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Are-there-more-Trancendental-or-Non-Trancendental-numbers-NOTE-Trancendentals-are-numbers-that-cannot-be-written-algerbraically-e-g-x-2-2-2-is-non-trancendental-pi-is-transendental-




Question Number 3994 by Filup last updated on 26/Dec/15
Are there more  Trancendental  or Non-Trancendental numbers?    NOTE:  Trancendentals are numbers that  cannot be written algerbraically.    e.g. x^2 =2  ∴(√(2 )) is non-trancendental    π is transendental
$$\mathcal{A}\mathrm{re}\:\mathrm{there}\:\mathrm{more}\:\:\mathcal{T}{rancendental} \\ $$$${or}\:\mathcal{N}{on}-\mathcal{T}{rancendental}\:{numbers}? \\ $$$$ \\ $$$$\mathcal{NOTE}: \\ $$$$\mathcal{T}{rancendentals}\:{are}\:{numbers}\:{that} \\ $$$${cannot}\:{be}\:{written}\:{algerbraically}. \\ $$$$ \\ $$$${e}.{g}.\:{x}^{\mathrm{2}} =\mathrm{2} \\ $$$$\therefore\sqrt{\mathrm{2}\:}\:\mathrm{is}\:\mathrm{non}-\mathrm{trancendental} \\ $$$$ \\ $$$$\pi\:\mathrm{is}\:\mathrm{transendental} \\ $$
Commented by prakash jain last updated on 26/Dec/15
Set of algebraic numbers is countable.  Set of trancsdental numbers is not countable.
$$\mathrm{Set}\:\mathrm{of}\:\mathrm{algebraic}\:\mathrm{numbers}\:\mathrm{is}\:\mathrm{countable}. \\ $$$$\mathrm{Set}\:\mathrm{of}\:\mathrm{trancsdental}\:\mathrm{numbers}\:\mathrm{is}\:\mathrm{not}\:\mathrm{countable}. \\ $$
Commented by Filup last updated on 26/Dec/15
How can we prove it?
$$\mathrm{How}\:\mathrm{can}\:\mathrm{we}\:\mathrm{prove}\:\mathrm{it}? \\ $$

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