Is -3.14 an irrational number

Answer 1

-3.14=(-314/100), so -3.14 is a rational number (since it can be expressed as p/q, with p and q being integers).

If you mean -pi (which is approximately -3.14), then it is irrational. pi is irrational (for a proof, which is fairly complicated, see: http://www.lrz-muenchen.de/~hr/numb/pi-irr.html).

And an irrational number times a rational number (which -1 is since it can be expressed as -1/1) is irrational. This can be proved by assuming the product is rational. Let x be a rational number, which can be expressed as m/n with m and n integers), and let y be the irrational number. Let S=xy. Assume S is rational, and can be expressed as t/u, with t and u being integers. Then:

S=xy

t/u=(m/n)y [Divide both sides by m/n, which is the same as multiplying by its reciprocal, n/m)

(t*n)/(u*m)=y

Since t, n, u, and m are integers, tn and um are integers. (t*n)/(u*m)=y implies y is rational, which is a contradiction. Therefore, xy=S is irrational.