Monday, June 3, 2013

Real Numbers

I received quite a few questions, including some very reflective and cerebral ones.  Since I am a teacher, and I sometimes need to be circumspect to avoid offending people, I will let my friend Anjo answer the questions that delve into the dangerous areas of religion and philosophy.

Question #1.  Why are are numbers real numbers, even the imaginary ones?

A real number is a number that exists on the number line.  It includes positive numbers, negative number and irrational numbers.  An irrational number is essentially a number that cannot be expressed as a simple ratio.  The best-known irrational number is pi, which also called a "transcendental number" because it cannot be expressed as the result of a mathematical formula.  This reminds me of a frustrating story that conveys an important life lesson.  My eighth-grade math teacher was named Mrs. Brownell.  She taught us that if we were calculating using pi, we could use 3.14 if we wanted, but if we wanted to be absolutely accurate we should convert our decimals to fractions and use 22/7 for pi.  She said that 22/7 was pi exactly.  A student pointed out that mathematicians try to calculate pi out to more and more decimal places.  Mrs. Brownell said that they do this by dividing 22 by 7 and carrying it out to more and more decimal places.  I figured that this is something that even I could do, so I did the division and soon realized that the number I got was both different than the standard expression of pi (starting in the thousandths place) and started to repeat (which everybody knows pi does not do).  I told Mrs. Brownell who said that I must have divided wrong.  I asked her to try it on the board to show me, but she got angry and said it was a waste of time to teach me how to divide in 8th grade.  I redid my calculations and showed it to her.  She refused to look at it and told me that I should probably argue with her husband, who is a mathematician at Los Alamos lab.  The bell rang and she stormed out of the class.  Instead of looking at my calculation, she became defensive and angry.  The moral of the story is that children are expected to question assumptions unless those assumptions are made by people in authority.  Also, adults tend to dig themselves into incredible intellectual ruts - try your best not to that.  I realize that teachers are the worst, because we want everything we say to be authoritative, and we don't like to analyze our own assumptions.

Back to the number.  Pi is considered a real number because it sits somewhere on the number line.  While nobody knows pi in its entirety, we know that it sits on the number line between 3 and 4 (closer to 3), and it can be used like a regular number.  The square root of 2, likewise, is an irrational number because it can't be expressed as a ratio, but we know it is sitting there somewhere.

Imaginary numbers are something else, and they are NOT real numbers.  The term "imaginary" was invented to distinguish it from real numbers.  The problem that they address is that negative numbers cannot have a square root, because any number multiplied by itself yields a positive result (a negative number multiplied by a positive number shows that you have a certain number of that negative number, so the solution is always negative).  In some cases you want to know what the square root of a number would be if it were not impossible to have one, so they came up with imaginary numbers.  Imaginary numbers yield a negative result when multiplied by themselves.  They are mathematically useful when you are dealing with things like non-linear forecasting or calculating impedance.  More on this later.