Roots and Directed Numbers

In working with numbers one often works with fractional exponents, “roots.”  This works well when one is working with “absolute values, ” unsigned numbers; however, it runs into complications as soon as one starts to operate with “signed” numbers.  The problem probably arises from the inherent fact that absolute number values do not have a directed motion automatically assigned to them.  A positive number is by convention associated–usually–with a motion upward, to the right, or forward, with a negative number associated with motion  downward, to the left, or backward. When we take the square root of one, unsigned, we’re are talking about a number which multiplied by itself gives the original number, one. When we take the square root of 4, we realize that it is the number two, when we place two units down twice we get four. When we are working with signed numbers we have a different situation. If we are taking the square root of +4, we are actually asking the question, “What is the directed side length of a square which we consider to have the area “Positive Four” when we operate according to the conventions associated with signed numbers?” By those conventions we can see that both +2 times +2 and -2 times -2 fit this criterion, so we say, quite correctly that the square root of +4 is either +2 or –2,  Perhaps we would, however, have been more accurate in saying that there are two sets of square roots to the number, +4, the set, +2,+2, {+2, =2} and the set ,{-2,-2}.  

The reason that this last was said will become clear when we discuss the situation for the “square root of -4.”  Let us analyze this problem  as we did above. The question we are asking is what two directed numbers will produce an area which by our conventions of directed numbers will be assigned a value of -4?” This occurs again in two cases, producing two sets, {+2, -2} and {-2,+2} . As these are directed  numbers the set,  {+2,-2} is not identical to the set {-2, +2} as they represent opposite directions of sequential motion. With the “Positive Area” we find that we can create what we call a positive area by going in a positive direction then turning in another positive direction, or going in a negative direction and then turning in a negative direction again.  For a negative area we can start out in a positive direction, then “turn negative” or start in a negative direction and “turn positive.” By this analysis, the square root of “Negative One” is not an imaginary number but can be said to be not as in the other case, “Plus or Minus One” implying each “operating” on itself, but “Plus and minus one” the two operating on each other. The concept of imaginary numbers arises because of the ignoring of this fact of the directed action factor inherent to signed numbers.

This can, of course, be extended to higher roots.  For the cube root of  +8, one may write the sets, {+2, +2, +2}, {-2,-2,+2}, {+2, -2, -2}, and    {-2,+2,-2}. Noting four sets that can be considered the “cube root ” of +8. A similar group of 4 sets represents the “cube root” of -8. A fourth root would presumably continue the pattern developing eight sets of 4 units each. This is left to be proven, or disproven by the reader.

While the idea of imaginary numbers as successive even roots of “Minus One,” is an interesting concept, but by the above analysis appears to be based on a misunderstanding of the significance of signed numbers. The use of a signed number indicates a motion in a direction and can be considered to define a “dimension.”