The idea of negative and positive means that we have two opposite conditions that describe (for example) the direction of a number. In mathematics, we apply a sign (+ or -) to an integer to depict this condition and therefore we think of numbers as positive or negative. However, I choose to separate the two ideas of number and sign, and consider the sign as merely a descriptor. This descriptor is most commonly used to define the direction of a number. Therefore, an integer by itself, has no sign. However, for clarity, I will retain the words positive integer to mean integer. If I use the term integer alone, I will mean unsigned integer.
To further illustrate the descriptive nature of signs, consider the
following.
The selection of only positive and negative integers is actually very
restrictive, since they only allow two directions (for example) to be used.
There is no
reason why we could not have North, South, East and West numbers. These too
are descriptors of the numbers in an identical way that positive and negative
describe numbers. If we use N S E W for the signs of these directions, then
such numbers would appear as N1, S2, E3 or W4.
I will focus mainly on the use of the classical descriptors ( + and - ) in
the following. When using signed integers, it is important that the
operations between these numbers are logical.
Addition of signed integers remains unchanged from the way we are taught.
I have purposely omitted the negative operations of subtraction and division
and address this topic on a separate page under inverse operators.
So (+4) + (+5) = (+9) and (+4) + (-5) = (-1) and so on.
Remember that the sign is simply a descriptor and is not part of the integer.
The signed integers can obviously create a result of zero. For example, (+5) + (-5) = 0. Zero is a valid result, but zero is NOT a number, and simply reflects the absence (or elimination) of numbers. It serves no other purpose. If the descriptors are opposite in polarity, then zero can signify the reference location that separates the two sets of integers. Origin is the usual term used. A similar analogy exists when you view the coordinate system in 2 dimensions, for example. Here we have the integers and four different descriptors, and zero is the origin. Classically we have subdivided the axes into positive and negative, as well as x and y axes. However, since each axis describes a distinct direction, therefore there should be four distinct descriptors (N,S,E,W) or similar. So an operation such as N1 + S1 will produce 0 since the polarity (or sign) is opposite. However N1 + W1 will produce a value of NWsqrt(2). This is identical to the way that vectors behave. Using this logic, the nature of the descriptors determines the result of operations between signed integers and therefore provides much more flexibility than simply positive and negative numbers.
Multiplication is a concise form of addition, and behaves as follows.
The operations (+3) x 4 = (+12) and (-3) x 4 = (-12) are valid.
For clarity, in expanded form we have : (+3) x 4 = (+3)+(+3)+(+3)+(+3).
In other words, multiplication duplicates the original number
(including its sign). Notice that one of the numbers in each product has a
sign and the other is simply the multiplier and therefore has no sign.
Remember that a number without a sign is not positive - it is
unsigned. Therefore, an operation such as (-3) x (+4), is invalid.
As a result, the common idea (that we have struggled to understand),
of multiplying two negative integers to create a positive integer, is
an invalid operation using this logic.
In summary, adding two signed integers is valid, but multiplying two
signed integers is not. The square root of (-9) is therefore (-3), since
(-3) x 3 = (-9). The operation here is simply taking the square root of 9
and then attaching the sign to the result as a descriptor. As a result of
this logic, you will not find complex (or imaginary) numbers mentioned on
these pages.
The sqrt(-1) is therefore (-1), just as the sqrt(+1) = (+1).
Enjoy.