Mathematical discovery has been at the forefront of every civilized society since the dawn of recorded history, and math has been used by even the most primitive and earliest cultures. Since ancient times, particularly the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology. In more recent times, it has assumed a similar role in the quantitative aspects of the life sciences. It became necessary due to the increasingly complex demands of societies worldwide, which necessitated more advanced mathematical solutions.
Math is present in everything we do. It serves as the foundation for everything in our daily lives, including mobile devices, computers, software, ancient and modern architecture, art, money, engineering, and even sports. Each school grade taught us some math, beginning with counting, as you may have noticed. Then, using quadratic equations and solving 90-degree triangle problems with sines and cosines, we learned how to add and subtract, then how to memorize multiplication tables, then how to do long division, and so on. Math is truly present in every action we do.
Math is the science and study of numbers, formulas, the logic of shapes, quantities, and arrangement, structure, space, and change. There are different definitions as to what mathematics really is. There is no general agreement on its precise scope or epistemological status. Today, mathematics is found essential in many fields like social sciences, natural sciences, engineering, medicine, and finance.
The majority of mathematics has developed since the 15th century CE, as a result of science’s exponential growth, and it is a historical fact that from the 15th century to the late twentieth century, new developments in mathematics were largely concentrated in Europe and North America.
This is not to say that developments elsewhere have been insignificant. Indeed, in order to comprehend the history of mathematics in Europe, one must first understand its history in ancient Mesopotamia and Egypt, ancient Greece, and Islamic civilization from the 9th to the 15th centuries.
India made significant contributions to the development of contemporary mathematics through its significant influence on Islamic mathematics during its formative years. South Asian mathematics is concerned with the origins of mathematics on the Indian subcontinent and the development of the modern decimal place-value numeral system on the subcontinent. East Asian mathematics encompasses the majority of mathematics development in China, Japan, Korea, and Vietnam.
It is critical to understand the nature of the sources used to study the history of mathematics. The history of Mesopotamian and Egyptian mathematics is based on surviving scribe-written original documents. Although these documents are few in number in Egypt, they are all of a similar nature and demonstrate unequivocally that Egyptian mathematics was, on the whole, elementary and profoundly practical in orientation. On the other hand, Mesopotamian mathematics is represented by a large number of clay tablets that reveal mathematical achievements far superior to those of the Egyptians. The tablets indicate that the Mesopotamians possessed extraordinary mathematical knowledge, though they provide no evidence that this knowledge was organized deductively. Further research may shed light on the early development of mathematics in Mesopotamia or on its influence on Greek mathematics, but it appears likely that this picture of Mesopotamian mathematics will endure.
There are no preserved Greek mathematical documents prior to Alexander the Great, and even for the subsequent period, it is worth noting that the earliest copies of Euclid’s Elements are in Byzantine manuscripts dating from the tenth century CE. This is diametrically opposed to the previous situation for Egyptian and Babylonian documents. Although the current account of Greek mathematics is secure in broad strokes, historians have offered competing accounts based on fragmentary texts, quotations from early writings culled from nonmathematical sources, and a fair amount of conjecture.
Numerous important treatises from the early period of Islamic mathematics have been lost or survived only in Latin translations, leaving many unanswered questions about the relationship between early Islamic mathematics and Greek and Indian mathematics. Additionally, the amount of surviving material from later centuries is so vast in comparison to what has been studied that it is not yet possible to make any definitive judgments about what later Islamic mathematics did not contain, and thus it is not yet possible to assess with certainty what was novel in European mathematics from the 11th to the 15th centuries.
In modern times, the invention of printing has largely eliminated the problem of obtaining secure texts, allowing historians of mathematics to focus their editorial efforts on mathematicians’ correspondence or unpublished works. However, the exponential growth of mathematics means that historians can treat only the major figures in detail for the period beginning in the nineteenth century. Additionally, as the period approaches the present, there is the issue of perspective. Mathematics, like any other human activity, has fashions, and the closer one gets to a particular era, the more likely it is that these fashions will become the wave of the future.
Until the 1920s, it was widely accepted that ancient Greeks were the first to use mathematics. At best, the Rhind papyrus provided a sliver of a precedent for earlier traditions, such as Egyptian mythology. As historians deciphered and interpreted Mesopotamian technical materials, this impression gave way to a very different one.
As clay tablets used by Mesopotamian scribes are extremely durable, there is substantial evidence of this culture’s existence. Mathematics specimens from all the major eras can be found, including Sumerian kingdoms of the 3rd millennium BCE, Akkadian/Babylonian era (2nd millennium), Persian empire (6th–4th century BCE), and Greek empire (early 1st millennium BCE) (3rd century BCE to 1st century CE). After Hammurabi, the lawgiver-king of the Old Babylonian dynasty (c. 18th century BCE), there were few notable advancements in the level of competence. However, during the Persian and Seleucid (Greek) periods, mathematics was used extensively in astronomy.
