Question

For the following, give constructions using a straightedge and a compass with memory. You must prove that your construction works.

We say that a positive real number a is constructible if whenever we are given a line segment of length c , we can construct a line segment of length ac . Suppose that a and b are constructible real numbers. Show that ab is also constructible.

Answer 1

Answer:

By the given definition, we say that a positive real number a is constructible if whenever we are given a line segment of length c , we can construct a line segment of length ac .

Given that a and b are constructible real numbers. We are to show that ab is also constructible real numbers.

since a and b are constructible real numbers, by the definition of a constructible real number, given line segments of length k and l we can construct a line segment of length ak and bl respectively.

Now since ak and bl are magnitudes , we have ,

or, since l and k are lengths of line segments , then 2lk = m is also a lenth of some line segment .So,we have ,

  

So, for given reals a and b ab is also a real numner. Now for real number ab given a line segment of length m=2lk , we can construct a line segment of length abm whose length is given as

.

Hence, proved.