Works Cited

[1] O'Connor, J., Robertson, E., Rafael Bombelli.  JOC/EFR, 2000.  http://www-history.mcs.st-and.ac.uk/Biographies/Bombelli.html

[2] Helden, A., Bombelli Rafael.  The Galileo Project, 1995. http://galileo.rice.edu/Catalog/NewFiles/bombelli.html

[3] http://www.math.wichita.edu/~richardson/timeline.html

[4]  Freeman, L., Rafael Bombelli.  Fermat's Last Theorem, 2006.  http://fermatslasttheorem.blogspot.com/2006/11/rafael-bombelli.html

[5] http://www.und.edu/instruct/lgeller/complex.html

[6] Rafael Bombelli, Wikimedia Foundation, Inc., 2014.  http://en.wikipedia.org/wiki/Rafael_Bombelli#Bombelli.27s_Algebra

Example of Bombelli's Algebra

Bombelli frequently used fractions to approximate square roots.  Here is an example:

Suppose we want to find the square root of 13.  The nearest square is 9, which has root 3.  The approximation of the square root of thirteen is 3 + an unknown.

3 + x = √13

Its square is 9 plus 6 unknowns plus 1 power.  We set this equal to 13

(3 + x)² = 9 + 6x + x² = 13

Subtract 9 from either side of the equation, and we are left with 4 equal to 6 plus 1 power

6x + x² = 4

If you ignore the power and set 6 unknowns equal to 4, the unknown (x) equals ²/₃

6x = 4 gives x = ²/₃

The approximate value of the root is 3 ²/₃ since it has been set equal to 3 plus 1 unknown.

√13 = 3 + x = 3 ²/₃

However, taking the power into account, if the unknown equals ²/₃ , the power will be  ²/₃ of an unknown which, added to the 6 unknowns, will give us 6 and  ²/₃ unknowns.  This equals 4.

6x + x² = 4 implies 6x + ²/₃ x = 4

The unknown will be equal to ³/₅  and since the approximate is 3 plus 1 unknown, it comes to 3 ³/_5.

x = 4/(6 + ²/₃) = implies 3 + x = 3 ³/₅

But if the unknown equals ³/₅ the power will be ³/₅ of an unknown and we obtain 6 ³/₅ unknowns equal to 4

6x + x² = 4 implies 6x + ³/₅ x = 4

Then the unknown comes out to be ²⁰/₃₃

6x + ³/₅ x = 4 implies x = 4/(6 + ³/₅) = ²⁰/₃₃

And this process can continue.

Bombelli's Imaginary Numbers

In Bombelli's book, Algebra (1572), he gave a complete account of the algebra known at the time.  He was the first European to write down how to perform computations with negative numbers, and the following is an excerpt from his 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" [6]

In our time, any student with little algebra education can comprehend these ideas.  However, in Bombelli's time little was known about negative numbers, so it was a huge breakthrough in mathematics.  As previously stated, Bombelli used simple language so that any person, with or without prior algebra history, can understand it, but he was thorough.

Also in his book, he includes his monumental contributions to complex numbers.  Before Bombelli dives into imaginary numbers, he makes it known to his reader that they do not hold the same arithmetic rules as real numbers.  This breakthrough was huge considering many mathematicians at the time could not grasp the topic of imaginary numbers.  To avoid confusion, Bombelli gave a special name to the square roots of negative numbers.  This made it clear that these numbers were neither positive nor negative.  This imaginary number, i, "plus of minus" or "minus of minus" for -i.

Bombelli used his knowledge to foresee that imaginary numbers were essential to solving quartic and cubic equations.  Bombelli was able to get solutions using Scipione Del Ferro's rule, where other mathematicians, like Cardan, had given up.  In his book, Bombelli explains this 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." [6]

After dealing with the multiplication of real and imaginary numbers, Bombelli talks about the rules of addition and subtraction.  He carefully adds that real add to real parts and imaginary parts add to imaginary parts [6].

References:
[6] Rafael Bombelli, Wikimedia Foundation, Inc., 2014.  http://en.wikipedia.org/wiki/Rafael_Bombelli#Bombelli.27s_Algebra

Bombelli's Formulation of Complex Numbers Using Cardan's Formula

Bombelli's work was mostly inspired by Cardan's formula, which gave solutions to the cubic: x³ = ax + b and was given as:

[5]

 

When this was used to solve the classic example x³ = 15x + 4, the formula yields:

 

[5]

Cardan claimed the general formula couldn't apply to this case because of the square root of -121, more questions needed to be answered.  This cubic has real solutions,

[5]

 

Now, here's where Bombelli comes in.  He was challenged with the task of solving this problem almost thirty years after Cardan published his work.  Bombelli justified Cardan's formula by the introduction of complex numbers.  He basically assumed that numbers of the form

[5]

 

existed and applied the normal operations rules of algebra.  In this case, Bombelli thought that since the radicands:

[5]

differed only in sign, the same could be true of their cube roots.  He set:

[5]

and solved for "a" and "b".  He found the answers a = 2 and b = 1, and this showed that:

[5]

 

The method gave him a correct answer to the equation, and it convinced him that his initial ideas about complex numbers were valid.  From here on out, Bombelli laid the groundwork for complex numbers.  He developed some rules for complex numbers, and he also worked with examples involving addition and multiplication of complex numbers [5].

References:
[5] http://www.und.edu/instruct/lgeller/complex.html