Meta: Mathematics symbols are a familiar asset for most people, yet only a few truly grasp their implications. This article will shed light on such inquiries.
Intro
Mathematics symbols are a consistent presence in all Math formulas and structures. We encounter them almost every day, even if we are not students anymore. But have you ever wondered why they were invented – or the implications behind their usage?
This article will introduce their origins and explain the meaning of certain prevalent symbols.
What Are Mathematics Symbols?
They symbolize a mathematical object – Source: Wikimedia.
Mathematical symbols are figures (or collections of icons) used to symbolize a mathematical object, a mathematical action on a mathematical item. They reflect the relationship among mathematical objects and structure other symbols in a formula.
Since complete formulas are often composed of numerous icons, several symbols are required to represent all types of mathematics aspects.
The most fundamental indicators are decimal digits (0 to 9) and the Latin alphabet’s letters. These decimal units thrive in the Hindu–Arabic system, which is to portray numbers. Also, in our history, mathematicians employed upper-case letters to indicate points in geometry, whereas their lower-case counterparts strive to denote variables and constants.
Letters are also a popular instrument to illustrate multiple mathematical items. Due to the remarkable increase in contemporary math, Greek letters and a few Hebrew characters are also applied.
The typical font in mathematical formulae is italic (for Latin and lower-case Greek characters) and upright (for upper-case Greek). In the same vein, other fonts help accommodate extra signals, most notably boldface or script typeface.
What Is Their Early History?
Mathematical figures and symbols have undergone a rather long and turbulent history. Let’s look back at some of the most memorable milestones:
Ancient Greece
The first mathematical representations for arbitrary values arose in ancient Greece in the fifth and fourth century B.C. The line lengths symbolized arbitrary variables (areas, volumes, and angles), while rectangles denoted the combination of two such numbers with three sides representing the corresponding components.
Quantities are signified in Euclid’s Elements (3rd century B.C.) by two letters – the starting and terminal letters of the relevant section – and sometimes by a single letter. The latter mechanism, which dates from Archimedes, became more mainstream.
This notational system may have evolved into a geometry of letters, but unfortunately, no operations on letters carried on back then, so such calculus did not exist.
Post-Hellenistic Period
Algebra was liberated from its geometric context – Source: Free Images Live.
The post-Hellenistic period saw the emergence of the fundamentals of the letter system and calculus, which occurred once algebra was liberated from its geometric context.
Diophantus denoted unknown variables – or their powers with coefficients to the right. For instance, 3×5 was denoted by δκυ~γ¯(with γ¯=3). Term additions were as straightforward as their juxtapositions, whereas subtracting procedures needed a more specific sign. Meanwhile, equality was represented by the letter “ιι” in Greek.
Mathematics in India
Hundreds of years later, Indian mathematicians – who had created a quantitative algebra – introduced numerous symbols for certain unknown factors, such as squares, square roots, and the subtrahend. As a result, the equation 3 x 2+10x-7=x2+1 was expressed as follows in Brahmaputra’s nomenclature (7th century):
ya -va- 3-ya-10-ru-7
ya-va-1-ya-0-ru-1
Renaissance
Symbols became more popular – Source: pixabay.
The development of contemporary algebraic symbols transpired during the 14th and 15th centuries, influenced by advances in pragmatic arithmetic and equation analysis.
In several nations, symbols representing eclectic operations and powers of an unknown magnitude developed spontaneously. However, decades – if not centuries – passed before any of them finally got acknowledged as practical instruments for computations.
Thus, around the end of the 15th century, L. Pacioli and N. Chuquet exploited the symbols p and m (derived from the Latin minimum and maximum) for adding and subtracting correspondingly.
At the same time, German mathematicians invented the contemporary “+” (possibly an acronym for the Latin “et:”). As of the seventeenth century, around twelve distinct multiplication symbols co-existed.
