Question 120076
A rational number is one that can be expressed as the ratio (hence the name) of two integers.  So if you can find any two integers p and q such that {{{x=p/q}}}, then x is rational.  Otherwise, x is irrational.


A couple of rules:  Any integer is rational.  That's because you can express any integer as that same integer divided by 1.


Any repeating decimal is rational, conversely, all irrational numbers are non-repeating decimals.


Problem 1) Squaring an integer results in an integer.  64 is an integer, so {{{64^2}}} is an integer.  All integers are rational, therefore {{{64^2}}} is rational.


Problem 2) All repeating decimals are rational, so -1.2... is rational.  But what are the integers p and q?  Hint:  Any time the repeating part of the decimal is the same number, try 9 as a denominator ({{{1/3=3/9}}} so the rule holds).  Since the absolute value of the given number is slightly greater than 1, you need a numerator that is slightly greater than the denominator.  Put the minus sign on either the numerator or the denominator.  {{{(-11)/9}}} does quite nicely.


Here's a couple more rules for the other side of the question:


The square root of any number that is not a perfect square is irrational.  The cube root of any number that is not a perfect cube is irrational.  And so on...


{{{pi}}}, the ratio of a circle's circumference to its diameter, is irrational.


{{{e}}}, the base of the natural logarithms, is irrational