Math is an essentially universal element used since prehistoric times by our ancient ancestors. Most believe prehistoric people were able to use math to differentiate between a certain number of like objects. Ancient tribes didn’t need a number system as they only really farmed for however much was there. They understood tallying and keeping track of certain things within the world such as moon phases and the changing of the weather. The concept of symbols, numbers, and naming for these numbers didn’t appear till much later in history. People understood the concept of geometric shapes, although not particularly mathematical yet, and used that to create bags and baskets to carry objects.

Ancient Mesopotamia, home of Babylonia and Sumer, was the originator of many concepts and ideas we use today such as the wheel and writing. The Sumerians earliest concept of writing used a clay object with symbols inscribed on it. Some historians believe the Mesopotamians taught basic mathematical and geometric concepts. The Sumerian government needed mathematics to measure land, keep an accurate count of tax on people, and keep track of money in general. Sumerians and Babylonians attempted to create a lunar calendar and chart the sky using the concepts they understood.

They created the concept of associating symbols with a certain amount of numbers, an example of this is using a small clay ball to convey an amount of 10. They used this system to keep stock on all their agriculture, jars, oil, etc. Later they start using symbols, similar to their system of writing, to replace the objects being used to convey a certain amount. The Sumerians and Babylonians where the first to create a numeric system similar to the Roman numerical system. There system started with a base of 60 and used symbols to represent 10 units and 1 unit almost exactly like the Roman numeral system. This started the use of the number 60 through much of our history, one example being our concept of time where one minute is 60 seconds and 1 hour is 60 minutes.

Babylonians also created a symbol for 0 similar to circle and used it as placeholder. There have been tablets discovered that display basic algebra, fractions, square root, and quadratic equations. The earliest estimate of pi also derives from ancient Mesopotamia. Quadratic equations where solved by splitting up shapes and rearranging them until a final answer was reached. There geometry equated to simple shapes to design buildings and games such as dice. Babylonians used geometry to calculate the area and volume of simple shapes. Some also believe the Babylonians created a Pythagorean triangle centuries before Pythagoras.

In 6000BCE Egyptians settled the Nile and started recording moon phases and seasons. They used measurements for land and buildings very early in their history and developed a numerical value by counting on 10 fingers. Near 2700BCE Egyptians created a base 10 numeration system that used symbols to represent a number. A stroke was used to represent 1 unit, a heel bone for 10 units, a coil of rope represented 100, a lotus plant to represent 1000, etc. The Rhind Papyrus showed the Egyptians had knowledge of fractions, prime numbers, and algebraic equations. They multiplied by repeatedly doubling the number. Fractions derived from the eye of Horus being split into fractions and being halved each time. They used fractions to divide and distribute food to people in equal amounts. There is also evidence that Egyptians used the formula for the volume of a pyramid to create exact stonework for their great pyramids.

The Greeks are responsible for some of the most important mathematical revolutions and innovations. They took Babylonian and Egyptian ideas and added on to them. In 450BCE the Greeks created a numeral system with symbols to represent 1, 5, 10, 50, 100, 500, 1000, and so on for other values. The Greek mathematicians primarily focused on geometry and the understanding of different shapes. Thales is one of these mathematicians, he created many theorems and set the basic groundwork for geometry as we know it today. Pythagoras was another great philosopher who discovered many important mathematical innovations such as the Pythagorean theorem. Hippocrates created a book called the “Elements” that described geometrical principles and elements. The Greeks also understood the concept of infinity, and Zeno attributed to this in his many paradoxes. Democritus is a very important pioneer of mathematics and geometry, creating many works and geometrical innovation. Plato and his apprentice Aristotle defined logic for thousands of years, creating an academy that birthed many great philosophers. Euxodes, a student of Plato, was the first to think of the theory of exhaustion and general theory of proportion.

In 3rd Century BCE the city of Alexandria gave rise to many great mathematicians. Euclid is one of the most influential teachers in our history, and basically created the modern geometry as we know it today. Archimedes is considered by some the greatest mathematicians in history. He created the idea of longitude and latitude, calculated the degree of the Earth, and created an algorithm for calculating prime numbers. Heron defined the idea of root and created a formula for finding the area of triangle with its side lengths. During the 3rd Century Diophantus was a pioneer in algebra, recognizing fractions as number and creating many equations.

The Romans only mathematical innovation was the creation of Roman numerals. They were largely based on Babylonian and Greek ideas. The Mayans made great advancements into calendars and astronomy. They created a number system, produced a concept for 0, and in 36BCE calculated the length of a solar year to be about 365 days. The Chinese used bamboo rods to represent numerical values, that eventually became the first decimal point system. Liu Hui made an algorithm to calculate the volume of π to around 3.14159. In the 13th Century Qin Jiu Shao solved quadratic and cubic equations.

