Mathematics is the study of structure, order, and relation based on the basic concepts of counting and measuring. We’ve come a long way from just using it to count and measure things, however, as math now forms the basis for many advanced fields of scientific study, from thermodynamics to rocket science to bionanotechnology. All of these advances have been made possible by groundbreaking discoveries and inventions by thinkers, scientists and mathematicians throughout history.

## 10. Cartesian Geometry

Also sometimes called analytical geometry, Cartesian geometry was pioneered by René Descartes in the 17th century. It was a groundbreaking moment in mathematics that, for the first time, brought algebra and geometry together.

Descartes first introduced the concept of Cartesian coordinates in his *Discours de la méthode*, or the ‘Discourse on Method’ published in 1637. He proposed that each point in a two-dimensional plane can be described by two numbers, which could then be used to represent its horizontal and vertical locations, respectively. This coordinate system provided a way to solve equations both algebraically and graphically, playing a key role in the subsequent development of calculus by Newton and Leibniz.

## 9. Binary System

With a base of ‘2’ and using only two digits, 0 and 1, the binary system is the most simple number system still in use today, though it’s perhaps the most influential. It’s used in computer memory, where calculations are only made up of two states – off and on – or bits. This basic system forms the basis for almost all digital communication today, making it one of the most important inventions in the history of mathematics and computers.

The concept was proposed and perfected by many scientists and mathematicians, primarily Gottfried Wilhelm Leibniz – a 17th-century German mathematician and philosopher who is also known for his co-invention of calculus. In 1701, Leibniz published his binary system in the paper *Essay d’une nouvelle science des nombres*, which ensured his election to the Paris Academy.

## 8. Difference And Analytical Engines

Charles Babbage is also known as the ‘father of the computer’ for his pioneering discoveries in mathematics, primarily the 19th century invention of computing devices called the Difference Engine and Analytical Engine. The Difference Engine was designed to calculate mathematical tables using finite differences, allowing basic arithmetic operations like addition, subtraction, multiplication, and division. It laid the foundation for modern computers and was the first automatic calculating engine in history.

The Analytical Engine, on the other hand, was a much more ambitious and versatile machine. Invented in 1834, it was a general-purpose programmable computing device that went beyond the limited capabilities of the Difference Engine, as it used punched cards for data manipulation and storage – a concept borrowed from the Jacquard loom. The Analytical Engine also introduced innovative features like conditional branching and looping, which allowed it to execute different instructions based on previous results and repeat specific series of operations.

## 7. Turing Machine

The Turing machine was a hypothetical device conceptualized by English mathematician Alan M. Turing in 1936. It’s an infinitely extendable tape divided into squares, each bearing a blank or finite number of symbols. With a tape head that can read, write, move, and modify squares, the Turing machine can – theoretically speaking – follow instructions stored in its control mechanism.

Although the Turing machine was a theoretical tool, it laid the groundwork for all modern digital computers. Turing’s abstraction of computation and logical processing into distinct steps formed the basis for all subsequent digital computational devices. The architecture of input-output devices – like the tape and reader – memory, and central processing units closely mirror the essential features of the Turing machine. It introduced the basic elements of information processing and opened the door for building programmable machines capable of complex calculations and logical operations.

## 6. Infinity

The concept of infinity has a long and complex history within both mathematics and philosophy. Greek philosophers like Aristotle first explored the idea in ancient times, but it was the Indian mathematician Surya Prajnapti who first categorized different types of infinities into enumerable, innumerable, and infinite sets. It wasn’t until the 17th century, however, that European mathematicians like John Wallis and Georg Cantor began incorporating it in their work, giving it the form modern mathematicians use till today.

John Wallis, an English mathematician, invented the symbol for infinity in 1655, followed by Georg Cantor – a German mathematician – who made major contributions to understanding infinity. He defined different levels of the concept, proving that not all infinities are the same size or type. Cantor would go on to introduce the concept of ‘infinity in infinities’ and develop the theory of infinite sets in calculus.

## 5. Newton’s Calculus

Isaac Newton is still regarded as one of the most important figures in the history of mathematics, largely due to his invention of calculus in the mid to late 1660s that revolutionized the field and paved the way for further advancements in science and engineering.

Calculus, made up of differential and integral calculus, allows for the study of the rate of change of quantities over time. Newton’s work on gravity and laws of motion in physics directly led to his development of this new mathematical framework. Today, calculus has applications across disciplines, including physics, chemistry, biology, engineering, and pure mathematics, among other specialized sub-fields. Fields of study like trend analysis, engineering, medical sciences, and motion-related studies heavily rely on calculus for precise calculation and prediction.

## 4. Pythagorean Theorem

The Pythagorean theorem is a fundamental geometric theorem that has had a major impact on mathematics and other scientific fields of study throughout history. It states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side.

While the discovery is commonly associated with the Greek mathematician-philosopher Pythagoras, some evidence suggests that the concept predates him, with Babylonian tablets from 1900-1600 BC featuring some understanding of the theorem and Pythagorean triples. The concept also appears in the Baudhayana Sulba-sutra of India – an important mathematical treatise written between 800 and 400 BC. Despite its earlier origins, Pythagoras is often credited with the discovery due to the influence of his school in Croton, Italy.

## 3. Zero

The invention of zero was another huge milestone of mathematics we don’t really think about, revolutionizing our understanding of numbers and leading to every major advancement in various fields that rely on math. It first showed up in the Bakhshali manuscript – an ancient Indian mathematical document dating back to the third or fourth century. Thinkers and mathematicians in previous civilizations did employ positional number systems before it, though the lack of a proper definition of zero led to difficulties in calculations and made the representation of fractions nearly impossible.

Since its discovery, zero has become a crucial placeholder digit in number systems, allowing for easier arithmetic and complex mathematical calculations. Zero facilitated the development of graphing techniques, calculus, and algorithms, among many other concepts fundamental to our modern understanding of mathematics, thanks to its dual role as both an abstraction and a tangible concept.

## 2. Algebra

The invention of algebra is often attributed to Abu Abdullah Muhammad ibn Musa al-Khwarizmi – a Persian mathematician born in the eighth century AD in the city of Khwarazm in modern-day Uzbekistan. He lived during a time when the Muslim empire, under the rule of the early Abbasid caliphs, was at its height during a period called the Golden Age of Islam, with Baghdad as its scientific and cultural capital.

He’d go on to make many important contributions to various mathematical fields, including arithmetic, trigonometry, and astronomy. Al-Khwarizmi’s most significant achievement, however, was his seminal work, *al-Kit?b al-Mukhta?ar f? ?is?b al-Jabr wal-Muq?balah*, or ‘The Compendious Book on Calculation by Completion and Balancing’, which is where the word ‘algebra’ comes from.

## 1. Euclidean Geometry

Euclidean geometry is named after Greek mathematician Euclid, living around 300 BC in Alexandria, Egypt. At its most basic, it’s the study of plane and solid figures based on axioms and theorems, introduced and detailed in his most famous work *Elements*. It largely consists of Euclid’s compilation of all the known mathematics of his time, along with some of his own theories and discoveries.

Because of the importance of his work, Euclid is still referred to as the ‘father of geometry’. Until the development of non-Euclidean geometry in the 19th century, *Elements *formed the standard for all mathematical and geometrical thinking around the world, covering a wide range of topics like plane geometry, number theory, and the study of solid figures.