Prove or disprove if a and b are rational numbers than a b

Prove or disprove: if a and b are rational numbers than a + b is also a rational number. Show work. it a and b are irrational numbers than a + b is also an irrational number. Show work if a is rational and b is an irrational number than a + b is irrational. Show work

Solution

Solution :

( a ) The proof is very simple. By definition of a rational number, a = p/q and b = s/t for some quadruple of integers p, q, s, and t and such that q and t are nonzero. The sum a+b = p/q + s/t = (p t + q s)/(q t), a rational number by definition. What would the converse say? It would assert \"If a and b are real numbers such that a + b is a rational number, then a and b are rational numbers.\" But this is false. Just let a = sqrt(2) + 1, where sqrt means the square root, and b = - sqrt(2). Neither a nor b are rational numbers, but a + b = 1, which is rational.

( c )

Proof :

Suppose not. [we take the negation of the theorem and suppose it to be true.] Suppose there is a rational number a and an irrational number b such that (a + b) is rational. [We must deduce a contradiction.] By definition of rational, we have

                                               a = r/s     for some integers r and s with s 0  

and                                    a + b = p/q    for some integers p and q with q 0  

By substitution, we get

                                                                  r/s + b = p/q

and so

                                                                          b = p/q r/s

                                                                            = (sp rq)/sq

Now (sp rq) and (sq) are both integers [since r, s, p, q are all integers, and since products and differences of integers are integers], and sq 0 [by the zero product property].

Hence, s is a quotient of the two integers (sp rq) and (sq) with sq 0. So, by the definition of rational, b is rational. This contradicts the supposition that b is irrational.