The Islamic empire in the 8th to 13th century took Greek and Indian ideas and fused them together. Because Muslims intricately designed their buildings with various shapes they had a deep understanding of geometry. The Quran, the Islamic holy book, inspired science and mathematics in the House of Wisdom in Baghdad. Al-Khwarizmi contributed the 1-9 numerical system that most of the world uses today. He also contributed to algebra by figuring out the methods of reduction and balancing and basically the base for the math for we use today. Muhammed Al-Karaji introduced the theory of algebraic calculus and he used mathematical induction to solve the binomial theorem. Omar Khayyam discovered that there where multiple cubic equations and solved them. Nasir Al-Din Al-Tusi used astronomy to separate trigonometry as its own mathematical field. He used sine from India to advance spherical trigonometry.

In 16th century Europe science and art thrived from many great figures like Leonardo Da Vinci. Math soon followed when Luca Pacioli published a book that which depicted the plus and minus signs for the first time. He also researched the golden ratio,1:1.618…, and stated that it was “The Divine Proportion” sent by god to unlock the secrets of this world. Simon Stevin was one of the first to use decimal point notation and recognized that all numbers are equal. At Bologna University Niccolò Fontana Tartaglia, created a formula to solve all cubic equations. Tartaglia, Gerolamo Cardano, and Lodovico Ferrari where the first to use imaginary numbers and set the basis for a mathematical revolution in the 17th century.

The 17th was a revolutionary age for science and mathematics alike. The logarithm, invented by John Napier, helped advance science and math by making difficult equations easier to calculate. A logarithm is basically exponents where 10 squared is 100, this made many problems much simpler. Logarithm’s helped Kepler and Newton create their significant innovations. Napier also used the decimal point and improved decimal notation. Marin Mersenne is responsible for Mersenne primes, prime numbers that are less than a power of 2. Rene Descartes developed analytical geometry to allow the orbit of the planet to be plotted and developed the basis for calculus and multi-dimensional geometry. Probability theory was developed by Pierre de Fermat and Blaise Pascal. Sir Isaac Newton defines all of modern physics and mathematics and is one of the greatest mathematicians of all time. Gottfried Leibniz’s publication on his findings of calculus notation, although Issa Newton discovered it first, contributed to multiple fields of science including medicine.

The 18th century was deeply influenced by Newton and Leibniz who used their ideas to advance technology as we know it. The Bernoulli’s were a family that spawned many mathematicians, more specifically Jacob and Johann. They together are credited with infinitesimal calculus otherwise known as calculus of variations. Leonard Euler was the greatest 18th century mathematician who introduced many theorems and methods. The 19th century introduced more elaborate and complex mathematical ideas. Frenchman Evariste Galois introduced development in abstract algebra, algebraic geometry, group theory, vector spaces, and many more including a method for solving polynomial equations.

Carl Friedrich Gauss was considered one of the 3 greatest mathematicians, alongside Newton and Archimedes. Gauss was ahead of his time touching on basically all fields of mathematics including number theory, calculus, and probability. Bernard Riemann thought about geometry on a higher dimension the just 2D and 3D, he explored zeta functions and multi-dimensional complex numbers. In the mid-19th century George Boole devised an algebraic idea that the only operators were AND, OR, and NOT. Boole made the binary system of 1 and 0 which led to the development of computer science. Georg Cantor devised set theory, which enabled the notion of infinity. In the late 19th Century Henri Poicare created the “three body problem” which eventually led to Chaos theory.

The 20th century continued to develop more complex and intricate mathematical problems. G.H Hardy and his apprentice Srinivasa Ramanujan solved problems of the previous century like the Riemann hypothesis. David Hilbert introduced the Hilbert problems, a set of 23 problems, some of which haven’t been solved today, to introduce to future generations. Austrian mathematician Kurt Gobel stated that there are mathematical problems which are true but cannot be solved, this is called incompleteness theory. Alan Turing elaborated on incompleteness theory by stating that there was no way to tell if the problems were unsolvable or solvable. Andre Weil created theorems that connected topology, number theory, geometry, and algebra.

From the Prehistoric age to the 20th century, mathematics has advanced and grown into what we know it today. Throughout all of these time periods math kept advancing into something greater than the last century. Almost every country around the world contributed to mathematics in some form. But within these countries the great mathematicians developed math into what we know today.