Anyone who has explored the realm of the science beyond what is taught in the classroom will undoubtedly run across the term scalar, without any consistency of application. Scalar waves, scalar motion, scalar this, scalar that… it appears the term is popular to describe something that the author does not quite understand himself. So, let us start with a clear definition of what the term "scalar" means:

Scalar

"a quantity possessing only magnitude."

Quantity

"an exact or specific amount of measure"

Magnitude

"greatness of size or extent"

From the definitions, a scalar is simply the "specific amount of
greatness." Sounds nebulous, but it is fairly precise and a good definition of
"scalar." First, consider the term amount. It comes from the old trading
days, where people would barter for one "exact or specific amount of measure"
for another. "I'll trade you this sack of sugar for two bags of flour." Amounts
were the counting numbers. There are three attributes of the counting
numbers that make them unique:

1. There is no zero. Suppose I came up to you, and said, "I'll trade you nothing
   for your new Lexus." Sound like a good deal? If so, please contact the author
   ASAP. If not, then you understand why zero is not included in the counting
   numbers. Since they are based in measures, and measures are used in trade, you
   can only trade what you have and if you have zero of it, then it cannot be used
   in trade.
2. There are no fractional parts. "I'll trade you two and a half necklaces for
   three-quarters of your mule." Possible, but pointless.
3. There are no negative amounts. With counting numbers, there must be something
   to count, and there is no such thing has having "-4" cats in the house.

Now that the idea of quantity is understood to be the whole or
counting numbers, consider the term magnitude. How does a magnitude
differ from an amount? In simplest terms, and amount is actually
an amount of something. You can't have just six. You need six
somethings. Amounts qualify other concepts.

But what about magnitude? The magnitude refers to the "greatness of size or
extent", which means that it is the quantity specified in the amount of
measure, the "six" in "six somethings." The "somethings" is not included in the
magnitude, because it doesn't matter what it is, only how many
there are.

And there you have the definition of scalar: "A quantity possessing
only magnitude", which is one of the non-zero, non-negative, non-fractional,
whole counting numbers, without any identification of what they are a quantity
of. The minimum scalar magnitude is therefore one and the maximum is unlimited.

Some people may say that zero and negative amounts are valid, but they are
not part of the counting number system. If the computer at "Cars-R-Us" says they
have "-2" brake pads in stock for you, are you going to walk home with anything?
A promise won't stop your car. Until you have them, for all practical
purposes, "promises" don't exist, and cannot be counted as an item up for trade.

Since we will be dealing totally with the natural systems of reference in
this work, we have to stick to what is "real", not "promises" created by the
inventive mind of man. They don't exist in Nature. Can you have "-1" ocean?