1. **State the problem:** We have a complex number $$z = (a - 1) + (a^2 + a - 2)i$$ where $$a$$ is a real number. We want to find the values of $$a$$ for which $$z$$ is purely imaginary. 2. **Recall the definition:** A complex number $$x + yi$$ is purely imaginary if its real part $$x = 0$$. 3. **Apply the condition:** The real part of $$z$$ is $$a - 1$$. For $$z$$ to be purely imaginary, set: $$a - 1 = 0$$ 4. **Solve for $$a$$:** $$a - 1 = 0 \implies a = 1$$ 5. **Check the imaginary part for $$a=1$$:** $$a^2 + a - 2 = 1^2 + 1 - 2 = 1 + 1 - 2 = 0$$ 6. **Interpretation:** When $$a=1$$, the imaginary part is zero, so $$z = 0 + 0i$$ which is purely imaginary (also purely real, but zero is considered purely imaginary). 7. **Check other candidate $$a=-2$$:** Real part: $$-2 - 1 = -3 \neq 0$$ so not purely imaginary. 8. **Conclusion:** The only value of $$a$$ making $$z$$ purely imaginary is $$a=1$$. **Final answer:** $$a=1$$