Complex numbers representing Real numbers

[DrummingAtom]

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:

$$\sqrt[3]{2+\sqrt{-121}} = a+b\sqrt{-1}$$

$$\sqrt[3]{2-\sqrt{-121}} = a-b\sqrt{-1}$$

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.

 

[DoctorBinary]

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]

(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.

 

[DoctorBinary]

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).

 

[CellCoree]

I would like to clarify the concept of complex numbers representing real numbers. Complex numbers are numbers that can be expressed in the form a+bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1. Real numbers are a subset of complex numbers, where b=0, meaning there is no imaginary component. In other words, real numbers are a special case of complex numbers where the imaginary part is equal to 0.

Now, let's look at the example provided in the content. The equation x^3 = 15x + 4 has three real solutions, as mentioned: 4, -2±sqrt(3). However, when using the Cardan formula, which is a method for solving cubic equations, we get imaginary solutions. This is because the Cardan formula involves taking the square root of negative numbers, which is where the imaginary unit comes into play.

In the book "An Imaginary Tale: The Story of Sqrt(-1)," the author explains how Bombelli used complex numbers to solve this equation and arrived at the real solutions of 4, -2±sqrt(3). This was done by manipulating the equation and separating it into real and imaginary parts. By setting these real and imaginary parts equal to known values, Bombelli was able to solve for the unknown variables and ultimately arrive at the real solutions.

So, while the Cardan formula may give us imaginary solutions, it is through the manipulation and understanding of complex numbers that we can arrive at the real solutions. Complex numbers are a powerful tool in mathematics and science, and they allow us to solve problems that would not be possible with just real numbers.