# If $a$ and $b$ are constructible numbers and $a ≥ b > 0$, give a geometric proof that $a + b$ and $a - b$ are constructible.

This problem was taken from Joseph Gallian's "Contemporary Abstract Algebra", 8th edition.

Chapter 23, Exercise 1, Page 402: If $a$ and $b$ are constructible numbers and $a ≥ b > 0$, give a geometric proof that $a + b$ and $a - b$ are constructible.

Honestly, I'm not able to actually start on this problem because the textbook does not provide many examples of geometric proofs. Also, the definition of a "constructible number" does not come with much to go on. In the textbook (page 400) it says:

To begin, we call a real number $\alpha$ constructible if, by means of an unmarked straightedge, a compass, and a line segment of length $1$, we can construct a line segment of length $|\alpha|$ in a finite number of steps.

I suspect the answer to this is actually really trivial, I just haven't seen an axiomatic treatment of any "geometric proof" in this text, so I'm lost on where to start.

## 1 Answer

Draw a circle of radius $b$ at one of the ends of your line segment of length $a$.Extend that line segment so that it (also) meets the circle beyond your chosen end. The two points of intersection give you line segments of length $a-b$ and $a+b$.

• Can you explain why you are allowed to "draw a circle of radius $b$"? It doesn't mention that in my book. – farleyknight Sep 14 '18 at 19:37
• Proposition 2 in Book 1 of Euclid's Elements: mathcs.clarku.edu/~djoyce/java/elements/bookI/propI2.html – hartkp Sep 14 '18 at 19:40
• @farleyknight You have available a segment of length $b$ (that's what "$b$ is constructable" means). That means you are able to draw a circle centered at one end of the segment and with radius equal to the length of the segment. – Arthur Sep 14 '18 at 19:53
• I figured that's what it meant. I was only worried because I didn't see it explicitly stated in the text, and I want to try and reinforce my proofs with material from the text rather than saying "that what X means". – farleyknight Sep 14 '18 at 20:02