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.