Final answer:
To find which complex number is closest to the origin, calculate the moduli of \((1/2)-(1/4)i\) and \((2/3)+(1/6)i\). To determine which is closest to \(1+i\), subtract \(1+i\) from each and then compute the moduli of the results.
Step-by-step explanation:
The student is asking which complex number, \((1/2)-(1/4)i\) or \((2/3)+(1/6)i\), is closest to the origin and which one is closest to \(1+i\). To determine the distances, we can calculate the modulus of each complex number (the distance from the origin) and then compare these values. To find the nearest complex number to \(1+i\), we can subtract this point from each complex number and again compute the modulus.
To calculate the modulus, use the formula \(|z| = \sqrt{x^2 + y^2}\) where z is a complex number of the form \(x+iy\). So for the first complex number, \(\sqrt{(1/2)^2 + (-1/4)^2}\) is computed to determine its distance from the origin. The second distance is calculated similarly. To ascertain proximity to \(1+i\), subtract \(1+i\) from each complex number, yielding \((1/2)-(3/4)i\) and \((2/3)-(1/3)i\), respectively, before calculating the modulus of these new complex numbers.