Irrational+Irrational=Rational (deeper)

[Monkfish]

Obviously one can create the "mirror" to an irrational number which itself is irrational and add these two irrationals to get a rational: e.g. pi and (4-pi) are both irrational and add together to make the rational number 4. However, I would call this a trivial solution because the second "mirror" irrational is generated from the first, which is cheating.

So my question is...

Are there any two irrational numbers that are entirely independent, that add together to make a rational? By "independent" I mean that their generation is not related.

This is a far deeper question as it is not always easy to determine if the irrational nature of one number is dependent upon the irrational nature of another. I suppose two such irrational numbers can only be said to be "independent" if their relationship is itself irrational, with all three irrationalities being distinct and unrelated.

For example, can the irrational number (4-pi) be generated in a way that is not related to pi?

If all mirrors to irrational numbers have no independent generators, then although in theory the addition of two irrational numbers can make a rational, in practice there are none where the irrationality of the second isn't spawned directly from the irrationality of the first.

 

[g_edgar]

If x+y=r, r rational, then y=r-x, so it is a "mirror" of x in your sense.

 

[Monkfish]

That's right, but the irrationality of y derives from the irrationality of x (or vice versa) so both x & y have the same irrationality. I'm asking whether two independent irrational numbers can ever add to make a rational.

Thinking about it, I'm sure there are no examples. Because for a completely independent irrational number to add to another and make a rational, every single digit in the second would have to match with every digit in the first to make the rational result. As irrational numbers extend to infinity, the chance of finding an independent match (mirror) to the first is zero. If there is a match (mirror), then its irrationality must be spawned by the irrationality of the first. In other words, it's the same irrationality... as in your example above.

 

[HallsofIvy]

Then YOU are going to have to define what YOU mean by "independent". The very fact that x+ y is equal to some specific number gives them a kind of "dependence". You question makes no sense without an a-priori definition of "independent".

 

[Monkfish]

Yes, I think you're right. I think I am talking nonsense. If two irrational numbers (or even their generating functions) add to form a rational, then by definition those numbers (or functions) are mirrors. So how did I get sucked down this road? I guess, I wanted someone to come up with an example where say the root of a prime added to pi squared produced a rational, or some such example of a pair of disparate irrationals that produce a rational that aren't just linked by the fact that x+y=r. But I think that even in this case, if x+y=r then f(x)+f(y)=f(r), so they can't be disparate functions. If they are disparate functions then we simply have the case where the two irrationals sum to form a third. So yes, you're right, I'm talking nonsense.

 

[Hurkyl]

For fun, let's apply the cubic formula to the polynomial

x³ + 6x -20​

Via the cubic formula, one of the roots is

$$\sqrt[3]{10 + \sqrt{108}} - \sqrt[3]{-10 + \sqrt{108}}$$​

more commonly known as

2​

 

[FaustoMorales]

An algebraic number is a number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients. Numbers such as pi that are not algebraic are said to be transcendental and ¨almost all¨ irrational numbers are transcendental.

Since pi is not a root of a polynomial with rational coefficients, there´s no way to express pi in closed form in terms of anything rational and/or irrational unrelated to pi. Anything reasonable you want to write on the RHS of the equation will have to contain at least one number from a field extension of the rational numbers containing pi.

Likewise, in order to express a rational number in terms of irrational numbers, it will be necessary that the ¨irrationality¨ involved cancels out as described in the previous paragraph.

 

[Monkfish]

Thanks FaustoMorales. I think that is what I was trying to grasp with my fuzzy brain. I will investigate in the direction you have pointed.

 

[HallsofIvy]

The problem with that is that it requires as precise definition of the "irrationality" of a number- and I doubt that it can be done.