Evolution of the Concept and Usage of Negative Numbers
Mathematics is a fascinating and enthralling discipline in the current world. It forms the human branch of knowledge concerned with numbers and has a historical-epistemological foundation that elucidates the evolution and its applicability. In the contemporary world, the negative numbers present little complications as exhaustive research done by different individuals at different times facilitated the concept of universalization. The set of natural numbers today provides the basis on which the negative numbers can be constructed with ease. The paper focuses on the evolution of the concept of negative numbers and their application.
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Some mathematicians deliberated on the likelihood of the existence of the negative numbers. The outstanding individuals who are appreciated for the evolution and the usage of the conception include Brahmagupta, Francis Maseres, William Friend, and De Morgan Peacock among others. The rules that defined the use of numbers might trace its origin back to the 7th century. It is supposed that Brahmagupta was responsible for that definition and in 1758 Francis Maseres made a remark concerning negative numbers. Other mathematicians during that time believed that the application of the negative numbers in mathematics could be possible but only if the negative digits were eliminated from the calculations. Proper investigation of the concept of negative numbers started in the 19th century when mathematician De Morgan Peacock and others focused on the laws that govern arithmetic.
Negative numbers started to make sense in 200 BCE when the Chinese worked on the concept. They had a special representation of the positive and the negative numbers primarily using symbols. The Chinese applied their concept of the rod system to solve problems that had elements of negative numbers in them (Shell-Gellasch & Thoo, 2015). Most fields of application of this concept dealt with money. Although the Chinese appreciated the existence of negative numbers and used them in various ways, they did not completely acknowledge negativity as a real value. They would reject any negative outcome in their commercial or tax computations. The Chinese developed the theory of rule of sign around the 1300s. Astronomy tended to appreciate the existence of the negative numbers due to its ideas of "weak" and "strong" to indicate the relative position of a number as below or above. Astronomers assigned positive numbers as strong and negative as weak.
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Italy began the study of equations around the 16th century. Mathematician Cardano acknowledged the existence of negative roots. He was given the credit of starting the negative rules in numbers. Around 620 CE, Brahmagupta, an Indian Mathematician, produced his work where the use of debts and fortunes represented negative and positive. This was the time when the system of place values was introduced in India. Their number system included zero as well. Brahmagupta suggested a special sign for the negative value and developed various rules that defined negative numbers. The rules included simple arithmetic that involved additions, subtractions, multiplications, and divisions where he referred to some of them as debts and fortunes.
During 300 CE, Diophantus presented his work where he came up with symbols that represented what was termed “unknown” in a given problem. He is credited with the modern quadratic and linear equations. Diophantus called the negative outcome of a problem absurd. Khwarizmi lived around 850 CE in Baghdad. He came up with six standard forms for quadratic and linear equations and suggested and solved the equations using geometry and algebra concepts (Corry, 2015). His work borrowed heavily from the Indian Brahmagupta. He also represented the negative numbers as debts though he acknowledged the fact that negative numbers had no meaning. Around 998 CE, Abul-–Wafa used the negative numbers to denote debts. He argued that if seven were subtracted from 5, a debt of 2 would result and if to multiply the result by a factor of 10, it would give a debt of 20.
The use of negative numbers seemed to revolve around fortunes and debts where they were considered negatives. Negative numbers were used mostly in areas that involved currency or money as these were the possible spheres where debts could occur. The work of the initial mathematicians underpinned the current laws of mathematics such as solutions to quadratic and linear equations.
Leading Mathematicians of Different Centuries
In the history of mathematics, there are certain individual mathematicians who stood out during their entire period of existence due to their exceptional discoveries. The following accounts present the greatest mathematicians of their time.
Eratosthenes may be considered the greatest pure mathematician during the third century BCE. He produced a logical and reliable methodology that aided in the discovery of prime numbers (Atkinson et al., 2013). His work formed the basis of this concept in the contemporary world. Eratosthenes was the first mathematician to design the system of latitude and longitude and obtained the earth's circumference with a certain degree of accuracy. He also calculated the earth's axis tilt since he knew that the earth completed a full rotation once a day. However, his greatest legacy as a mathematician is “Sieve of Eratosthenes" which led to the discovery of prime numbers.
