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Algebraic Numbers

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Algebra is a study of playing with numbers and its patterns. For example: if we add 6 to a number it becomes 11, what is the number? It can be written as x + 6 = 11. The letter x is denoted to the missing number so that we can derive an equation to this pattern. Every time a number is added to 6 it gives 11. Solving this we get the number to be 5.

This x will be known as a variable.
A polynomial is an expression consisting of variables in different powers and coefficients. 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: $\sqrt{2}$ is algebraic number of transcendental number. So $\sqrt{2}$ is Square root number and $\sqrt{2}$is the solution to $p^{2} – 2 = 0$; therefore the number is algebraic number. All rational numbers are algebraic numbers but not all irrational numbers are. The irrational numbers such as pie are not algebraic as they can never come as a root of a polynomial expression.

Definition

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A number x is said to be an algebraic number if the given conditions are satisfied:

1) The polynomial $ax^{n} + bx^{n-1} +..+ c = 0$ is a non-zero polynomial.

2) x is the root of the given polynomial.

3) The co-efficient a, b, c... are all rational numbers.

An algebraic number is any real number that is the root of a polynomial equation with single variable and rational coefficient. The set of algebraic numbers is mostly represented by A, and sometimes by Q.

For the expression, $x^{2} - 3 = 0$ $\sqrt{3}$ is the root. Hence, it is an algebraic number. A complex number is also algebraic.

Properties of Algebraic Numbers

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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: 4$x^2$ - 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 $4x^{2}$ - 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 $\sqrt{2}$ is algebraic number of transcendental number.So $\sqrt{2}$ is Square root number and $\sqrt{2}$ is the solution to $x^{2}$ - 2 = 0; so number is algebraic number.

Let us discuss some basic Properties of Algebraic Numbers:

1) All algebraic numbers are definable, means they all can be computed.

2) We can easily count the algebraic numbers making them countable.

3) i is an imaginary number which is an algebraic number.

4) All rational numbers are algebraic but only some irrational numbers are.

Evaluation

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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: $\frac{X*4Y}{7(Y+Z)}$, if X = 3 and Y = -2

and Z = 4.

Substituting the values of X, Y we Z we get the following result:

$\frac{X*4Y}{7(Y+Z)}$= $\frac{3*4*-2}{7(-2+4)}$ = $\frac{12 (-2)} { 7 (2)}$ = $\frac{-24}{14}$.

Hence,

$\frac{X * 4Y} { 7 (Y + Z)}$ = $\frac{- 24}{14}$ = $\frac{-12}{ 7}$.

Similarly we can solve Algebraic Numbers when they are in exponents or powers.

Like:

$4^{2} + 8^{0} + 5^{3} = 142$.

Examples

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Example 1: Is 2i an algebraic number? Explain the answer.

Solution: Yes, it is an algebraic number. If we take a polynomial $x^{2} + 2 = 0$, then 2i will be the root of this non-zero polynomial with rational coefficients.

Example 2: Is e an algebraic number? Example.

Solution: No. There is no such non-zero polynomial which has e as its root.

Example 3: If a number is added to 11 it gives 5, write the algebraic expression and find the number.

Solution: Taking the variable as x, x + 11 = 5. Adding 11 on both sides, x = 5 - 11 = -6.