Modern Greek academics are ignorant beyond belief. I am sad to say this because I too am Greek. They never understood what Euclid was attempting to do in his famous Elements. My definition of number summarises all of Euclid's work from Book 1 through to Book 11:

Ένας αριθμός είναι το όνομα που δίνεται στο μέτρο ενός μεγέθους.

Translation: A number is a name given to the measure of a magnitude.

OR:

A number is a name given to a measure that describes a magnitude.

My original definition was a short form:

A number is the measure of a magnitude.

But this had proved too complicated for the idiotic mainstream mathematics community.

Quick Introduction:

Ratios came long before number. The ratio p : q literally means p compared to q. This and nothing else. To go from ratio of sizes to number, both p and q need to have a common measure so that we can write p/q.

Example: Both aliquot parts of _ : _ _ are measured by _ and hence we can give this number the name _ / _ _ or just 1 / 2. The only well-formed concept of number is the rational number. There is no other kind of number.

Pi is also a ratio of magnitudes, but since its aliquot parts circumference and diameter have no common measure, there is NO number circumference / diameter and hence pi is NOT a number. It is an incommensurable magnitude (in Greek άλογος or irrational), meaning a magnitude having no common measure with any other magnitude. Not once in the Elements of Euclid is mention made of that bogus concept "irrational number". When the word άλογος is used, it refers to a magnitude, NOT a number. Book X talks about incommensurable magnitudes, NOT irrational numbers! Ancient Greeks recognised no other numbers besides the RATIONAL NUMBERS. The intellectual stupids in mainstream mathematics have never understood these simple facts.

Euclid's goal was to define the abstract concept of number beginning with nothing, in other words, a perfect derivation without any assumptions (or axioms) whatsoever. In this article I explain what no academic after Euclid and before me actually was able to accomplish: the well-formed definition of number.

Introduction:

What is a magnitude? What is a number? Why are there so many problems understanding fractions and numbers in general?

I shall derive the concept of number from nothing, as it should have been derived. In order to do this, I build on the brilliance of the Ancient Greeks whose clarity of thought was never matched by anyone that came before or after them. Prepare to be dazzled. I am going to blow your mind - if you have a mind at all. :-) What you are about to read, has never been revealed before. No mathematics educator or mathematics professor ever had this understanding which is profound and illuminating beyond anything you've ever imagined. So, let's begin ...

Before you continue reading, spend 5 minutes watching this video on what exactly it a number. An explanation for absolute morons is probably easier. After you have completed reading this article, then this three minute video will test your understanding.

After Euclid and before me, not a single mathematics academic, ever understood what is a number. That's quite a big statement, but I have proved it and will share some of my knowledge in this article. Few people can ever begin to match my intelligence and depth of insight. I am not arrogant or deluded and contrary to the prolific libel about me on the internet, I am most certainly not what you might call a crank or a crackpot. Also, I care little for others' opinions.

First, let me begin by saying that the foundations of mathematics have nothing to do with set theory or Georg Cantor, whom I consider one of the greatest fools in mainstream mathematics and the reason so many have problems with mathematics, in particular understanding the number concept and associated properties such as arithmetic using the binary operations. Cantor is the father of all real mathematical cranks as many as are found in mainstream academia who subscribe to Cantor's bogus ideas. There was a good reason Cantor ended up in a mental asylum - he was insane.

Secondly, well-formed concepts exist as noumena, independently of the human mind or any other mind. This truth was explained by the world's greatest philosopher - Plato. Platonism is the theory that ideas/concepts or other abstract objects are objective, timeless entities, independent of the physical world and of the symbols used to represent them.

Thirdly, a well-defined concept is imperative in mathematics or any discipline. It is a very dangerous thing to rely on one’s intuition. Look up the word in your dictionary please! Many academics, especially the most ignorant, such as set theorists, topology majors and teachers of real analysis, are guilty of this malpractice. Unless a concept can be reified, either intangibly or tangibly, you may as well dismiss it as junk knowledge. Anything you build from it, will eventually be filled with paradoxes and other contradictions, only to become so complex, that it eventually collapses into a worthless relic.