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The Persian scientist called Al-Ḵwārizmī might be the leading mathematician in the 9th century BCE. He is responsible for various crucial mathematics books that spread Indian mathematics to the Western world. His first mathematical works were written in Hindu-Arabic. The current algorithm studied in mathematics was Al-Ḵwārizmī work and the title was a Latinization of his name. Modern algebra as a result of his work and its name originated from the title of his publications: Al-ğabr wa’l-muqābala. The mathematical work Al-Ḵwārizmī presented during that time underpinned the current mathematics concepts. He spread Indian mathematics to the Western countries that formed their mathematics basis, particularly algebra and algorithms.
In the 16th century, the Italian Luca Pacioli can be called one of the leading mathematicians. Among the great mathematics contributions Pacioli made, there are the geometry and arithmetic puzzles that became so popular. He also introduced the symbols for minus and plus and publicized them in a printed book for the first time in history. These symbols were later standardized and formed the basis for notations. However, people have associated the plus and minus signs with the works of other mathematicians such as Johannes Widmann. Pacioli is also credited for his investigation of the Golden Ratio in the book The Divine Proportion that was published in 1509. His conclusion regarding the ratio was God’s secret knowledge about the internal beauty of things.
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John Napier could be the leading mathematician of the 17th century having discovered what formed the foundation for the subsequent work by great researchers and scientists. He invented a logarithm (Hobson, 2012). Napier particularly revolutionized mathematics, astronomy, and science by simplifying complex mathematical calculations. His invention marked one of the critical mathematical advancements during the 17th century. His work received wide appreciations among physicists such as Sir Isaac Newton and Johannes Kepler. They would not have carried out their innovative' complex calculations if Napier's logarithm did not exist. Two centuries later, Pierre-Simon Laplace appreciated Napier's work regarding the significant contributions the invention made to astronomy. In the contemporary world, John Napier's invention is a mathematical concept applicable to many spheres of human knowledge, regarding the use of the power of 10 or whichever base used.
Joseph Louise Lagrange was among the greatest mathematicians in the 18th century BCE. His work on the calculus of variation was instrumental to the development of mathematics. His contribution to the number theory and differential equations was also significant. Lagrange created various theories as well, including a group theorem which later defined mathematics in the 19th/20th century. In this theorem, he was able to demonstrate that the sub-group elements of a set can have an even division in the finite set original number of items. Lagrange developed the mean value theorem which is currently very useful to the mathematicians who are interested in numerical integration. He came up with the theorem of four-square where he showed that natural numbers can be written as the cumulative addition of four squares, for instance, 7=22+12+12+12, 3=12+12+12+02 (Hosch, 2010). Lagrange’s contribution to mathematics is significant and helped to define some of the crucial concepts today.
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During the 19th century, there was an overall increase in the complexity and extensiveness of the concepts that defined mathematics. Joseph Fourier may be among the leading mathematicians of the century. He is credited with the advancement of the trigonometry that developed the mathematical analysis. The sums of cosines and sines that define the infinite series of the periodic functions are Fourier's work and have been called after his name. These functions are useful in the contemporary world’s applied and pure mathematics. Fourier also contributed to the definition of a function. His invention in the 19th century, particularly the trigonometric functions, marked a breakthrough and were to be refined to form the basis of the current trigonometric ratios which are instrumental in many fields.
After the 1789 French revolution, higher education underwent a major transformation. There was a particular focus on engineering. The École Polytechnique was formed and had the inputs of some of the greatest mathematicians and scientists as the Polytechnique professors who included Lagrange and Laplace. Students registered for studies out of talent as opposed to birth. The events led to the emergence of new mathematicians, scientists, and engineers since mathematics was taught at the college besides thorough research in the mathematics field.
The period from 1768 to 1830 saw Joseph Fourier advanced mathematics through his analysis. He developed knowledge about heat diffusion. Fourier employed the differential equations and infinite series to further his analysis. By the 1920s, his work was the center of attraction both for those who were interested in finding the correctness of his work and for heat theory. One of the college’s graduates, Augustin-Louis Cauchy, handled a different approach to calculus and also series and functions (Corry, 2015). His method was an advancement of the prior works done by Newton. It was known as the calculus' fourth approach. This method was later referred to as the initial three approaches. Jacobi then advanced Cauchy's mathematics ideas and came up with the complex variable theory with integral. This Jacobi theorem developed to be regarded as a significant mathematics branch, especially by Germans.
