Investigate whether the real and imaginary parts of a constructible complex number are necessarily constructible | Step-by-Step Solution

What We're Solving:

We need to determine whether the real and imaginary parts of a constructible complex number are always constructible themselves. This is a fundamental question about how constructibility behaves when we decompose complex numbers into their components.

The Approach:

To tackle this problem, we'll use the geometric interpretation of constructible numbers and examine how we can perform operations with compass and straightedge. The key insight is that if we can construct a complex number geometrically, we should be able to "project" it onto the real and imaginary axes to find its components.

Step-by-Step Solution:

Step 1: Recall what constructible means A complex number z = a + ib is constructible if we can construct the point (a, b) in the coordinate plane using only compass and straightedge, starting from the points (0,0) and (1,0).

Step 2: Visualize the geometric setup If z = a + ib is constructible, then we can construct the point P = (a, b) in the plane. This point represents our complex number geometrically.

Step 3: Think about projecting onto axes If we have constructed the point P = (a, b), we can construct points on the real axis at distance |a| from the origin, and points on the imaginary axis at distance |b| from the origin.

Step 4: Use perpendicular construction We can:

• Drop a perpendicular from P to the real axis (x-axis), which gives us the point (a, 0)
• Drop a perpendicular from P to the imaginary axis (y-axis), which gives us the point (0, b)

Step 5: Verify these are basic constructions Dropping perpendiculars is a fundamental compass-and-straightedge construction:

• Draw the line through P parallel to the y-axis (perpendicular to x-axis)
• This intersects the x-axis at (a, 0)
• Similarly for the y-axis intersection at (0, b)

Step 6: Handle signs carefully The real part a could be positive or negative, and same for b. But since we can construct the actual points (a, 0) and (0, b) with their correct positions, we get both the magnitude AND the sign of each component.

The Answer:

Yes! If a complex number a + ib is constructible, then both its real part a and imaginary part b are constructible real numbers. This follows from the fact that we can always drop perpendiculars from a constructed point to the coordinate axes using compass and straightedge.

Memory Tip:

Think of it as "projection preserves constructibility" - if you can build the point, you can always project its shadow onto each axis to get the components! It's like shining light from above and to the side to cast shadows on both the x and y axes.

This result is quite intuitive when you think about it geometrically - constructibility shouldn't be "lost" when we simply look at the individual coordinates of a point we've already constructed!