Hatch marks (also called hash marks or tick marks) are a form of

mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathema ...

. They are used in three ways as: * Unit and value marks — as on a ruler or number line * Congruence notation in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

— as on a

geometric figure A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...

* Graphed points — as on a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...

Hatch marks are frequently used as an abbreviation of some common units of measurement. In regard to distance, a single hatch mark indicates feet, and two hatch marks indicate inches. In regard to time, a single hatch mark indicates minutes, and two hatch marks indicate seconds. In

and

trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...

, such marks are used following an elevated circle to indicate degrees, minutes, and seconds — ( ° ) ( ′ ) ( ″ ). Hatch marks can probably be traced to

hatching Hatching (french: hachure) is an artistic technique used to create tonal or shading effects by drawing (or painting or scribing) closely spaced parallel lines. (It is also used in monochromatic representations of heraldry to indicate what the ...

in art works, where the pattern of the hatch marks represents a unique tone or hue. Different patterns indicate different tones.

Unit and value marks

Unit-and-value hatch marks are short vertical line segments which mark distances. They are seen on

ruler A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long ...

s and

number line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...

s. The marks are parallel to each other in an evenly-spaced manner. The distance between adjacent marks is one unit. Longer line segments are used for

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

s and

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...

s. Shorter line segments are used for

fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

s. Hatch marks provide a visual clue as to the value of specific points on the number line, even if some hatch marks are not labeled with a number. Hatch marks are typically seen in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...

and

. <----, ----, ----, ----, ----, ----, ----, ----, ----, ----, ----, ----, ----, ----, ----> -3 -2 -1 0 1 2 3

Congruency notation

, hatch marks are used to denote equal measures of angles, arcs, line segments, or other elements.{{Cite web, title=Introduction to Geometry {{! SkillsYouNeed, url=https://www.skillsyouneed.com/num/geometry.html, access-date=2020-09-01, website=www.skillsyouneed.com Hatch marks for congruence notation are in the style of

tally mark Tally marks, also called hash marks, are a unary numeral system (arguably). They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate r ...

s or of

Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...

– with some qualifications. These marks are without

serif In typography, a serif () is a small line or stroke regularly attached to the end of a larger stroke in a letter or symbol within a particular font or family of fonts. A typeface or "font family" making use of serifs is called a serif typeface ( ...

s, and some patterns are not used. For example, the numbers I, II, III, V, and X are used, but IV and VI are not used, since a rotation of 180 degrees can make a 4 easily confused with a 6. For example, if two triangles are drawn, the first pair of congruent sides can be marked with a single hatch mark on each. The second pair of congruent sides can be marked with two hatch marks each. The patterns are not alike: one pair uses one mark while the other pair uses two marks (Figure 1). This use of pattern makes it clear which sides are the same length, even if the sides cannot be measured. If the sides do not appear to be congruent, as long as hatch marks are present and are the same number of hatch marks, then the sides are congruent. Note that the inverse situation should not be assumed. That is, while sides that are hatch marked identically must be assumed to be congruent, it does not follow that sides hatch marked ''differently'' must be ''incongruent''. The different hatch marks simply signal that the length measurements may (in this case) be considered to be ''independent'' of each other. So, for example, while we are not allowed to conclude that the triangles in the accompanying figure ''must be''

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

s or

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

s, we yet remain obliged to allow that they ''could be'' either of those things.

Graphed points

Line chart A line chart or line graph or curve chart is a type of chart which displays information as a series of data points called 'markers' connected by straight line segments. It is a basic type of chart common in many fields. It is similar to a s ...

s may sometimes use hatch marks as graphed points. In the early days of computers, monitors and printers could only make charts using the characters available on a common typewriter. To graph a line chart of sales over time, symbols such as *, x, or , were used to mark points, and various characters were used to mark the lines connecting them. While computers have advanced considerably, it is still not unusual to see x or , used as the points of interest (or points of change) on a graph.

References

Mathematical notation Geometry

Unit and value marks

Congruency notation

Graphed points

See also

References