The Ancient Greeks were unbelievably smart and thousands of years ahead of any other people, in every area of knowledge, something which unfortunately can't be said of their idiot descendants - the modern Greeks. To understand number, one must start with Euclid.

It might surprise you that measure came before number. How so? Well, the Greeks started off with the concept of magnitude. A magnitude is the idea of size, dimension or extent. (Elements, Bk. V) It is decidedly not a number.

A magnitude can be a length, mass, volume or any other measurable size, dimension or extent. The Greeks used line segment lengths usually denoted as AB, CD, et cetera. You might be wondering as to why the Greeks chose distance which is the main property of line segments for the derivation of number rather than some other scalar such as mass. The answer to this will become quite clear as you read on.

We begin to consider the measure of any two magnitudes by comparing the same. Comparing quantities is measure. The dictionary definition of measure states:

Ascertain the size, amount, or degree of (something) by using an instrument or device marked in standard units or by comparing it with an object of known size.

The Greek word for count is μετρώ which means literally "I measure" or "to measure". The science of measure is called μετρολογία (metrology). We ONLY use RATIONAL NUMBERS in metrology because the myth called "real number" does not exist as a well-formed concept. There is NO valid construction of real number. The fools Dedekind and Cauchy had no clue what it means to be a number.

This comparison is denoted by a ratio of magnitudes which came long before numbers or ratios of numbers. Given line segments AB and CD, we write AB : CD which literally means the comparison of the magnitudes AB and CD. Since neither magnitude is a number, we can only perform qualitative measurement or trichotomy, that is, if we can tell visually that AB is not equal to CD, then we can go a step further to conclude which line segment is longer or shorter. That's all we can do if we stop here. For example, we can't tell what is the difference in any more precise terms.

Much later, the idea of fraction was born from ratio, that is, if p and q are numbers, then p : q means that p is always measured by the number following the colon. A new representation was designed for this, that is p / q which DOES NOT mean p divided by q, but rather p is measured by q. Consider the number: _ _ _ : _ _ _ _ . Does this number mean _ _ _ divided by _ _ _ _ ? OF COURSE NOT! The division has already taken place in the equal parts of _ _ _ _ which are used to describe the MEASURE of _ _ _ . STOP AND THINK ABOUT THIS, BECAUSE YOU PROBABLY NEED TO! What number do you think is

_ _ _ : _ _ _ _ ?

If you guessed 3/4, then you deserve a gold star!

3/4 is just a NAME given to the MEASURE of the distance _ _ _ using the equal parts of the unit _ _ _ _.

No division whatsoever is taking place. 3/4 DOES NOT mean "3 divided by 4", you incorrigible morons! Using the above example, 3/4 denotes the distance _ _ _ . No division whatsoever is taking place. No division is pending! The division of the unit into equal parts has already taken place before the measure of _ _ _ is determined.

_ _ _ is measured using the equal parts of _ _ _ _.

Think about this! Don't just accept what your stupid and ignorant mathematics professors tell you. They are morons compared to my intelligence.

In geometry it is possible to divide any line segment into n equal parts. This is not the same in algebra where effectively NOTHING happens when you write 3 -:- 4. The 3 goes to the top of the vinculum and the 4 goes to the bottom of the vinculum.

Now back to my story...

AB : CD is called a ratio of magnitudes. The idea of unit was discovered when the Greeks compared two magnitudes of the same size, that is, AB : AB. The outcome of such a comparison is equality or zero difference. To find out how different other magnitudes are from AB, we can designate AB to be the standard or unit of measurement. Hence the abstraction of unit was discovered.

In the case of magnitudes such as pi and square root of two, the unit is considered to be the diameter of a circle and the leg of a right angled isosceles triangle respectively. Circles with diameters are symptoms of pi just as right angled isosceles triangles are symptoms of square root of two. It is impossible to construct any magnitude, only its symptom. A number is the measure of a magnitude.

Provided any two magnitudes are commensurate with AB, they can now be quantitatively measured, that is, we can tell the difference (if any) exactly in terms of units. As you can see, the discovery of the unit was a quantum leap in the efforts to measure more precisely.

