Algebraic Numbers

A number which is root of a non – zero polynomial with rational coefficients is known as algebraic number.
For example: If we have a polynomial
⇒3p² – 4p + 2 = 0; then ‘p’ is algebraic number because it is a non- zero Polynomials and ‘p’ is a root value which gives the result of zero for the function 3p² – 4p + 2;
The coefficients 3, 4 and 2 are Rational Numbers.
If any number which is not algebraic is known as transcendental. Any number which is not algebraic is known as transcendental number.

For example: √2 is algebraic number of transcendental number.
So √2 is Square root number and √2 is the solution to p² – 2 = 0; therefore the number is algebraic number.
Now we will see some properties of algebraic numbers.
All the Algebraic Numbers are computable so we can say that the algebraic numbers are definable. We can easily count the algebraic numbers so all the algebraic numbers are countable.
‘I’ is Imaginary Number which is algebraic.

All the rational numbers in the mathematics are algebraic and irrational number may or may not be algebraic number.
Some of the Irrational Numbers are also algebraic.
Let ‘r’ is a root of a non zero polynomial equations then:
   P_n xⁿ + P_{n – 1}x^{n – 1} + ……+ p₁ x + p₀ = 0;
Where ‘p_i’ are integers and ‘r’ satisfies only different equation of degree < n, then ‘r’ is said to be an algebraic number of degree ‘n’.
Suppose ‘r’ is an algebraic number and pⁿ = 1 then the number is known as algebraic number. In general the algebraic numbers are taken as Complex Number but the algebraic number also is real number. The Set of algebraic numbers is represented by ‘A’ and sometimes it is represented by ‘Q’.