Following Leonardo Pisano (1220), the sign R.R. (from the Latin “radix,” i.e., root) was extensively utilized to denote “square root.” Chuquet signified squares, cubes, and so on by RR2 and RR3.
Some academics repeated the similar symbol for higher-level functions; others added a suitable character after it (an abbreviation for the exponent’s name). Others marked a fitting illustration in a circular pattern (or between square brackets or parentheses) to differentiate it from the number beneath the radical sign.
It was not until the early 18th century that people started writing the exponent above the radical sign. Nonetheless, this norm seemed to date back to the 16th century (A. Girard, 1629), which means about 500 years had passed before the official acceptance of radical signs.
Leibniz and The Development of Contemporary Notions
G. Leibniz coined the contemporary notation for differential and integral calculus. He devised the differentials dx, d2x, d3x, and the integral ydx.
Most experts lauded Leibniz’s integral symbol’s primary benefit over Newton’s proposal, particularly the addition of x. While Leibniz’s notation ydx implies the process of yielding an integral sum, it also contains explicit identification of the integration variable.
Hence, the notation ydx is well suited for formulae involving variable transformations – and is often used for repeated and line integrals. Newton’s terminology does not express such possibilities explicitly. In the same fashion, Leibniz’s differential signs are superior to Newton’s for fluxions and minuscule increments.
Euler’s Refinements
L. Euler – Source: Flickr.
L. Euler is credited for his impressive development of a significant quantity of contemporary mathematical notation.
He forged the first universally recognized sign for a variable action – the function indicator fx (derived from the Latin function; 1734), though reports claimed that J. Bernoulli used the sign x earlier (1718).
Following Euler, the standardization of symbols for numerous individual equations (including trigonometry) occurred. Euler became the first person to use the expression e (the base of the natural log, 1736) to popularize the notation (probably derived from the Greek, i.e., circumference, 1736; Euler borrowed the notation from H. Jones.).
He also incorporated the illusory unit I, which quickly gained widespread acceptance.
Why Are They Important?
In a way, we could claim that symbols are ageless. While languages undergo constant changes and modifications, most mathematical signals persist for hundreds of years – and hence, streamline the process of passing down your knowledge from generation to generation.
Some symbols encompass intricate and abstract notions, which we must describe in thousands of paper pages (sometimes even books). Suppose people opted for words and sentences to showcase formulas; the number of books and volumes to accommodate all of them would be insane.
Another palpable upside is their effective and straightforward layout, facilitating much faster comprehension.
Let’s think of a familiar formula and try to express it in full sentences. What do we have here? It is no surprise that the whole concept gets convoluted and extremely challenging. On the other hand, a few shortened symbols will familiarize readers with these configurations in only a few seconds.
Thus, discarding signals and icons in favor of literary words is not a sensical approach. Are you one of those students who already struggle with Math? Then this subject would crush you even worse if there were no existing math symbols at all.
What Are Some Common Mathematical Symbols?
Algebraic Operators
Algebraic operators – Source: pixabay.
+
1. It demonstrates addition and is pronounced as “plus” (example: 3+ 2).
2. Sometimes, we might use it to represent a disjointed fusion of sets.
–
1. It implies subtraction and is pronounced: “minus” (example: 3 – 2).
2. It also denotes the multiplicative inverse and is interpreted as “negatives” or “the inverse of”; for instance, –2.
3. In some scenarios, this symbol might depict the set-theoretic counterpart; see Set theory.
x
1. In basic mathematics, x implies multiplication and is spelled as “times” (example: 2×3).
2. The icon denotes the scalar product in topology and linear algebra.
3. In the set and category theory, most experts employ x to refer to the Cartesian and direct products.
.
1. The icon portrays multiplication and is interpreted “multiple times” (examples: 3.2).
2. In geometry and algebra, the dot product is denoted by “.”
3. It is a placeholder element that serves as a replacement for an uncertain factor. By illustration, “the optimum value is indicated by |. |” is more precise than writing “| | signifies the optimum value.”
