Algebraic number

An algebraic number is any real or complex number that is a solution of an equation of the form

a_nxⁿ + a_n-1x^n-1 + ... + a₁x¹ + a₀ = 0

where n >= 0 and every a_i is an integer.

All rational numbers are algebraic because every fraction a / b is a solution of bx - a = 0. Some irrational numbers such as 2^1/2 (the square root of 2) and 3^1/3 (the cube root of 3) are also algebraic because they are the solutions of x² - 2 = 0 and x³ - 3 = 0, respectively. But not all real numbers are algebraic. Examples of this are π and e (the natural logarithm base).

If an algebraic number satisifies such an equation with a polynomial of degree n and not such an equation with a lower degree than the number is said to be an algebraic number of degree n.

It can be shown that if we allow the coefficients a_i to be any algebraic number then the solution of the equation will also be an algebraic number.

If a complex number is not an algebraic number then it is called a transcendental number.

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