Answer to Question #342076 in Algebra for Keotshepile

Question

1.2.Explain how and who came up with these algebraic concepts or used these algebraic concepts deloped over time.

1.2.1 decimal number system

1.2.3 abstract symbol s

1.2.3 negative numbers

1.3 Describe the stages in the development of symbolic algebra. Give at least one example of equation for each of these stages.

Answer 1

1.2.1 Decimals as they look today were used by John Napier, a Scottish mathematician who developed the use of logarithms for carrying out calculations. The modern decimal point became the standard in England in 1619.

1.2.2 Humans convert virtually all direct and indirect impressions into symbols. The symbols that humans create are words. Every word consists of two components, a denotative and a connotative

1.2.3 The English mathematician, John Wallis (1616 - 1703) is credited with giving some meaning to negative numbers by inventing the number line, and in the early 18th century a controversy ensued between Leibniz, Johan Bernoulli, Euler and d'Alembert about whether log (-x) was the same as Log(x)

1.3

Symbolic algebra, in which full symbolism is used. Early steps toward this can be seen in the work of several Islamic mathematicians such as Ibn al-Banna (13th-14th centuries) and al-Qalasadi (15th century), although fully symbolic algebra was developed by François Viète (16th century). Later, René Descartes (17th century) introduced the modern notation and showed that the problems occurring in geometry can be expressed and solved in terms of algebra.

Equally important as the use or lack of symbolism in algebra was the degree of the equations that were addressed. Quadratic equations played an important role in early algebra; and throughout most of history, until the early modern period, all quadratic equations were classified as belonging to one of three categories.

x²+px=q

x²=px+q

x²+q=px

Where p and q are positive. This trichotomy comes about because quadratic equations of the form x²+px+q=0, whis p and q positive, have no positive roots.

In between the rhetorical and syncopated stages of symbolic algebra, a geometric constructive algebra was developed by classical Greek and Vedic Indian mathematicians in which algebraic equations were solved through geometry. For instance, an equation of the form x²=A}x was solved by finding the side of a square of area A.