In the predynastic period (c. 3000 BCE), when writing first appeared in Egypt, a new class of literate professionals was born: the scribes. The scribes took on all the responsibilities of a civil service because of their writing abilities, including keeping records, tax accounting, and overseeing public works (such as building projects). They even assisted in the conduct of war by monitoring military supplies and payrolls. Students at scribal schools were required to learn the basics of reading and writing, as well as the fundamentals of mathematics.
While studying ancient Egyptian literature in the 13th century BCE, students in the New Kingdom were required to copy an amusing letter written by Hori in which the scribe mocks rival Amen-em-opet for his lack of expertise as an advisor or manager. Horatio: You’re the scribe in charge, Hori tells me. Three similar problems in the same letter cannot be solved without additional information. As Hori challenges his rival with these difficult but typical tasks, however, the point of the humor is clear.
The tests posed by the scribe Hori match up with what is known about Egyptian mathematics. In the past, scribal schools used two long papyrus documents as textbooks to teach their students. There is a copy of a two-century-old papyrus in the British Museum’s Rhind papyrus. 84 specific problems in arithmetic and geometry are solved after a long table of fractional parts to aid division. Golenishchev papyrus (in the Moscow Museum of Fine Arts) presents 25 problems of a similar type dating from the 19th century BCE. For example, the scribes would have to figure out how to distribute beer and bread as wages, or how to measure fields and the volumes of pyramids. These problems are a good fit for their duties.
Archimedes is considered as the Father of Math.
There were several civilizations that contributed to the invention of math that is used in today’s time. This includes the civilizations of China, India, Egypt, Central America, Mesopotamia, and Sumerians. In fact, the Sumerians were known to be the first group of people to develop the counting system with a system of base 60. From there, as civilizations develop, more and more branches of mathematics emerge. Really, math is hard to figure out where it came from. There were also articles that thought that the Pythagoreans came up with math in the early 6th century. After that, Euclid came up with the axiomatic method, which includes a definition, an axiom, a theorem, and a proof.
Pure and applied math are the two broad categories of mathematics.
Pure math is the study of math for the sake of math. It is the study of mathematical concepts without regard to their application in other fields.
The following is a list of the branches of Pure Math.
- Number Theory
- Mathematical Analysis
Pure mathematics can be simply defined as the study of mathematical concepts that are entirely based on mathematics and are unrelated to any other concept. Its seven branches are algebra, arithmetic, combinations, geometry, mathematical analysis, number theory, and topology.
Applied math is the study of math with the goal of resolving real-world problems. It is used to construct skyscrapers, create computers, forecast earthquakes, and explain how the economy works, among other things.
The following is a list of the branches of Applied Math.
- Statistics and Probability
- Set Theory
Applied math is a branch of math that looks at how different fields of study can be used together with math concepts. It is just the use of math and specific knowledge together. It has four branches namely: calculus, set theory, statistics & probability, and trigonometry.
Foundations, Arithmetic, Geometry, Calculus, Statistics & Probability, Algebra, Trigonometry, Number Theory and Topology are the main branches of Mathematics.
Arithmetic is a branch of mathematics that deals with numbers and their wide application. The fundamental operations of addition, subtraction, multiplication, and division lay the groundwork for more complex concepts such as exponents, limits, and a variety of other types of calculations. This is a vital branch because its fundamentals are used in daily life for a variety of purposes ranging from simple calculations to profit and loss computations.
These are the fundamental basic operations of arithmetic.
- Addition: It is the process of adding two or more numbers together; it is the sum of two or more numbers.
- Subtraction: It is the process of removing something from a group or a collection of things. When something is subtracted from a group, the number of items in the group decreases or becomes smaller.
- Multiplication: It is a mathematical operation that shortens the process of adding an integer to zero a certain number of times. It is then used to add other numbers in the same way that integers are added to zero.
- Division: It is a process to distribute a group of things into equal parts. It is the opposite of multiplying. The main goal of the division is to figure out how many equal groups there are or how many people are in each group when everyone shares equally.
    
Geometry is one of the earliest branches of mathematics. It is concerned with spatial properties pertaining to the distance, shape, size, and relative position of figures. A geometer is a mathematician who specializes in geometry.
Calculus is the branch of mathematics concerned with the discovery and properties of functions’ derivatives and integrals via methods based on the summation of infinitesimal differences. It is originally called infinitesimal calculus or “the calculus of infinitesimals.” The development of calculus was led by Sir Isaac Newton in the 17th century.
Differential calculus and integral calculus are the two main types of calculus.
- Differential calculus: It is a branch of mathematics that studies the rates of change of quantities. It is concerned with determining the derivatives and differentials, as well as their properties and applications.
- Integral calculus: It is a branch of mathematics that is concerned with the area beneath a curve. Also, it is concerned with the theory and applications of integrals and integration (such as determining lengths, areas, and volumes and solving differential equations).
  
Statistics is the study of the processes of data collection, analysis, interpretation, presentation, and organization. Simply put, it is a mathematical discipline concerned with the collection and summarization of data.