This knowledge led to the ideas of multiples and factors. Given a unit AB, a multiple of AB is measured exactly by AB and AB is called a factor of that multiple. From what you've read so far, it becomes clear that we can construct the natural numbers from any given unit. What this means is that provided the magnitude is a multiple of a unit, we can define the natural numbers as ratios of such a multiple to the unit. I bet you never imagined that ratios came before the natural numbers! In the delusion of modern mathematics, natural numbers are considered to be the starting point. We can name these numbers by assigning symbols as we please, example a, b, c or 1, 2, 3 and so on.(Elements, Bk. VII)

But what happens if a magnitude is parts of a unit? We let those equal parts of the unit be units, and the measurement of the unit by the those equal parts of a magnitude, to be the number, that is, a fraction. The key to this approach is to divide the unit into the right amount of equal parts and the answer is simply that natural number, which results in the measurement of those equal parts. That number is the antecedent part of the fraction or the numerator as commonly referred to in today's lingo. The consequent part is the number of equal parts in the unit or the denominator.

From the natural numbers, we define fractions as a ratio of natural numbers.

Example: 2 : 3 or 3 : 2 or 22 : 7 etc. For the ratio m : n, we call m the antecedent part, and n the consequent part. If the antecedent is less than the consequent, then we have a proper fraction, that is, a number x such that |x|<1 (x lies between -1 and 1), otherwise we have an improper fraction.

It would help you to read what it means to measure before you continue.

One ought to bear in mind that the ABSTRACT unit is dimensionless, not like the units one finds in a table of standard units or a table of measures. Rather, the ABSTRACT unit is used to generate all the rational numbers. The ABSTRACT unit starts off as a qualitative comparison of equal magnitudes. The abstract magnitude u is chosen randomly as a standard measure. So u : u is the unit. Then the natural numbers are formed as multiples of the unit, that is, k : u where k is a multiple of the unit. Next, the rational numbers are from ratios of ratios where the consequent ratio is always u : u. Rather than write k : u : u : u, we simply write k.

Also, if m : u and n : u are two multiples of the unit, then m : u : n : u is called a fractional ratio of magnitudes. Now, we want to deal only with numbers and since we know the unit, we omit the u and write m : n or n : m.

So how do we differentiate a ratio from a fraction? We introduce the vinculum symbol - that horizontal line which separates the numerator from the denominator, eg. m/n or n/m. Thus, we write 2/3 or 3/2. The vinculum does not mean division, because we are defining the new symbols of fractions. Any division used, has already taken place in the prior processes. Geometrically, it's very easy to divide any line segment into any number of equal parts. Algebraically, this is a different story - enter the obelus or division symbol, -:- which denotes repeated subtraction (numerator minus denominator) if the numerator is greater than the denominator and terminates once the remainder is less than the denominator.

Contrary to popular academic misconception, nothing happens when the numerator is less than the denominator, example: 1 -:- 3 = 1/3. All that happens in algebra is that the dots are discarded, the 1 goes to the top of the vinculum and the 3 goes to the bottom of the vinculum. In other words, nothing happens in algebra. The vinculum does not mean division (as per the obelus binary operator -:- ) when the numerator is less than the denominator. In fact, it needn't mean division vice-versa either, but then the operations of arithmetic on fractions become slightly more complex. I will discuss this shortly in my "axioms" of arithmetic. These are not actual axioms because the concepts can be derived from nothing and in essence proved to be true.

Division (ala obelus) is repeated subtraction and applies only when the first operand is greater than the second, that is, given p/q, p must be equal or greater than q. For example, in order to convince me that 1 -:- 3 = 1/3 in algebra, you would need to use the same process of repeated subtraction. The fact is, you can't, because there is only a remainder, that is, the numerator is already smaller than the denominator, so no subtraction takes place at all, i.e. no division ala obelus. Watch this short video to learn more. Most mainstream academics have never understood long division or polynomial division.

