Problems in Mathematics–Signs and Signed Numbers

The signs, + , and,  – , are used throughout mathematics and physics in a number of ways,and with several meanings which are often not carefully checked. The plus or positive sign has its initial use in addition in the sense of increasing a pile, of no particular dimensions, by a certain amount described by a counting number written after it. The negative sign represents the opposite operation of removing a certain specified amount.

In Physics, the positive sign represents a “charge” associated with the proton, the negative sign represents and “opposite” charge associated with the electron and other species having a characteristic in common with the electron. [This writer suspects that this characteristic is a counter-clockwise spin.] In this usage, the signs do not represent reversed operations but characteristics which are considered opposites.

A third usage shows up in mathematics where the signs are associated with counting numbers to form sets of “signed numbers” which seem to be able to be added, subtracted, multiplied and divided like counting numbers. However, this turns out to have problems when one does multiplication and division processes.  What is overlooked is that the addition of the sign to a number gives it both a magnitude and a direction. Something having both magnitude and direction is not a true number, it is what is called a “vector.”

The signed number may represent movement away from a zero point, a line, a plane, a three dimensional figure formed of planes or some “higher order figure” depending on where it occurs in a sequence of operations. Signed numbers are handled according to a convention wherein the positive sign is considered as being to the right of an origin, upward from an origin or forward from an origin, and if one multiplies three positive signed numbers together, say plus two times plus two times plus 2, ( +2 x + 2  x +2) what one has really described is moving two units to the right of the zero point, moving this “two-units-line”  upward to form a square, then moving this square two units forward to create a cube which is situated to the right, above and in front of the origin point.   This “eight-cubic-units entity is called a positive volume because we say + x + x  = + as we consider that the + sign represents travel ” in the same direction” while the negative sign represents travel in the opposite direction.   This signed unit, like the line and the square, has a direction associated with it which would be at right angles to the last upward motion. Labeling this unit as positive continues the vector content but is truly in accord with the “reversal idea upon which it is based, for, as we have seen above, each operation represents a change in direction, but of  90 degrees, not a reversal.  If we go, + 2, +2, -2, in our sequence of operations, we will go to the right first, up second, and back from the “center-plane”  third to form another eight unit volume which will be above, to the right, but  behind the point of origin. This will be considered a “negative volume” purely by convention as it has one negative sign associated, however, this convention does preserve the vector designation by the conventions observed.

As the positive numbers are associated with “positive values,” right–as in handedness, upward–toward the Heavens, and forward–“progress.”  Negative numbers are associated with the reverse, left–“sinister” or left-handed, downward, and backward.  The operation, -2 x -2 x -2 , would create a cube, which was to the left of the origin point, below the ‘origin line,” aka, the “x-axis” and behind the “origin plane,” aka, the “xy-plane.” Note that the first square formed would be considered a “positive number as it is “minus x minus = plus”  but the third operation, adding another direction considered “minus” labels the resulting volume as a “negative number volume.”

Summing the above, a signed number represents a line, two signed numbers multiplied together represent a plane and three signed numbers multiplied together represent a volume and, in  the order of the multiplication process will determine what plane or volume is described.  The operation, ” +2 x -2 ” represents, by the conventions used, a square which is to the right and down from the origin. while the reverse operation, “-2 x +2” forms the representation of a square which is to the left and above the origin.

It can be seen then that while 1 x l x l as counting numbers still represents the original one.  Plus-one times Plus-one times Plus one represents one whole, but it is one whole cube, one length to a side, not one line….!

Similarly, it can be seen that the cube root of eight as a counting number is simply the number two. The “cube root of +8, as a “vector cube” has the “absolute value” of 2 but this two can be either a positive or negative vector depending on which of the “generating sets” it belongs to and the order in which it falls in the set. A  positive-volume-vector, +8, can be generated by any of four sets of three “signed twos.” These sets are as follows:  {+2, +2, +2}, {-2, -2, +2}, {-2, +2, -2}, or {+2, -2, -2}. A negative-volume-vector,-8,” can be generated by any one of  the sequenced-operation sets {-2, -2, -2}, {-2, +2,+2}, (+2, +2,-2} or {+2,-2,+2}. Assigning a “signed-root” to a signed number, is therefore a difficult and tricky business which would actually require a knowledge of the history of the signed number in question!   It is no wonder that the mathematicians seem to ignore “odd-number” roots of signed numbers and consider that the square root of minus one is “plus or minus ‘i,’ an imaginary number, which truly it is for in the most basic unit of “minus one” we are considering a line vector of a unit length and how does one take a root of a line vector “running backwards?”  Actually the root would have absolute dimension of one, either plus or minus, as one would have to be speaking of the “second-order-vector-square” which can be generated by either of the sets,        {+1, -1} or {-1, +l}.  Mathematicians have no trouble with saying that the square root of +1 is plus or minus one as it is generated by the two sets, {+1, +1} and {-1, -1}, sets within which the internal values appear to be identical to one another. As one can see from previous discussions that the internal elements are not identical but represent different directions of the vector depending on their position in the sequence.

The use of the two signs with different meanings of operation, reversal, or direction causes some interesting problems in understanding mathematics.