Yes, rational and irrational numbers can be negative. Te only thing that is desired is that they could be mapped to a place on a real number line. Negative numbers are to the left of 0 on number line.
By definition, rational numbers are a ratio of two integers p and q, where q is not equal to 0. Hence, if p is negative and q is positive (or vice versa but q!=0), p/q could be negative.
Examples of negative rational numbers are -3.14159, -17/4, -2/3 or -3.bar(142857) (here bar(142857) indicates these numbers are repeating infinitely). These are equivalent to -314159/100000, -17/4, -2/3 or -22/7 (in form p/q).
Similarly there could be negative irrational numbers too like -pi, root(3)(-80), -sqrt2 etc. These are equivalent to their positive irrational numbers like pi, root(3)(80), sqrt2 but towards left of 0 on real number line.
Similarly, there could be irrational numbers like 6-3pi which are 3pi units to the left of 6, but as this would lie to the left of 0. Some other such numbers are 1-sqrt3, root(3)2-sqrt5, -sqrt17+2 and -7.239135113355111333555...... (The last number is non-terminating non-repeating decimal number and hence an irrational number.)