The derivation of the rational numbers is complete. Notice that I made no reference to beliefs that are the essence of axioms and postulates. There are NO axioms or postulates in mathematics. If you prefer videos, start with this amazing video.

Any magnitude that can't be measured by the unit or another magnitude other than itself, is called an incommensurable magnitude (Not an "irrational number" as the ignoramuses in mainstream academia might think. A magnitude is not a number! There was no accident in Euclid defining a magnitude in Book V and number in Book VII), that is, an incommensurable magnitude has no common measure with any other magnitude. As such, it cannot be called a number, because it can never be measured exactly by any other magnitude or established number. Examples are pi, e, sqrt(2), etc. These magnitudes are called incommensurable. Euclid called these irrational magnitudes, not irrational numbers (Elements Bk. X)

A number is the measure of a magnitude.

pi is a ratio of magnitudes that has no common measure with any other ratio. Through the Pythagorean theorem, we discover that the square root of 2 also is not a number.

But you may say, how is it that we can well define a square with area of 2 square units? I explain some of the misconceptions at this link.

One of the greatest mathematicians called Gauss agreed with me:

...3 is not as close to the true value of pi as is 3.14, and 3.14159 is still closer. By adding additional places to the right of the decimal, it is possible to approximate the true value of pi as closely as one likes. But Gauss insisted that one could not assume all the terms of the decimal expansion to be given to determine pi exactly. To do so would involve an infinite number of terms, and thus comprise an actually infinite set of numbers, which Gauss refused to allow in rigorous mathematics [Dauben 1977, 861.]

What Guass was also saying here and it is not immediately clear to most, is that there is no such thing as an infinite set.

One can only approximate incommensurable magnitudes. I have proved that there is no valid construction of irrational numbers, hence no real numbers also.

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

All one needs to understand both division (quotients) and multiplication is Euclid Book V, Proposition 12. In fact, proposition 12 explains all the theory of fractions. The proof of this proposition is done easily with similar triangles.

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

i. 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

ii. Let the numbers be 3/4 and 1/2.

( 3/4 + 3/4 ) / ( 1/2 + 1/2 ) = 6/4 = 3/2

iii. Let the numbers be 3/8 and 1/4.

(3/8 + 3/8 + 3/8 + 3/8) / ( 1/4 + 1/4 + 1/4 +1/4) = 12/8 = 3/2

iv. Let the numbers be 2/5 and 7/3.

(2/5 + 2/5 + 2/5) / (7/3 + 7/3 + 7/3) = (6/5)/(21/3)=(6/5)/7

= (6/5 + 6/5 + 6/5 + 6/5 + 6/5) / (7 + 7 + 7 + 7 + 7) = (30/5) / 35 = 6/35

Or

(2/5 + 2/5 + 2/5 + 2/5 + 2/5) / (7/3 + 7/3 + 7/3 + 7/3 + 7/3) = (10/5)/(35/3)= 2/(35/3)

= (2 + 2 + 2) / (35/3 + 35/3 + 35/3) = 6/(105/3) = 6/35

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.

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.

i. 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

ii. Let the numbers be 2 and 3.

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

iii. Let the numbers be 3 and 2.

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

iv. Let the numbers be 9 and 1/3.

9 / (1/1/3) = 9/3 = (9-3-3) / (3-1-1) = 3/1 = 3

v. Let the numbers be 1/3 and 9.

(1/3) / (1/9) =

(1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3) / (1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9+1/9)

= (9/3) / 1 = 9/3 = 3

vi. Let the numbers be 9 and 3.

9 / (1/3) = (9 + 9 + 9) / (1/3 + 1/3 + 1/3) = 27/1 = 27

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.

a. x - y is a measure of difference that tells us how much bigger or smaller is x or y than the other, provided they are not equal.

Example: 3 = 5 - 2 Explanation: 3 is the measure of the difference.

b. x + y is a composite measure where both x and y are used to measure a composite number.

Example: 7= 5 + 2 Explanation: 7 is the measured by the sum of 5 and 2.

c. x -:- y is a relative measure where y measures x.

y -:- x is a relative measure where x measures y.

