Strange real numbers requiring use of complex numbers to exist

In summary, the conversation discusses the possibility of a real number that can only be expressed in exact form by including complex numbers. The participants also discuss the existence of cubic equations with real solutions that involve complex numbers in the intermediate calculations. The conversation concludes with the question of whether it is possible to express certain numbers in exact form without using transcendental functions or imaginary units.
  • #1
daniel.e2718
10
0
I couldn't really think of a good title for this question, lol.

Is it possible that a real number exists that can only be expressed in exact form when that form must includes complex numbers?

For example, the equation

[itex]2 \, x^{3} - 6 \, x^{2} + 2 = 0[/itex]

has the following roots

[itex]x_1 = -\frac{1}{2} \, {\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} {\left(i \, \sqrt{3} +
1\right)} - \frac{-i \, \sqrt{3} + 1}{2 \, {\left(\frac{1}{2} i \,
\sqrt{3} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}} + 1[/itex]

[itex]x_2 = -\frac{1}{2} \, {\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{2}
i \, \sqrt{3} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} - \frac{i
\, \sqrt{3} + 1}{2 \, {\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}} + 1[/itex]

[itex]x_3 = {\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} +
\frac{1}{{\left(\frac{1}{2} i \, \sqrt{3} +
\frac{1}{2}\right)}^{\frac{1}{3}}} + 1[/itex]

which have numerical approximations of

[itex]x_1 \approx 0.65270364[/itex]

[itex]x_2 \approx -0.53208889[/itex]

[itex]x_3 \approx 2.8793852[/itex]

When I run these through Sage's simplify_full() command (five times, even), they just become single fractions.

Is this a CAS simplification thing or are there real numbers that simply cannot exist without complex numbers?
 
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  • #2
But you just expressed those numbers in a form without complex numbers.

Anyway, I don't think such numbers exist, since any complex algebra can be replaced with geometrical operations in polar space, involving just real numbers.
 
  • #3
It's entirely possible for a polynomial with coefficients in the reals to have one or more complex roots, but this probably isn't what you're asking. The simple answer is that it's not necessary to reference the complex numbers in order to describe or obtain the reals; the complex numbers are what is known as an extension of the real number field, created for reasons that have to do largely with polynomials (at least, as far as mathematicians are concerned).
 
  • #4
The interesting question was the one at the start of the OP:
Is it possible that a real number exists that can only be expressed in exact form [by using] complex numbers?
That is, can there be an expression using rationals, surds and complex numbers that evaluates to a real but cannot be expressed using only real rationals and surds?
I suspect the answer is yes. You could take any such complex expression, whether or not it evaluates to a real, and add its complex conjugate (by changing i to -i everywhere) to produce one that does evaluate to a real.
I took the specific case here of (1+i√3) and wrote it as (a+b√3+ic+id√3)3. It looked like it might be possible to show that a, b, c and d cannot all be rational.
 
  • #5
Hello daniel.e2718 !

The roots written without any complex term :
x1 = 1-2cos(4pi/9)
x2 = 1-2cos(2pi/9)
x3 = 1-2cos(pi/9)
 
  • #7
Just to be clear, real numbers "exist" irrespective of any way of writing them. However, it is true that there exist cubic equations, having real solutions, such that if you use Cardano's cubic formula, which involves taking square roots and then cube roots, you can wind up with complex numbers in the intermediate calculations- the imaginary parts then eventually cancelling out. It was this discovery that led to complex numbers being accepted as "numbers".
(I almost wrote "as real numbers"!)
 
  • #8
JJacquelin said:
Hello daniel.e2718 !

The roots written without any complex term :
x1 = 1-2cos(4pi/9)
x2 = 1-2cos(2pi/9)
x3 = 1-2cos(pi/9)

Ahh this. I didn't think about that at all. Then again, de Moivre's formula...

But also, cosine is a transcendental function. None of the exact numbers in my original post had infinite series.

Is it possible to express the above numbers in an exact form without using transcendental functions or using an imaginary unit? Or transcendental numbers like pi.

This is interesting :eek:
 

1. What are complex numbers?

Complex numbers are numbers that have both a real part and an imaginary part. They are typically written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit (equal to the square root of -1).

2. Can you give an example of a strange real number that requires the use of complex numbers to exist?

One example is the complex number 1 + i, which cannot be expressed as a real number. It requires the use of the imaginary unit i to exist.

3. How are complex numbers used in scientific research?

Complex numbers are used in a variety of scientific fields, such as physics, engineering, and mathematics. They are particularly useful in modeling and analyzing systems with both real and imaginary components, such as electrical circuits and quantum mechanics.

4. Are there any practical applications for complex numbers?

Yes, complex numbers have many practical applications in fields such as signal processing, image processing, and control systems. They are also used in financial modeling, weather forecasting, and fluid dynamics.

5. How does the use of complex numbers impact our understanding of the world?

The use of complex numbers allows us to describe and analyze phenomena that cannot be fully represented by real numbers. This expands our understanding of the physical world and allows us to make more accurate predictions and solve complex problems.

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