Is π/2 rational or irrational? Please explain

2 Answers

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

Jan 26, 2018

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.