Abstract

A complex number is a number that can be written in the form of a+bi where a and b are real numbers and i is the value of the square root of negative one. In the form a + bi, a is considered the real part and the bi is considered the imaginary part. The goal of this project is show how the use of complex numbers originates in the history of mathematics.

Introduction

Complex numbers are very important component of mathematics. They enable us to solve any polynomial equation of degree n. Simple equations like x3+1 would not have solutions if there were no complex numbers. The complex number has enriched other branches of mathematics such as calculus, linear algebra (matrices), trigonometry, and you can find its applications in applied sciences, and physics. In this project we will present the history of complex numbers and the long road to understanding the applications of this truly powerful number.

I) Ancient History

Russian Egyptologist V.S. Glenishchev traveled to Egypt in 1893 on routine business little did he know that what he would purchases would change how the world viewed Ancient Egyptian civilization, and shape mathematical landscape for years to come

Stolen from the valley of kings in 1878, at Deir el-Bahri, the Moscow Mathematical Papyrus which was sold to the Museum of fine arts in Moscow in 1912 by V.S. Glenishchev, and where it remained a mystery until its translation in 1930; is the oldest Egyptian papyrus remaining Based on the Carbon dating and orthography of the text, it was most likely written in the 13th dynasty and based on older material probably dating to the Twelfth dynasty of Egypt, roughly 1850 BC. Approximately 18 feet long and varying between 1½ and 3 inches wide, its format was divided into 25 problems with solutions by the Soviet Orientalist Vasily Vasilievich Struve in 1930.

In particular the 14th problem on the Moscow Mathematical Papyrus is a good example of how to find the volume of a truncated pyramid. And strongly suggests that the ancient Egyptians had knowledge of complex formula. V=13h(a2+ab+b2) (Notice: that if a and b are not specific numbers then when factoring we will procure a negative square)

A and b being real and positive numbers are empirical or observable, meaning that anybody can take a measurement of these quantities, but h on the other hand is not a empirical value and must be solved for. Which leads us to believe that Egyptian’s also had knowledge of this formula as well? h=c2-2(a-b2)2

The earliest derivation of this formula dates back to the 1st century A.D. by a well-known mathematician and engineer Heron of Alexandria (who was possibly Egyptian). This is where we find the earliest examples of the square root of a negative number. In Herons book titled Stereometria we see the example of finding the volume of a truncated pyramid or frustum: where the side of the lower base is 28, the upper 4 and the edge 15 instead of taking the square root of 81-144 which would have resulted in a negative square we see that Heron replaced the value with a positive property. This error cost Heron deeply in lost fame, it would be 1000 plus years until the subject of negative roots would appear again.

II) Cubics

In the late 15th century the two most sought after problems that mathematicians pondered many hours over where a) the quadrature of the circle and b) solving cubic equations. The latter being of much interest to us in our study of the history of complex numbers. It marks a very strange chapter in mathematical history and sad to say for the most part was a dead end. The following is an approximate historical account.

In the 15-16th century at the University of Bologna, the Italian mathematician Scipione Del Ferro (1465-1526) did, in fact discover how to solve a depressed cubic, a special case of general cubic in which the second-degree term is missing. Because his solution to the depressed cubic is central to the first progress made...