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, if 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 to the function 2x2-4x+2)
- the coefficients are rational numbers
The polynomial could, of course, be simpler or more complicated than this example, just so long as the coefficents are rational.
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 is therefore algebraic
Properties
All algebraic numbers are computable and therefore 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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