Consider the following theorem and proof. theorem: the number âš2 is not rational number. proof: let's suppose âš2 is a rational number. then we can write âš2 = a/b where a, b are whole numbers, b not zero. we additionally assume that a/b is simplified to lowest terms, i. e., a and b have no common factors. both of the numbers a and b cannot be even. from the equality âš2 = a/b it follows that 2 = a2/b2 .so, a2 = 2 â· b2. showing that the square of a is an even number since it is two times something. from this we know that a itself is also an even number. in symbols, a = 2k where k is a natural number. if we substitute a = 2k into the original equation 2 = a2/b2, this is what we get: 2=(2k)2/b22=4k2/b22*b2=4k2b2=2k2sho wing that b2 is even because it is 2 times a number. this implies that b itself is also even. therefore the supposition made in the first line cannot be true and we can conclude that the number âš2 is not rational number. thereforethe statement "both of the numbers a and b cannot be even." is justified by the fact that the assumption that âš2 is a rational number. the assumption that a/b is simplified lowest terms. the assumption that âš2 is an irrational number. the fact that b divides a evenly. the fact that a and b are whole numbers.