Finally one can understand exactly what is an arithmetic mean in my video at:

http://youtu.be/_RLRMGFBZBsYou can download the Geogebra applet here from Google Drive:

https://drive.google.com/open?id=0B-mOEooW03iLQjN3WnVKLURjOFE&authuser=0Before me, no one ever understood what it means to be an arithmetic mean. That's a big statement, but it's true.

Learn more about me at:

http://thenewcalculus.weebly.comAlso follow me on Space Time and The Universe:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html#post21409What is a quotient? What is an obelus?

13 -:- 5 and 13/5 do NOT mean the same thing at all.

The obelus, that is, -:- , means repeated subtraction and only applies when the numerator > denominator. As far as a proper fraction is concerned, true division is only possible using geometry. In algebra, the statement 1 -:- 3 is a NO OPERATION. The 1 dots are discarded, the 1 goes to the top of the vinculum (horizontal bar or slash) and the 3 to the bottom of the vinculum. No repeated subtraction of any kind takes place.

Here is the definition of quotient that works for ALL numbers:

The quotient (or division) of two positive numbers is that positive number, that measures either positive number in terms of the other.

Let the numbers be 2 and 3.

2/3+2/3+2/3 = 6/3 = (6-2-2)/(3-1-1) = 2/1 = 2

3/2+3/2=6/2= (6-3)/(2-1)= 3/1 = 3

In order to understand what it means to be a quotient, you first need to understand what it means to be a "number".

For this, you must be able to derive the number concept from scratch in 5 easy steps, as demonstrated in the axioms of magnitude:

The Gabrielean Axioms of Magnitude:

1. A magnitude is the concept of size, dimension or extent.

2. The comparison of any two magnitudes is called a ratio.

3. A ratio of two equal magnitudes is called a unit.

4. A magnitude x that is measurable by a unit magnitude u, is a natural number in the ratio x:u or a proper fraction in the ratio u:x.

5. If any magnitude or ratio of magnitudes cannot be completely measured, that is, they have no common measure, not even the unit, then it is called an incommensurable magnitude or ratio of magnitudes.

Now you can formally state the axioms of arithmetic:

The Gabrielean Axioms of Arithmetic:

1. The difference (or subtraction) of two numbers is that number which describes how much the larger exceeds the smaller.

2. The difference of equal numbers is zero.

3. The sum (or addition) of two numbers is that number whose difference with either of the two numbers is the other number.

4. The quotient (or division) of two numbers is that number that measures either number in terms of the other.

5. If a unit is divided by a number into equal parts, then each of these parts of a unit, is called the reciprocal of that number.

6. Division by zero is undefined, because 0 does not measure any number.

7. The product (or multiplication) of two numbers is the quotient of either number with the reciprocal of the other.

8. The difference of any number and zero is the number.

Observe that all the basic arithmetic operations are defined in terms of difference.

Gabrielean Axioms of arithmetic explained:

1. The difference (or subtraction) of two positive numbers, is that positive number which describes how much the larger number exceeds the smaller.

Let the numbers be 1 and 4.

4 - 1 = 3 or |1 - 4| = 3

2. The difference of equal numbers is zero.

Let the numbers be k and k.

|k - k| = 0

3. The sum (or addition) of two given positive numbers, is that positive number whose difference with either of the two given numbers produces the other number.

Let the numbers be 1 and 4.

1 + 4 = 5 because 5 - 4 = 1 and 5 - 1 = 4

4. The quotient (or division) of two positive numbers is that positive number, that measures either positive number in terms of the other.

Let the numbers be 2 and 3.

2/3+2/3+2/3 = 6/3 = (6-2-2)/(3-1-1) = 2/1 = 2

3/2+3/2=6/2= (6-3)/(2-1)= 3/1 = 3

5. If a unit is divided by a positive number into equal parts, then each of these parts of a unit, is called the reciprocal of that positive number.

Let the positive number be 4.

The reciprocal is 1/4 and 1/4+1/4+1/4+1/4 = 1

6. Division by zero is undefined, because 0 does not measure any magnitude.

Since the consequent number is always the sum of equal parts of a unit, it follows clearly that no such number exists that when summed can produce 1, that is, no matter how many zeroes you add, you never get 1.

7. The product (or multiplication) of two positive numbers is the quotient of either positive number with the reciprocal of the other.

Let the numbers be 2 and 3.

1/2+1/2+1/2+1/2+1/2+1/2=3

1/3+1/3+1/3+1/3+1/3+1/3=2

8. The difference of any number and zero is the number.

Let the number be k.

|k-0|=|0-k|

Observe that all the basic arithmetic operations are defined in terms of the primitive operator called difference.

These are the true axioms of arithmetic and the definition of the arithmetic operators.

Axioms for negative numbers are easy to define with some trivial modification.