# 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.