From 1810 to 1830, the French introduced other branches of mathematics that included mathematical physics. The major contributors who revolutionized mathematics during the French revolution included Sim?on Poisson, who lived in 1781 to 1840 and made major contributions to electrostatics and magnetism. André Ampère was another mathematician who advanced electro-dynamism. His contributions were developed during the period from 1775 to 1836. The wave theory was initiated by Augustin Fresnel, who lived around the 1800s. Their inventions and contributions to the knowledge-centered on mathematics. The works were based heavily on mathematics; it played a central role in the advancement of other sciences. Cauchy, the ex-student of French-based college, later developed the continuum mechanics.
After the revolution, geometry was widely advanced since the French colleges majored and taught it. The descriptive geometry was introduced by Gaspard Monge and was established as one of the acceptable subdivision of mathematics. The École Polytechnique in France adopted the theory and spread it further through its curriculum. It was seen to be a significant contribution to engineering particularly drawing. Later, Victor Poncelet furthered knowledge through intensive research around the 1860s.
As the mathematicians worked and advanced theories in France, other scientists from other countries were inspired and continued with the advancements. They included G?ttingen University director C. F. Gauss, who performed excellent work in the theory of number, the theory of probability, and celestial mechanics. George Green was another mathematician outside France, who was inspired and presented work on mathematical analysis. His contribution was a manifestation of the revolution of mathematics after the French revolution.
The work that was performed in France was significant and led to a series of research and invention outside France. The graduates of the first college also became great as they spread the mathematical knowledge to other parts of Europe and other regions. It can be argued that the French revolution was responsible for the nurturing of the new great mathematicians such as Cauchy who was a student of Laplace and Lagrange at École Polytechnique. The influence of the French started to create a perceptible imprint in the US immediately after the American War of Independence. The academies founded in the 1780s followed the French methods and the Royal Academy of Teachings rather than that of England. The French Revolution played an important role in the advancement of mathematics in America.
The Most Interesting Moments in the History of Mathematics from My Perspective
This course has had a lot of teachings regarding the discoveries and inventions of various branches of mathematics. It is great to realize how inventions in mathematics were done by different individuals who lived in different regions of the globe, yet, keen scrutiny of their proposals and works seem to have a convergence that may be taken to be a confirmation that whatever they expressed had truths. The French Revolution shaped the field of mathematics in the world. Napoleon was concerned with the practical helpfulness of mathematics. His military determinations boosted mathematics in France. This was exemplified by Lagrange, Legendre, and Laplace. Gause used curved space to researched non-Euclidean geometry but did not finish his work and publish it due to the fear of controversy. His completion of the ideas would have seen the advancement of the avant-garde. His failure was an opportunity for Bolyai and Lobachevsky from Hungary and Russia respectively. The two worked autonomously and came up with perfect ideas concerning curved spaces and geometrical hyperbola.
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The British mathematicians were successful in coming with new inventions having seen renaissance during the mid-19th century. The English Charles Babbage made a device that was capable of performing calculations following the stored commands. The computations on logarithms discovered by John Napier could now be performed electronically by the Babbage's device. The machine also computed trigonometric functions. This device underpinned the invention of the modern-day computer. It can be seen how the work of various mathematicians was interrelated and could be applied in various ways. George Boole was another British mathematician who has impacted positively on the lives in the 21st century. He came up with a theory of operators consisting of AND, NOT, OR referred today as the Boolean logic. His discovery was widely applied in finding the solution for functions and logical problems. Boole’s work further led to the discovery of TRUE and FALSE that had an interpretation of on and off, electronically represented in zeros and ones. This knowledge formed the basis of the current mathematical logic and computer science advancement. Boole’s theory has guided most of the modern-day applications in electronics. The operations of today’s electronic devices are founded on the principle of on and off. Although Babbage and Boole advanced their research and studies separately and independently, the two individuals have one thing in common; they have contributed to the modern-day simplicity of life due to the presence of machines that include computers and calculators. They founded the functionalities of the gadget currently being used though they never lived to see the potential of their work.
In conclusion, the early discoveries and inventions in mathematics have contributed to the way the world is functioning today. Almost every application in the current world traces its origin to some early years’ inventions. Every century was marked by unique inventions and discoveries, and the mathematicians seemed to have a convergence in their work though they worked independently. From the 19th century, the discoveries made exhibited special qualities of being extensive and complicated. Later mathematicians dealt with more complex inventions. One of the events that shaped the existence of mathematics was the French revolution. The established École Polytechnique in France brought together some of the world's great mathematicians such as Laplace and Lagrange in a formal institution where they took part in teaching the course. Their contribution to the institution led to a new breed of mathematicians and scientists. The graduates contributed heavily to the field of mathematics, too; Cauchy was one of these graduates.