College Algebra & Trigonometry, 2018a

1.4. COMPLEX NUMBERS 39
This hierarchy of numbers is often represented in the following diagram:
R
Q
Z
N
One of the best ways to conceptualize the Real number system is on the number
line - every point on the number line corresponds to a unique Real number and
every Real number corresponds to a unique position on the Real number line.
-4 -3 -2 -1 0 1 2 3 4
√
2
After the development of the printing press in the 15th century, Fibonacci’s Liber
Abaci was translated into Italian from Latin and read throughout Italy. As a result,
Italy became a thriving center of mathematics until the 17th century, when the
center of European mathematics moved north to France, Germany and England.
Throughout the 1500’s Italian mathematicians such as Girolamo Cardano, Raphael
Bombelli and Niccolo Fontana Tartaglia worked to extend the ideas in Fibonacci’s
book. They produced formulas to solve cubic (x 3 ), and quartic (x 4 ) degree equations.
In solving some of these equations they found that their formulas sometimes
produced negative values under a square root. None of the known number
systems could accomodate this possibility. In Cardano’s book on algebra Ars
Magna, he encounters a problem which involves the square root of a negative
number. He says, ”It is clear that this case is impossible. Nevertheless we will
work thus...” and he proceeds to compute a valid complex solution to the problem.
Mathematicians eventually defined the complex unit √ −1 = i and then