The difference between Algebraic number and Irrational number
When used as nouns, algebraic number means a complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers, whereas irrational number means any real number that cannot be expressed as a ratio of two integers.
check bellow for the other definitions of Algebraic number and Irrational number
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Algebraic number as a noun (algebra, number theory):
A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers.
Examples:
"The golden ratio (φ) is an [[algebraic number]] since it is a solution of the quadratic equation <math> x^2 + x - 1 = 0 </math>, whose coefficients are integers."
"The square root of a rational number, <math>\textstyle\sqrt{\frac m n},</math> is an [[algebraic number]] since it is a solution of the quadratic equation <math>n x^2 - m = 0</math>, whose coefficients are integers."
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Irrational number as a noun (mathematics):
Any real number that cannot be expressed as a ratio of two integers.
Examples:
"The square root of a positive integer is an [[irrational number]] unless the positive integer is a perfect square, in which case the square root will also be a positive integer."