1. We are given the equation $$|z| - z = 1 + 2i$$ and need to find the value of complex number $$z$$. 2. Let $$z = x + yi$$ where $$x, y \in \mathbb{R}$$. 3. The modulus of $$z$$ is $$|z| = \sqrt{x^2 + y^2}$$. 4. Substitute into the equation: $$|z| - z = \sqrt{x^2 + y^2} - (x + yi) = 1 + 2i$$. 5. Equate real and imaginary parts: Real part: $$\sqrt{x^2 + y^2} - x = 1$$ Imaginary part: $$-y = 2$$ so $$y = -2$$. 6. Substitute $$y = -2$$ into real part equation: $$\sqrt{x^2 + (-2)^2} - x = 1$$ $$\sqrt{x^2 + 4} = 1 + x$$. 7. Note that $$\sqrt{x^2 + 4} \geq 0$$ and $$1 + x \geq 0$$ must hold for equality. 8. Square both sides: $$x^2 + 4 = (1 + x)^2 = 1 + 2x + x^2$$. 9. Simplify: $$x^2 + 4 = x^2 + 2x + 1$$ $$4 = 2x + 1$$ $$2x = 3$$ $$x = \frac{3}{2}$$. 10. Check validity: $$1 + x = 1 + \frac{3}{2} = \frac{5}{2} > 0$$. 11. Therefore, the solution is $$z = \frac{3}{2} - 2i$$. Final answer: $$\boxed{\frac{3}{2} - 2i}$$.