Are all objects with irrational lengths measureable?

I was deleting old emails a while ago and I came across with questions from some students reading my blog.  I have answered quite a number of questions from middle school and high school students via Email and Facebook since this blog started. I think some are worth publishing here, so I’ll probably post one from time to time.  Below is the first Q & A in this series.

Question

Are all objects with irrational lengths measureable?

Answer

Yes. In principle, they are measureable.

The number line represents all real numbers. It contains all the rational and irrational numbers.  In fact, there is a one-to-one correspondence between the set of real numbers and the set of points on the number line.  This means that every real number has a corresponding point on the number line, and every point on the number line has a corresponding real number. Therefore, since we can locate every irrational number on the number line, we can find its distance from 0.  This distance represents the irrational length.

Introduction to the Complex Plane

Imaginary numbers had no use when it was invented except for intellectual exploration until it was linked to coordinate geometry.  The complex number a + bi can be plotted as the ordered pair (a,b) where a is the real part and b is the imaginary part.  Therefore, the complex numbers 3 + 2i, 3- 2i, -2 + 0i,  and 0 - i can be plotted as (3,2), (3,-2), (-2,0), and (0,-1) in a “coordinate plane” respectively as shown below.  In the figure, we can see that the origin is 0 + 0i and that 3 - 2i, the complex conjugate of 3 + 2i, is its reflection along the x-axis (Can this be generalized?).

The plane where the complex numbers are plotted above is called the complex plane.

In the complex plane, we can observe that all numbers of the form (a,0) are real numbers and these numbers are represented by the horizontal axis. The numbers of the form (0,bi) are all imaginary numbers and it is represented by the vertical axis. Thus, we can call them  the real axis and the imaginary axis respectively.

In addition, we can conclude that all real numbers are complex numbers since  for all  real numbers  a, a = a + 0i. So, the set of real numbers is a subset of the set of complex numbers.

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