Algebraic Number

Algebraic Number

An algebraic number is any number that is a root of a non-zero polynomial with rational coefficients.

Put more simply, when you have a polynomial like (for example):

2x2 - 4x + 2 = 0

Then x is algebraic.

Because:

  • It is a non-zero polynomial
  • x is a root (ie x gives the result of zero for the function 2x2 - 4x + 2)
  • the coefficients (the numbers 2, 4 and 2) are rational numbers

 

The polynomial could, of course, be simpler or more complicated than this example, just so long as the coefficents are rational.

Not Algebraic? Then Transcendental!

If a number is not algebraic, it is called transcendental.

Example: is √2 (the square root of 2) algebraic or transcendental?

√2 is the solution to x2 - 2 = 0, so it is algebraic

Properties

All algebraic numbers are computable and so they are definable.

The set of algebraic numbers is countable.

The imaginary number i is algebraic (it is the solution to x2 + 1 = 0).

All rational numbers are algebraic, but an irrational number may or may not be algebraic.

 

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