Transcendental number

A real or com­plex num­ber is said to be tran­scen­den­tal if it is not the root of any (nonzero) in­te­ger-co­effi­cient polyno­mial. (“Tran­scen­den­tal” means “not alge­braic”.)

Ex­am­ples and non-examples

Many of the most in­ter­est­ing num­bers are not tran­scen­den­tal.

  • Every in­te­ger is not tran­scen­den­tal (i.e. is alge­braic): the in­te­ger \(n\) is the root of the in­te­ger-co­effi­cient polyno­mial \(x-n\).

  • Every ra­tio­nal is alge­braic: the ra­tio­nal \(\frac{p}{q}\) is the root of the in­te­ger-co­effi­cient polyno­mial \(qx - p\).

  • \(\sqrt{2}\) is alge­braic: it is a root of \(x^2-2\).

  • \(i\) is alge­braic: it is a root of \(x^2+1\).

  • \(e^{i \pi/2}\) (or \(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\)) is alge­braic: it is a root of \(x^4+1\).

How­ever, \(\pi\) and \(e\) are both tran­scen­den­tal. (Both of these are difficult to prove.)

Proof that there is a tran­scen­den­tal number

There is a very sneaky proof that there is some tran­scen­den­tal real num­ber, though this proof doesn’t give us an ex­am­ple. In fact, the proof will tell us that “al­most all” real num­bers are tran­scen­den­tal. (The same proof can be used to demon­strate the ex­is­tence of ir­ra­tional num­bers.)

It is a fairly easy fact that the non-tran­scen­den­tal num­bers (that is, the alge­braic num­bers) form a countable sub­set of the real num­bers. In­deed, the Fun­da­men­tal The­o­rem of Alge­bra states that ev­ery polyno­mial of de­gree \(n\) has ex­actly \(n\) com­plex roots (if we count them with mul­ti­plic­ity, so that \(x^2+2x+1\) has the “two” roots \(x=-1\) and \(x=-1\)). There are only countably many in­te­ger-co­effi­cient polyno­mi­als spell out why, and each has only finitely many com­plex roots (and there­fore only finitely many—pos­si­bly \(0\)real roots), so there can only be countably many num­bers which are roots of any in­te­ger-co­effi­cient polyno­mial.

But there are un­countably many re­als (proof), so there must be some real (in­deed, un­countably many!) which is not alge­braic. That is, there are un­countably many tran­scen­den­tal num­bers.

Ex­plicit con­struc­tion of a tran­scen­den­tal number

Liou­ville’s constant

Parents:

  • Number

    An ab­stract ob­ject that ex­presses quan­tity or value of some sort.