6.2 Transcendental Number (초월수)

Number System

integer (Z) = natural number (N) \(\cup\) zero \(\cup\) negative integer
rational number (Q) = integer \(\cup\) fractional number
real number (R) = rational number \(\cup\) irrational number
complex number = real number \(\cup\) imaginary number

decimal number = finite number \(\cup\) infinite number
infinite number = repeating(periodic) number \(\cup\) nonrepeating(nonperiodic, irrational) number
repeating number = pure repeating number \(\cup\) mixed repeating number

number = algebraic number \(\cup\) transcendental number

Algebraic number

any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently - by clearing denominators - with integer coefficients).

Number which can be a root of rational or integer coefficients polynomial equation.

Transcendental number

Number which is not an algebraic number.

\(\therefore\) irrational number = root number \(\cup\) transcendental number

\(\therefore\) algebraic number = rational number \(\cup\) root number

Examples of transcendental number

e
\(\pi\)
\(\pi\) + e
\(\pi\) - e
\(\pi \cdot e\)
\(\pi / e\)
\(\pi^{\pi}\)
\(\pi^e\)
\(e^e\)

Transcendental function

Function which can produce(return) transcendental number with algebraic input.

Examples of transcendental function

exponential function
log function
trigonometric function
\(x^{\pi}\)
\(c^x\)
\(x^x\)
\(x^{\frac{1}{x}}\)

Set theory

Countable infinite/denumerable set, 가부번(가산) 무한집합
Uncountable infinite, 비가부번(불가산) 무한집합

n(natural number) = n(integer) = n(rational number) = \(\aleph_0\)

n(real number) = \(\aleph_1\) = \(2^{\aleph_0}\)

Question

How many rational numbers between 0 and 1? (2007년 초등학교 4학년 1학기 수학익힘책 123쪽)