In general, statistics are classified into two categories. These are:
- Descriptive Statistics: It is a set of short descriptive coefficients that summarize a data set, which can be a representation of the population or a sample of the population.
- Inferential Statistics: It is used to describe and infer about a population by analyzing a random sample of data from the population. Inferential statistics are advantageous when it is impractical or impossible to examine every member of a population individually. For instance, it is impractical to measure the diameter of each nail manufactured in a mill.
Algebra is a broad field of math that deals with how to solve and manipulate algebraic expressions to get the results. An algebraic equation is solved for unknown quantities that are written in alphabets. The value of the variable is then found out. It is a branch of math that is very interesting because it uses complicated solutions and formulas to figure out how to solve the problems.
These are the five types of algebra.
- Elementary algebra: This branch talks about the basics of numbers, variables, constants, and how they relate to each other. It tackles equations, how to make them, how to evaluate them, equalities and inequalities, how to solve equations (algebraic and linear), and more.
- Abstract algebra: It talks about truths about algebraic applications that don’t matter what kind of application they are. Associativity, Binary Operations, Identity Element, Inverse Element, and Sets are the parts of this branch of algebra.
- Advanced algebra: This is a more in-depth and intermediate algebra step. The equations in this branch aid in the study of the following:
- Conic sections
- Rational expression
- Commutative algebra: This branch of mathematics is concerned with communicative rings, which include algebraic integer rings, algebra polynomial rings, and so on. Additionally, it can be considered a subfield of abstract algebra. It covers algebra ring theory, Banach algebra, and representation theory, among other topics.
- Linear algebra: This is a branch of mathematics concerned with linear equations and their vector and matrix representations. Linear algebra is a fundamental concept in all branches of mathematics. This branch of algebra covers the following topics:
- Introduction to linear algebra
- Linear equations
- Matrix decomposition
- Relations and Computations
- Vector spaces
Trigonometry is derived from the Greek words “trigonon” which means triangle and “metron” which means “measure.” It is concerned with the study of angles and sides of triangles in order to determine distance and length. Trigonometry is a branch of mathematics that is widely used in the fields of technology and science to develop objects. It is the study of the relationship between the angles and sides of a triangle.
There are three types of trigonometry. These are the following.
- Core trigonometry: This type of trigonometry is used to calculate the angles of triangles with a 90 degree angle.
- Analytical trigonometry: It is a subset of core trigonometry that seeks to determine values using the x-y plane of a triangle.
- Plane trigonometry: It is used to determine the angles’ heights and distances in a plane triangle. This triangle has three vertices (intersection points) on the surface, and its sides are straight lines.
- Spherical trigonometry: It is concerned with triangles drawn on a sphere, and it is frequently used by astronomers and scientists to determine the distances between objects in the universe.
Number theory is one of the earliest branches of mathematics. It found a relationship between real numbers and the numbers in the set of real numbers. The fundamental level of Number Theory teaches the properties of integers, like addition, subtraction, multiplication, and modulus. It then moves on to more complicated systems like cryptography, game theory, and more.
Topology talks about how different shapes change when they are stretched, crumpled, twisted, and laid down. It doesn’t include things like cutting and tearing, so they don’t include them. It can be seen in differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis where it is used.
It’s not just the big, main branches that have a lot of extra branches. There are also advanced branches that are studied at a high level and deal with complicated ideas that require strong computational skills. Lists of more advanced branches are below.
- Cartesian Geometry
- Complex Numbers
- Game Theory
- Matrix Algebra
- Numerical Analysis
- Operations Research
- Set Theory
Abstract algebra is the most difficult part because it deals with complicated and infinite spaces.
Carl Gauss is thought to be one of the best mathematicians of all time because of his work in number theory, geometry, probability theory, geodesy, planetary astronomy, the theory of functions, and potential theory (including electromagnetism).
Benjamin Peirce is an American mathematician, astronomer, and educator who calculated the planets Uranus and Neptune’s general perturbations. In mathematics, he is best known for proving that there is no odd perfect number (a number equal to the sum of its proper divisors) with fewer than four distinct prime factors.
Leonardo Fibonacci was an Italian mathematician from the Republic of Pisa who was dubbed “the Middle Ages’ most gifted Western mathematician.” He introduced the world to a variety of mathematical concepts, including what is now known as the Arabic numbering system, the concept of square roots, number sequencing, and even word problems in mathematics.
Al-Khwrizm gained prominence for his mathematical works. al-Khwārizmī’s algebra book was one of his works where he derived the term “algebra.” He also wrote a book on calculation in which he introduced the Hindu-Arabic numerals and how to perform arithmetic with them to Europe.
Some innovations in the field of mathematics are the following.
- Arabic numerals
- Negative numbers
- Decimal fractions
- Binary logic
- Non-Euclidean geometry
- Complex numbers
- Matrix algebra
These are some of the many innovations in the field of mathematics that have dramatically improved the efficiency, productivity, and outcome of people.