±
1. ± means a plus or negative symbol.
2. ± portrays the range of possible values for a measured quantity; by illustration, 10±2 indicates an unknown number between 8 and 12.
∓
When used with ±, the ∓ symbol will imply the polar opposite indicator (+ when ± means –, and – when ± means +).
÷
This icon thrives in anglophone nations to denote a division. However, its usage is not encouraged in certain countries. Other places employ it to denote subtraction at times.
1. ÷ indicates the relationship between the two quantities.
2. In some cases, this term may refer to the partition.
3. Regarding set-builder syntax, ÷ operates as a divider that means “in such a way”.
/
1. / represents division and is pronounced “divided by” or “over”.
(examples: 3/2). The icon is often omitted in favor of a horizontal bar, though.
2. It depicts the presence of a quotient structure. For instance, quotient groups and quotient categories are all examples of quotients.
3. The symbol denotes transition probabilities in probability theory.
√
1. √ represents the square roots and is read as such. The figure is seldom used in contemporary mathematics without a horizontal bar defining the argument’s breadth. We rarely see it in contemporary mathematics, either.
2. A left superscript larger than 2 signifies an nth root.
^
1. In typical formulas, exponentiation is framed by a superscript.
2. ^ is not to be used interchangeably with ∧.
Equality
Equality – Source: Free Svg.
=
1. = indicates that there is no difference.
2. It refers to a mathematical object inside a statement, such as “let x=E,” where E represents an expression. The symbol relates to the computer science notion of assignment, directed in numerous ways (which relies on the programming language).
≠
≠ alludes to inequity and translates as “not equal.”
≈
≈ means “roughly equivalent to”. For instance, π ≈ 3.14
~
1. When operated between two integers, it signifies either “approximately equal” (the same as ≈) or “has the same magnitude as”
2. ~ suggests that two functions or patterns are asymptotically equivalent.
3. ~ to refer to various sorts of similarity, such as matrix similitude or geometric shape resemblance.
4. It is a representation of equivalence relations using a standard notation.
5. The icon defines a random variable’s probability distribution in statistics and probability. For instance, X~ N (0,1) implies that the variable x has a standard normal distribution.
6. Sometimes, mathematicians employ ~ to demonstrate proportionality.
≡
1. ≡ highlights an identification, a pertinent equivalence regardless of the values assigned to the variables incorporated inside it.
2. In theoretical physics (more precisely, modular arithmetic), it signifies congruence mod integers.
≅
1. ≅ refers to an isomorphism between different mathematical structures and is interpreted as “is isomorphic”.
2. In geometry, the figure alludes to the conformance of two geometric forms (or their comparability up to a relocation). We read this symbol as “is congruent to”.
Comparison
Comparison – Source: edetaria.
<
1. < equates to a strict incompatibility between two numbers. We spell it as “less than”.
2. < also denotes any rigorous order.
3. Between two groups, it might indicate that the first is a legitimate subgroup of the second.
>
1. > points out a strict incompatibility between two numbers (translated as “more than.”)
2. Similar to <, it also indicates any rigorous order.
3. Between two groups, it implies that the second is a legitimate subgroup of the first.
≤
1. ≤ means an item is less than (or equivalent) to another. X ≤ Y means X < Y or X = Y.
2. When sandwiched between two groupings, it specifies that the first is a subdivision of the second.
≥
1. ≥ reveals that a value is approximately greater or equivalent to another. X ≥ Y means X > Y or X = Y.
2. Paired with two clusters, it hints that the second is a subdivision of the first.
Conclusion
This article has introduced to you the backgrounds and implications of some of the most prevalent mathematics symbols. A more exhaustive understanding of their functions will expedite your calculating procedures and support you in diverse Math issues.
Write to us if there are still questions. We are ready to tackle your inquiries!