# What are negative numbers?

The question really is “What are the positive and negative numbers ?”

First of all we have “the numbers”. 1, 2, 3, … and zero for completeness sake, and also that “none” is a number, and “nothing” is not a number.
“The numbers” are actually very different from one another. The counting numbers, 0, 1, 2, … , are very different from the measuring numbers. Measuring numbers have “quantity”, counting numbers have “counts”.
Measuring numbers are cakes, pizzas, watts, feet, mass, volume, area, and so on, where the quantity is “some” or “none”.
Examples of measuring numbers are “half of a foot”, “2/3 of a pizza”, “0.05 square feet”, and they are just “numbers”, attached to units of measurement.

Now there is a problem.

“Apply and extend previous understandings of numbers to the system of rational numbers.” (quote from CCSS)
Just like that !!!!!!
There is NO easy extension, “half of a foot” is NOT extendable to “three feet below sea level”.
The meaning of “positive and negative numbers”, or the “signed numbers”, is not a “some or none” situation at all.
The “signed numbers” are abstractions of “relative position” and “change of position”, and the position of “zero” is often, if not always, arbitrary.

A temperature scale has a zero, and temperatures above zero are “positive”, temperatures below zero are “negative”.

A different temperature scale has a different zero, and, worse still, the scale factors (scales) are different as well.

“Feet above sea level” and “meters above Mount St Helens” are similarly “different”.

An electrical circuit can have chosen a voltage value of zero at any point in the circuit.

In these and all similar situations the zero is chosen by a human, and not as the “none or nothing” value.
The value 9, or the value -5, is marked on the scale as a position relative to the zero on that scale.
Marks on the left, or the “down side”, are conventionally the “negative” marks, and marks on the right, or the “up side” are the “positive” marks.
The positive marks. “+”, are conventionally ignored, but at the start one should put them in.

… to be continued, when the confusion between “negative” and “subtraction” is resolved.

And this CCSS bit is so stunningly superficial.
Apply and extend previous understandings of numbers to the system
of rational numbers.
5. Understand that positive and negative numbers are used together
to describe quantities having opposite directions or values (e.g.,
temperature above/below zero, elevation above/below sea level,
credits/debits, positive/negative electric charge); use positive and
negative numbers to represent quantities in real-world contexts,
explaining the meaning of 0 in each situation.
6. Understand a rational number as a point on the number line. Extend
number line diagrams and coordinate axes familiar from previous
grades to represent points on the line and in the plane with negative
number coordinates.
a. Recognize opposite signs of numbers as indicating locations
on opposite sides of 0 on the number line; recognize that the
opposite of the opposite of a number is the number itself, e.g.,
–(–3) = 3, and that 0 is its own opposite.