The first week of daily history facts is over and I have to say that I enjoyed it a lot. For those of you that do not know, every day in December I am writing a fact about the history of mathematics, these go live on my Facebook page and Tumblr. Also, once a week I will post them on WordPress. And here is the first week ^_^

1st December:

Negative numbers were long denied legitimacy in mathematics. We have no evidence of negative numbers being recognized in Babylonian, Pharaonic, ancient Greek, or any other ancient civilization. On the contrary, the Greeks considered geometry the only acceptable form of mathematics and since distance cannot be negative, they had no use for negative numbers.

In the 7th century, negative numbers were used in bookkeeping in India; positive quantities denoted assets, negative ones debts. The Hindu astronomer Brahmagupta, in a chapter dealing with mathematics in his work on astronomy from about A.D. 630, shows a clear understanding of negative numbers.

The earliest documented evidence of the use of negative numbers in European mathematics is the “Arts Magna”, published in 1545 by the Italian mathematician Girolamo Cardano. In the early 17th century, mathematicians began explicitly to use “negative numbers” but met with heavy opposition. Descartes called negative roots “false roots”, and Pascal was convinced that numbers “less than zero” could not exist. Leibniz admitted that negative numbers could lead to absurd conclusions and misconceptions, but defended them as useful aids in calculations.

The general acceptance and algebraic use of negative numbers came during the 18th century, although there were still mathematicians who did not feel at home with them and quite often tried to avoid using them.

2nd December:

For the second day, we are still on the numbers. But I wanted to say thank you for everyone that found some time to share some extremely interesting things with all of us. Thank you ^_^

The symbol “e” was first used by the Swiss mathematician Leonard Euler in accounts of his results, in letters written in 1727 or 1728 from St. Petersburg, and again in 1731. In print, “e” appeared in 1736 in his “Mechanica”, possibly inspired by the word “exponential”; today it is generally regarded as a homage to Euler.

In 1737, Euler showed that “e” is irrational.

3rd December:

π is the symbol used today to represent the ratio of the circumference of a circle to its diameter.

This designation was not introduced until 1706, when used by William Jones in his “Synopsis palmariorum matheseos”, probably after the initial letter of the Greek “periphery”. Until then, instead of π, one had to content oneself with the quaint Latin phrase: “quantitas, in quam cum multiplicetur diameter, provenient circumferentia”, meaning “the quantity which, when the diameter is multiplied by it, gives the circumference”.

It is due to the great prestige of the Swiss mathematician Leonhard Euler that we use π with today’s meaning. In his early writings, Euler had frequently used p to denote circumference-to-diameter ratio, but changed to π in his textbook “Mechanica”, published in 1736.

4th December:

Transcendental numbers are not the roots of any algebraic equation. The existence of transcendental numbers was proven by Joseph Liouville in 1844.

In 1873 Charles Hermite proved that “e” is transcendental.

The German mathematician Ferdinand von Lindemann, in 1882, succeed in proving that π is transcendental. The area of a circle is π*r^2 (r = radius), that of a square is s^2 (s = side); consequently, the side of a square whose ares is equal to that of a circle with radius 1 is square root of π. A construction with straightedge and compass alone can give only lengths that are algebraic numbers, so Lindemann’s proof that π is transcendental was conclusive evidence that the age-old problem of squaring the circle is insolvable.

The transcendentality of e^π was proved in 1929. Yet, even if we know today that the number of transcendental numbers is infinite, there are still many irrational numbers, e.g. π^π, that defy our curiosity whether they are algebraic or transcendental.

5th December:

The idea of representing complex numbers with points in a coordinate plane originated with the English mathematician John Wallis in his “De Algebra tractatus” (1685), but it was put forward in a rather vague manner and had no influence on contemporary mathematics.

The first practical representation of complex numbers dates from 1797 when the Norvegian cartographer Casper Wessel read a paper “On the Analytic Representation of Direction”, before the Danish Academy of Sciences, published the following year in the “Memoires” of the Academy. Wessel’s work went essentially unnoticed, however, until a French translation appeared a century late, in 1897.

In 18016, the Geneva-born Parisian bookkeeper Jean Robert Argand published, anonymously, in a small privately printed edition, a method of representing imaginary numbers geometrically. This work might have suffered the same fate as Wessel’s paper, except for J.F. Francais, a professor of military engineering, who found a copy of it and invited the author to come forward and acknowledge his work.

As often happens when “the time is ripe”, Wessel and Argand had published their accounts of essentially the same idea independently and simultaneously; neither Wessel nor Argand explicitly mentioned a complex number plane, a name we owe to the German mathematician Carl Friedrich Gauss.

Source: Mathematics: From the Birth of Numbers by Jan Gullberg

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