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# Algebraic Numbers

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 Sub Topics A number which is root of a non –zero polynomial with rational coefficients is known as algebraic number. For example: A polynomial, 3p$^{2}$ – 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 3p$^{2}$ – 4p + 2; Here the coefficients 3, 4 and 2 are Rational numbers. 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: Pn xn + Pn – 1xn – 1 + ……+ p1 x + p0 = 0; Where ‘pi’ 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’.

## Evaluation of Algebraic Numbers

Algebraic expression in mathematics is a Combination of constants and variables that are connected together to each other by mathematical operators '+' representing addition, '-' as subtraction, '*' as multiplication and '/' as division. For instance, we have following expressions: X + 4, or X – 10Y. Let us suppose an example to understand how to do evaluation of algebraic Numbers.

A fact that must be known to all is that an Algebraic expression can be solved only when values of unknown variables in expression are known i.e. algebraic numbers are known.
Let us find the value of expression: 4X + 7Y, if X = 4 and y = 20.
Substituting the values of X and Y in equation 4X + 7Y we get result:
4 * 4 + 7 * 20 = 16 + 140 = 156,
Next let us do 12XY, if X = 5 and Y = 11.
Substituting the values of X and Y we get:
12XY = 12 (5) (11) = 60 (11) = 660 (simplify)
Let us take a more complex expression follows: (X * 4Y) / 7 (Y + Z), if X = 3 and Y = -2 and Z = 4.
Substituting the values of X, Y we Z we get the following result:
(X * 4Y) / 7 (Y + Z) = 3 * 4 (-2) / 7 (-2 + 4) = 12 (-2) / 7 (2) = - 24 / 14
Simplifying the fraction further we get:
(X * 4Y) / 7 (Y + Z) = - 24 / 14 = -12 / 7,
Similarly we can solve Algebraic Numbers when they are in exponents or powers. Like: 42 + 80 + 53 = 142.

## Properties of Algebraic Numbers

A number, which is a root of non – zero polynomial and have rational coefficients is called as algebraic number. For example: If we have a polynomial expression as:
⇒4x2 – 8x + 7 = 0;
Here ‘x’ shows the algebraic number because it is a non- zero Polynomials and ‘x’ shows a root value that gives result of zero for function 4x2 – 8x + 7;
The coefficients value 4, 8 and 7 are Rational Numbers.
Any number that is not algebraic is known as transcendental number. For example: number √2 is algebraic number of transcendental number.

So √2 is Square root number and √2 is the solution to x2 – 2 = 0; so number is algebraic number.

Let us discuss some basic Properties of Algebraic Numbers.
The entire Algebraic Numbers are assessable so we can say that, algebraic numbers are definable. We can easily count the algebraic numbers so entire algebraic numbers are countable.
The symbol ‘i’ is Imaginary Number that is algebraic.

Entire rational numbers in mathematics are algebraic and irrational number may or may not be algebraic number.
Some of the basic Irrational Numbers are also said to be algebraic.
Suppose ‘r’ is a root of a non zero polynomial equations then:
An xn + An – 1xn – 1 + ……+ a1 x + a0 = 0;
Here value of ‘ai’ shows integers and ‘r’ satisfies only different equation of degree < n, then ‘r’ is called as an algebraic number of degree ‘n’.
Let given ‘r’ is an algebraic number and an = 1 then number is said to be algebraic number. Basically the algebraic numbers are taken as Complex Number but they are real also. The Set of algebraic numbers is also denoted by ‘A’ and sometimes it is represented by ‘B’.