Number Theory/Irrational Rational and Transcendental Numbers

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Definitions[edit]

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 \frac{3}{4} \, , -\frac{27}{3} \, etc.


A rational number can also be expressed as a termininating or recurring decimal. Examples include
1.25 , -0.333333 , 0.999 \ldots

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. 5:4 \,

The entire (infinite) set of rational numbers is normally referenced by the symbol \mathbb{Q} \, .

Irrational numbers are all the rest of the numbers - such as \sqrt{2} , \pi , e \,

Taken together, irrational numbers and rational numbers constitute the real numbers - designated as \mathbb{R} \, .

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, \sqrt{2} is a root of the polynomial equation x^2 - 2 = 0 \, 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, \pi , e \, 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).