Perfect Squares & Irrational Numbers

An irrational number is a real number that cannot written in the form a/b with a ∈ ℤ (a whole number) and b ∈ ℤ₀ (a natural number or whole number excluding 0). A number is a perfect square if it is the square of a whole number. A few examples are 1, 4, 16 etcetera. One ‘interesting’ property of perfect squares is that they always end with an even number of zero’s (no zero is also an even number of zero’s). You can see this is true by looking at the squares of 1 to 10 which all end with zero 0s except 10² which ends with 2 zero’s. Number 11 to 19 can be written as 10 + b with b from 1 to 9. The square is then (10 + b)² to which the formula for (a + b)² = a² + 2ab + b² can be applied. In this case you get (10 + b)² = 100 + 20b + b². And b² will not end on a zero so all these numbers will also not end on a zero. Why do I call this property interesting ? Because you can use it as a ‘trick’ to prove for example that √10 is not a rational number, meaning it cannot be written as a/b with both a and b natural numbers. The proof is a straight forward reductio ad absurdum where you make use of this property.