Rafael Bombelli - Bombelli's Algebra

Bombelli's Algebra

In the book that he wrote in 1572, entitled Algebra, Bombelli gave a comprehensive account of the algebra known at the time. Bombelli wrote down the rules formulated by Brahmagupta regarding negative numbers. The following is an excerpt from the text:

"Plus times plus makes plus
Minus times minus makes plus
Plus times minus makes minus
Minus times plus makes minus
Plus 8 times plus 8 makes plus 64
Minus 5 times minus 6 makes plus 30
Minus 4 times plus 5 makes minus 20
Plus 5 times minus 4 makes minus 20"

These rules were discovered by Brahmagupta. (Reference Colebrooke, 1826). Bombelli in Algebra says that algebra is higher arithmetic invented in India. (Reference: Travelling Mathematics - The Fate of Diophantos' Arithmetic By Ad Meskens, Springer, page 143) As was intended, Bombelli used simple language as can be seen above so that anybody could understand it. But at the same time, he was thorough.

Perhaps more importantly than his work with algebra, however, the book also includes Bombelli's monumental contributions to complex number theory. Before he writes about complex numbers, he points out that they occur in solutions of equations of the form, given that, which is another way of stating that the discriminant of the cubic is negative. The solution of this kind of equation requires taking the cube root of some number and adding the square root of some negative number.

Before Bombelli delves into using imaginary numbers practically, he goes into a detailed explanation of the properties of complex numbers. Right away, he makes it clear that the rules of arithmetic for imaginary numbers are not the same as for real numbers. This was a big accomplishment, as even numerous subsequent mathematicians were extremely confused on the topic.

Bombelli avoided confusion by giving a special name to square roots of negative numbers, instead of just trying to deal with them as regular radicals as other mathematicians did. This made it clear that these numbers were neither positive nor negative. This kind of system avoids the confusion that Euler encountered. Bombelli called the imaginary number i “plus of minus” or “minus of minus” for -i.

Bombelli had the foresight to see that imaginary numbers were crucial and necessary to solving quartic and cubic equations. At the time, people cared about complex numbers only as tools to solve practical equations. As such, Bombelli was able to get solutions using Scipione del Ferro's rule, even in the irreducible case, where other mathematicians such as Cardano had given up.

In his book, Bombelli explains complex arithmetic as follows:

"Plus by plus of minus, makes plus of minus.
Minus by plus of minus, makes minus of minus.
Plus by minus of minus, makes minus of minus.
Minus by minus of minus, makes plus of minus.
Plus of minus by plus of minus, makes minus.
Plus of minus by minus of minus, makes plus.
Minus of minus by plus of minus, makes plus.
Minus of minus by minus of minus makes minus."

After dealing with the multiplication of real and imaginary numbers, Bombelli goes on to talk about the rules of addition and subtraction. He is careful to point out that real parts add to real parts, and imaginary parts add to imaginary parts.