 
4. ↑ Scott Pakin: The Comprehensive LaTeX Symbol List. (PDF, 8.7 MB) January 19, 2017, p. 61 , archived from the original on September 28, 2017 ; Retrieved on September 28, 2017 (English, linking the original results in a mirror of CTAN , the archive link compare file: Comprehensive LaTeX Symbol list.pdf ).