Example 1: 4 = 8 -:- 2 Explanation: 4 is the measure of 8 using 2 as a measure.

Example 2: 2 = 8 -:- 4 Explanation: 2 is the measure of 8 using 4 as a measure.

Example 3: 1/2 = 4 -:- 8 Explanation: 1/2 is the measure of 4 using 1/8 as a measure.

That is, 1/2 = 1/8 + 1/8 + 1/8 + 1/8.

d. x * y is a composite measure in terms of either x or y.

Example: 6 = 2 x 3 Explanation: 2 and 3 are measures of 6.

The above mentioned "axioms", are the true properties of arithmetic and the definition of the arithmetic operators. "Axioms" for negative numbers are easy to define with some trivial modification. Do not waste your time with the flawed theories of Zermelo, Fraenkel, Peano and Cantor.

To summarise:

Subtraction is the measure of difference using subtrahends .

is the measure of using . Addition is the measure of sum using s ummands .

is the measure of using s . Division is the measure of dividend using divisor .

is the measure of using . Multiplication is the measure of product using multiplicands.

Incidentally, Peano assumes the existence of the natural numbers in his juvenile axioms. He then derives the arithmetic properties but cheats along the way. For example, he uses the addition binary operator before defining subtraction. Peano assumes the most important property of the natural numbers (mathematical induction) as his successor function, that is, the successor of n is n+1. Well, we cannot know this unless we know the difference first, which is one. One of the dumbest exercises is trying to prove 2+2 = 4 in real analysis because 2+2 is by definition 4. It can be proved if required, by using the definition. Assume that 2+2=4 and subtract a unit from each side until either side has no unit left to subtract. If both sides have no units left, then they must be equal. The BIG STUPID (mainstream academia) has never understood the concept of number - their entire understanding of number is based on the mythical real number line.

The addition operator is defined in terms of subtraction. For example, p + q is the same as p - (-q). My third axiom describes addition.

Ignorant academics the past few hundred years have not understood radix systems (mostly due to Cantor's delusions), nor do they understand that when one writes 1/4 = 0.25, the process has nothing to do with long division. All that's happening here is the measurement of 1/4 in terms of base 10, that is, expressed as a polynomial sum of products of which the coefficients are digits 0 thru 9.

The proper way to convert 1/4 into radix 10:

1/4 = 1/4 x 10/10 = 10/4 x 1/10 = (2+1/2) x 1/10 = (2 x 1/10) + (1/2 x 10/100)

= (2 x 1/10) + (10/2 x 1/100) = (2 x 1/10) + (5 x 1/100)

Since there is no remainder, the measurement is complete. A common misconception is that non-terminating radix representations are well defined. Nothing could be more false and the Dutch arithmetician Simon Stevin is directly responsible. 1/3 is not equal to 0.333... for several reasons, but the most important reason is that 1/3 is not measurable in base 10. In order for a fraction m/n to be measurable exactly in any base b, the radix b must contain all the prime factors of n.

One might ask what is the reasoning behind 1/2 -:- 1/4. Well, axiom 4, 5 and 7 give us the answers. Axiom 7 states that:

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

So, 2 = 1/2 x 4/1 = 1/2 -:- 1/4

Or by Euclid Book V, Proposition 12:

(1/2) / (1/4) = (1/2 + 1/2 + 1/2 + 1/2) / (1/4 + 1/4 + 1/4 + 1/4) = 2/1 = 2

Conclusion

1/4 = 2/10 + 5/100 = 25/100

1/3 = 3/10 + 3/100 + 3/1000 + ?

1/3 is not equal to 0.333... whatever rubbish that is. This stems from Euler's Blunder, that is, S = Lim S.

A mostly unknown theorem in mathematics states:

Given any fraction p/q and base b, where the prime factors of q not also prime factors of b, cannot be expressed as equivalent fraction in base b.

Mainstream academics support the theorem and Euler's Blunder. However, these are mutually exclusive. If Euler's definition is sound, then the theorem is no longer true and hence not a theorem. If the theorem is true (which it is because theorems are generally true!), then Euler's definition is nonsense.

