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An Algebraic Number is a number EQN:x which satisfies an algebraic equation i.e. an equation of the form

EQN:a_0x^n+a_1x^{n-1}+...+a_n=0

where

where EQN:a_0,{\quad}a_1,{\quad}\ldots{\quad}a_n are integers, not all zero.

EQN:a_0,{\quad}a_1,{\quad}\ldots{\quad}a_n

Real numbers which are not algebraic numbers are called transcendental numbers.

are integers, not all zero.

The size of the set of algebraic numbers is the size of countable sets EQN:\aleph_0.

Exercise: By considering the case EQN:ax+b=0 prove that all rational numbers are also algebraic numbers.

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Exercise: By considering the equation EQN:ax+b=0 prove that all rational numbers are also algebraic numbers.