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# Is pi2 a rational or irrational number?

Rational number is a type of real number, which is in the form of p/q where q is not equal to zero. Any fraction with non-zero denominators is a rational number. Some of the examples of rational numbers are 1/2, 1/5, 3/4, and so on. The number “0” is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational, since they give us infinite values. Also, check irrational numbers here and compare them with rational numerals.

Rational Numbers and Irrational Numbers

There is a difference between rational and Irrational Numbers. A fraction with non-zero denominators is called a rational number. The number ½ is a rational number because it is read as integer 1 divided by integer 2. All the numbers that are not rational are called irrational.

Rational Numbers can be either positive, negative, or zero. While specifying a negative rational number, the negative sign is either in front or with the numerator of the number, which is the standard mathematical notation. For example, we denote the negative of 5/2 as -5/2.

An irrational number cannot be written as a simple fraction but can be represented with a decimal. It has endless non-repeating digits after the decimal point. Some of the common irrational numbers are:

Pi (π) = 3.142857…

Euler’s Number (e) = 2.7182818284590452…

√2 = 1.414213…

### Algebraic and Transcendental Numbers

The set of polynomials with coefficients in  Z, Q, R, or  C  is denoted by  Z[x], Q[x], R[x], and C[x], respectively.

An element  x∈R  is called an algebraic number if it satisfies p(x)=0, where p  is a non-zero polynomial in  Z[x]. Otherwise, it is called a transcendental number.

Algebraic Number: If r is a root of a non-zero polynomial equation

anxn+a(n-1)x(n-1)+…+a1x+a0= 0……..(1)

where all the coefficients are integers (or equivalently, rational numbers) and r satisfies a similar equation of degree <n, then r is said to be an algebraic number of degree n.

A number that is not algebraic is said to be transcendental.

In general, algebraic numbers are complex, but they may also be real. An example of a complex algebraic number is ‘i’, and an example of a real algebraic number is √2,

If, instead of being integers, the ai in the above equation are algebraic numbers bi, then any root of the given equation is an algebraic number.

bnxn+b(n-1)x(n-1)+…+b1x+b0 = 0……..(2)

If alpha is an algebraic number of degree n satisfying the polynomial equation

(x-alpha)(x-beta)(x-gamma)… = 0,

then there are n-1 other algebraic numbers beta, gamma, … called the conjugates of alpha. Furthermore, if alpha satisfies any other algebraic equation, then its conjugates also satisfy the same equation.

Transcendental Numbers

A transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best-known transcendental numbers are π and e.

Transcendental Function

In a similar way that a Transcendental Number is “not algebraic”, so a Transcendental Function is also “not algebraic”.

More formally, a transcendental function is a function that cannot be constructed in a finite number of steps from the elementary functions and their inverses.

An example of a Transcendental Function is the sine function sin(x).

### Is π2 a rational or irrational number?

Proof

π is transcendental, meaning that it is not the root of any polynomial equation with integer coefficients.

Hence,

π2 is transcendental and irrational too.

If π2 were rational, then it would be the root of an equation of the form:

ax + b = 0 for some integers a and b

Then π would be the root of the equation:

ax2 + b = 0

Since π is not the root of any polynomial with integer coefficients, let alone a quadratic, this is not possible.

Further, if π2 was the root of any polynomial equation with integer coefficients then π would be the root of the same equation with each x replaced by x2. So since π is transcendental, so is π2.

### Sample Questions

Question 1: Is 1.33 a rational number?

Yes, 1.33 is a Rational Number. As rational numbers can be expressed as decimals values as well as fractions.  The number can also be written as 133/100 which is the ratio of two integers.

Take a look at the below proof.

Proof:

The number 1.33 can be represented as shown below:

=>1.33/1

This can be further broken down as,

=>133/100

The number 133/100 is the ratio of two integers that are 133 integers divided by 100 integers and expressed in fraction form (as p/q where q is not equal to 0).

Question 2: Is (√2)2 a rational number?

Yes, (√2)2 is a Rational Number. As rational numbers can be expressed as decimals values as well as fractions.  The number can also be written as (√2)2 which is the ratio of two integers.

Take a look at the below proof.

Proof:

The number (√2)2 can be represented as shown below:

=> (√2)2 = 2

This can be further broken down as,

=> 2/1

The number 2/1is the ratio of two integers that are 2 integers divided by 1 integers and expressed in fraction form (as p/q where q is not equal to 0).

Question 3: Is 2.33 a rational number?

Yes, 2.33 is a Rational Number. As rational numbers can be expressed as decimals values as well as fractions.  The number can also be written as 233/100 which is the ratio of two integers.

Take a look at the below proof.

Proof:

The number 2.33 can be represented as shown below:

=>2.33/1

This can be further broken down as,

=>233/100

The number 233/100 is the ratio of two integers that are 233 integers divided by 100 integers and expressed in fraction form (as p/q where q is not equal to 0).

Question 4: Is (√3)2 a rational number?

Yes, (√3)2 is a Rational Number. As rational numbers can be expressed as decimals values as well as fractions.  The number can also be written as (√3)2 which is the ratio of two integers.

Take a look at the below proof.

Proof:

The number (√3)2 can be represented as shown below:

=> (√3)2 = 3

This can be further broken down as,

=> 3/1

The number 3/1is the ratio of two integers that are 3 integers divided by 1 integers and expressed in fraction form (as p/q where q is not equal to 0).

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