Number Theory/Irrational Rational and Transcendental Numbers

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Rational numbers are numbers which can be expressed as a ratio of two integers (with a non-null denominator).

This includes fractional representations such as etc.

A rational number can also be expressed as a termininating or recurring decimal. Examples include

However, a decimal which does not repeat after a finite number of decimals is NOT a rational number.

One other representation that is sometimes used is that of a ratio e.g.

The entire (infinite) set of rational numbers is normally referenced by the symbol .

Irrational numbers are all the rest of the numbers - such as

Taken together, irrational numbers and rational numbers constitute the real numbers - designated as .

The set of irrational numbers is infinite - indeed there are "more" irrationals than rationals (when "more" is defined precisely).

Algebraic numbers are numbers which are the root of some polynomial equation with rational coefficients. For example, is a root of the polynomial equation and so it is an algebraic number (but irrational).

Transcendental numbers are irrational numbers which are not the root of any polynomial equation with rational coefficients. For example, are not the roots of any possible polynomial and so they are transcendental.

The set of transcendental numbers is infinite - indeed there are "more" transcendental than algebraic numbers (when "more" is defined precisely).