Complex numbers representing Real numbers

In summary, the author explains how Bombelli used the Cardan formula to find the real solutions for the equation x^3 = 15x + 4, which are 4 and -2±sqrt(3). By manipulating the equation and splitting it into real and imaginary parts, Bombelli was able to show that the solution of 4 is correct. However, the process may seem complex and confusing, but it ultimately leads back to the simple solution of 4.
  • #1
DrummingAtom
659
2
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:

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

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

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

[tex] 2+\sqrt{-121} = a(a^2-3b^2)+b(3a^2-b^2)\sqrt{-1}[/tex]

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

[tex] a(a^2-3b^2) = 2 [/tex]

[tex]b(3a^2-b^2)\sqrt{-1}=11[/tex]

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.
 
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  • #2
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.)
 
  • #3
DoctorBinary said:
(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.
 
  • #4
Mark44 said:
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).
 
  • #5


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.
 

What are complex numbers representing real numbers?

Complex numbers are numbers that have both a real part and an imaginary part. The imaginary part is a multiple of the imaginary unit, i, which is defined as the square root of -1. Complex numbers representing real numbers are those that have an imaginary part equal to 0, making them purely real.

How are complex numbers represented on the complex plane?

Complex numbers can be represented on the complex plane, also known as the Argand diagram. The x-axis represents the real part of the complex number and the y-axis represents the imaginary part. The point where the two axes intersect is known as the origin (0,0).

What is the difference between real and complex numbers?

The main difference between real and complex numbers is that real numbers do not have an imaginary part, while complex numbers have both a real and imaginary part. Real numbers can be represented as points on a number line, while complex numbers are represented on a two-dimensional plane.

Why are complex numbers useful in mathematics?

Complex numbers are useful in mathematics because they allow us to solve equations that cannot be solved with real numbers alone. They are also useful in representing many physical phenomena, such as electrical circuits and quantum mechanics.

What is the conjugate of a complex number representing a real number?

The conjugate of a complex number is the complex number with the same real part but the opposite sign of the imaginary part. For example, the conjugate of 3+4i is 3-4i. Conjugates are useful in simplifying complex expressions and solving equations.

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