Question

Is π/2 rational or irrational? Please explain

Answer 1

`pi/2` is irrational

Explanation:

A rational number is any number that can be expressed as a fraction of two integers provided that the denominator is not zero.

Mathematically, a rational number can be expressed as

`rArrp/q, q!=0`

Examples are `0, 1,-2, 0.bar3, sqrt4, 1/5, 0.6, 7 8/9`

NB: Zero is a rational number because it can be expressed as a fraction: `0 = 0/1` and `0.6` because it can be expressed as `6/10`

An irrational number is any number that cannot be expressed as a fraction of two integers where the denominator is not zero.

Examples are `pi, e, sqrt2, -sqrt3, 4/0, sqrt5/3, 1.3578239412762....`

NB: The quotient of irrational numbers can be rational or irrational.

Now to the question at hand:

Is `pi/2` rational or irrational?

`rarr` We know that `pi` is an irrational number because its approxiamation is `22/7` which is not the exact value of `pi`.

`rarr` Next, we know that the quotient of irrational numbers can be rational or irrational.

`rarr` Here is the case where we have `pi/2`, an irrational number divided by a rational number giving a quotient that is irrational.

`rarr` Thus, `pi/2` is an irrational number.

An example of an irrational number divided by an irrational number giving a quotient that is rational is `sqrt12/sqrt3`

It can be simplified to `(sqrt4*sqrt3)/sqrt3`

`rArr(sqrt4*cancelsqrt3)/cancelsqrt3`

`rArrsqrt4=+-2`

Answer 2

`pi/2` is irrational

Explanation:

A rational number is expressible in the form `p/q` for integers `p, q` with `q != 0`.

Any real number that cannot be expressed in this form is called irrational.

The number `pi` is an irrational number, so cannot be expressed as a fraction, though there are some famous rational approximations to it, namely `22/7` and `355/113`.

Since `pi` is irrational, it follows that `pi/2` is also irrational.

`pi` is actually a transcendental number: It is not the zero of any polynomial with integer coefficients. This is a stronger condition than irrationality and implies it.

In fact, in a technical sense, most real numbers are transcendental, though much of the time we deal with rational and other algebraic numbers.

Proving that `pi` is irrational (or even better that it is transcendental) is beyond the scope of most Algebra courses. The simplest proofs involve the use of calculus.