# Complex numbers representing Real numbers

## Main Question or Discussion Point

I got this out of An Imaginary Tale: The Story of Sqrt(-1). In section 1.5 of the book, the author explains that Bombelli took x3 = 15x + 4 and found the real solutions: 4, -2±sqrt(3). But if you plug the equation into the Cardan forumla you get imaginaries. http://en.wikipedia.org/wiki/Cardan_formula#Cardano.27s_method

The author shows that if a and b are some yet to be determined real numbers where:

$$\sqrt{2+\sqrt{-121}} = a+b\sqrt{-1}$$

$$\sqrt{2-\sqrt{-121}} = a-b\sqrt{-1}$$

Then he takes the first equation and cubes both sides, does a bunch of Algebra and gets:

$$2+\sqrt{-121} = a(a^2-3b^2)+b(3a^2-b^2)\sqrt{-1}$$

And says if this is equal to the complex number, $$2+\sqrt{-121}$$ then the real and imaginary parts must be separately equal. Then he splits terms into:

$$a(a^2-3b^2) = 2$$

$$b(3a^2-b^2)\sqrt{-1}=11$$

To find that a = 2 and b = 1, then says "With these results Bombelli showed that the Cardan solution is 4 and this is correct."

The very last part is where I don't understand how all that complex stuff arrives back at 4. Even though, through simple Algebra with the very first equation with have real solutions.. Any help would be awesome. Thanks.

Isn't he just adding the two terms? (2 + sqrt(-1)) + (2 - sqrt(-1)) = 4. (Sorry, the tex formatting was acting weird in preview mode so I ditched it.)

Mark44
Mentor
(Sorry, the tex formatting was acting weird in preview mode so I ditched it.)
Known problem on this site. If you refresh your browser, the LaTeX will show up correctly. The problem seems to occur when there is already some LaTeX script in the browser's cache it will display what's in the cache, rather than what you are trying to preview.

Known problem on this site. If you refresh your browser, the LaTeX will show up correctly. The problem seems to occur when there is already some LaTeX script in the browser's cache it will display what's in the cache, rather than what you are trying to preview.
Thanks (I thought I was going crazy).