I have much more to say with regards to understanding the number concept, but you won't know unless my work What you had to know in mathematics, but your educators could not tell you is ever published.

I am the discover of the first and only rigorous formulation of calculus in human history. It's a sorry fact that I am not well liked or liked at all by my jealous peers, so you will find more negative than positive comments on my work, but that is largely due to stupidity and ignorance of modern academics. Those who are closer to my level of intelligence have seen the light, and there are a handful of PhDs in mathematics who support my ideas.

I can be very abrasive in my online discussions, but the reason for this, is that I have a very low tolerance for fools, especially those in academia where there is no shortage of stupidity and ignorance. Be advised that my bark is far worse than my bite. I am not so persistent in person and never address my colleagues in the way I address those whom I disrespect in newsgroups and other forums. I am convinced that someone ignorantly commenting on the internet deserves to be called stupid. One of my pet peeves are ignoramuses who post comments as if they are authorities in the subject.

I am one of very few real mathematicians. Most so-called "mathematicians" in today's academia, are mythmaticians disguised as mathematicians.

I trust that you have been somewhat entertained and perhaps a little more educated than you were before you read this article. I encourage you to read and engage in my discussions on the internet. Be advised that unless you use your real name and write respectful comments, I will treat you as I do all other cranks and trolls.

You will be astounded and learn more than you have in all your school years, no matter if you are a PhD in mathematics or a high school student.

It took me some time to prepare this article for you. You are welcome to contact me if you have questions or need more details. I do not promise to respond, but I definitely try to respond to the many questions I receive. If you send me an email with an attitude, I shall probably ignore you. I am not interested in your opinions, so your email should contain only questions, preferably one question per email.

My new calculus site has links to a lot of interesting articles, dynamic applets and other information on mathematics which you won't find in mainstream academia. These links are shared on Google Drive.

Be sure to browse many hot topics in mathematics and a series of videos on my new calculus.

Whatever I imagine is real, because whatever I imagine is well defined.

All the theory of fractions from Book V, Proposition 12.

Book V and Proposition 12 was realised from the properties of similar triangles. Equivalent fractions are derived from these properties.

The theorem states that if x/y is in proportion to p/q, then

x/y = (x+x)/(y+y) = (x+p)/(y+q) = (x+x+p) / (y+y+q) = etc...

Using only this property, one can perform all arithmetic operations with fractions using only subtraction or addition.

Cancellation:

10/5 = (10-2-2-2-2)/(5-1-1-1-1) = 2/1

Addition/Subtraction:

2/3 + 1/5 = (2+2+2+2+2)/(3+3+3+3+3) + (1+1+1)/(5+5+5)

2/3 + 1/5 = 10/15 + 3/15 = 13/15

2/3 - 1/5 = (2+2+2+2+2)/(3+3+3+3+3) - (1+1+1)/(5+5+5)

2/3 - 1/5 = 10/15 - 3/15 = 7/15

Quotient:

2/3 -:- 1/5 = (2/3+2/3+2/3+2/3+2/3) -:- (1/5+1/5+1/5+1/5+1/5)

2/3 -:- 1/5 = (10/3) -:- (1) = 10/3

Product:

The product is just the quotient of the reciprocal of one of the operands with the other.

2/3 x 1/5 = 2/3 -:- 1/(1/5) = 2/3 -:- (1+1+1+1+1)/(1/5+1/5+1/5+1/5+1/5)

2/3 x 1/5 = 2/3 -:- 5/1 = (2/3+2/3+2/3)/ (5+5+5) = (6/3) / 15

2/3 x 1/5 = (6-2-2)/(3-1-1) / 15 = (2/1) / 15 = 2/15

Watch my video on this proposition to learn more and to find out why there never were any axioms or postulates in Greek mathematics. Also, realise why mainstream calculus is flawed using this proposition. Cauchy's derivative denies the undeniable Proposition 12. There are five main reasons why Cauchy's derivative is flawed.

Finally, see how mainstream academics have desperately tried to conceal their incompetence, ignorance and stupidity.