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John Gabriel

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1:16  - I have always objected to 0.333... = 1/3. It's nonsense. If this were true, then it would imply that 1/3 is measurable in base 10, which is outright FALSE. In order for a fraction to be measurable in a given base b, the denominator d of that fraction must contain at least one prime factor in common with b.

That aside, non-terminating representations are garbage that began with the Dutch arithmetician Simon Stevin. Non-terminating representations are ill-formed concepts. 

As for the long division process, it has nothing to do with the measurement of 1/3 in base 10. Long division is by definition a finite process.

At 3:24, you attempt to show that long division produces 0.999..., but that's not long division, only bad arithmetic on your part.

To see why 1/3 is not equal to 0.333..., see: https://drive.google.com/open?id=0B-mOEooW03iLVEd3RW5DRDR4Y3c&authuser=0

Follow me at: http://thenewcalculus.weebly.com  and
http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-772.html#post40616
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John Gabriel

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How is this at all fascinating? It's mind-numbing junk. A mediant makes zero sense unless any two given fractions are in proportion Bk V. Prop. 12 (The Elements).

Stern-Brocot trees are garbage. No offense to Wildberger please!

Moreover, the vinculum as used in 1/0 is illogical and anti-mathematical rot. There is no number k, such that k x 0 = 1. But this is true for every VALID fraction. That is, 3/4 means that the unit has been divided into 4 equal parts say k each, so that 4 x k = 1 and the 3 denotes how many of those parts are being considered.

Algebraically, NOTHING happens when the numerator is less than the denominator, that is, we toss the dots away from the obelus  -:-  to get the vinculum (horizontal line) and then place the numerator on top of vinculum and denominator on bottom of vinculum. Geometrically, a lot happens because we can divide ANY line segment into ANY number of equal parts we like using only a compass and a straight edge.

Thus, 1/0 is invalid and no fraction or number at all. It is a nonsense concept.

Google john gabriel new calculus to read more. Also check out my posts at LinkedIn.
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+John Gabriel
I have demonstrated you are doing strawmen, do you even know what one is?
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John Gabriel

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It's not correct to think about numbers as either a "choice" or an "algorithm".

No valid construction of irrational numbers proved here:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html#post21409
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John Gabriel

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No. 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. 

http://thenewcalculus.weebly.com
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John Gabriel

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Müll. Es gibt nicht einen einzigen gültigen Beweis dafür, dass 0.999 ... ist gleich 1.

Lesen Sie meinen Artikel zu sehen, warum:

https://drive.google.com/open?id=0B-mOEooW03iLVEd3RW5DRDR4Y3c&authuser=0

Folgen Sie mir auf:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-772.html#post40616

Lesen Sie über meine  Neue Analysis hier:

http://thenewcalculus.weebly.com

And also in German language:

http://hanspeterguettinger.weebly.com/

And Chinese:

http://thenewcalculuscn.weebly.com
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John Gabriel

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You haven't really started from nothing because you've assumed that natural numbers are strings of units (ones). However, a lot of work went into the development of the abstract object known as the 'unit'.

1. A magnitude is the idea of size, dimension or extent.
2. A ratio is a comparison of magnitudes, eg. if AB and CD are line segments, then AB:CD literally means AB compared with CD. The result is qualitative trichotomy, that is, AB = CD or AB > CD  or AB < CD.  However, if we know that AB is not equal to CD, we still don't know how much larger or smaller is AB than CD.
3. A unit is the comparison of equal magnitudes, that is, AB : AB  or  CD : CD. We can choose any magnitude as the unit of measurement.
4. With the unit in place, we can now form the natural numbers and find the difference between any two magnitudes that are multiples of the unit, that is, quantitative measurement. 

The natural numbers are now established.

The next step are the rational numbers. These are ratios of natural numbers. So if AB and CD are multiples of the unit, then AB : CD and CD : AB are rational numbers. A proper fraction is a ratio of natural numbers where the antecedent (numerator) is less than the consequent (denominator).

And that's the derivation of numbers from nothing.

While I used line lengths to derive numbers, I could equally well have used mass, volume, area, etc because all these are magnitudes or quantities.

Your approach, Prof. Wildberger, is really not different from the set theoretical approach which presumes the natural numbers are already in place. You have also used the Peano approach which assumes all the properties of the natural numbers.
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John Gabriel

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If you want to understand exactly what is an arithmetic mean, then you can do it in less than 3 minutes here:  http://www.youtube.com/watch?v=_RLRMGFBZBs

This is one of the most important concepts in mathematics and the least understood right after that of the number concept.

Also check out the entire article at: http://www.linkedin.com/pulse/arithmetic-mean-john-gabriel
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John Gabriel

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Proof that no valid construction of irrational numbers exists:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html


I like Wildberger. 


I too am called a crank. If you are not part of that group called mainstreamers, you are a delusional crank. Well, I'll drink to that! :-)


http://thenewcalculus.weebly.com
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+njwildberger And by facts, I am certain you do not mean  "Valid construction of the real numbers" because there is none. Is this correct?  A simple 'yes' or 'no' response will do. :-)

As for the "broad picture of modern mythmatics", why should it be preserved? 
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John Gabriel

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Finally one can understand exactly what is an arithmetic mean in my video at: http://youtu.be/_RLRMGFBZBs

You can download the Geogebra applet here from Google Drive:

https://drive.google.com/open?id=0B-mOEooW03iLQjN3WnVKLURjOFE&authuser=0

Before me, no one ever understood what it means to be an arithmetic mean. That's a big statement, but it's true. Let's see some examples of useless arithmetic means:

1. In a class of 10 students, the scores are 30, 30, 30, 30, 30, 30, 30, 90, 100, 100. The arithmetic mean is 50. To infer that the ability of each student is average, is clearly in error. Ignorant academics will claim that this is due to the outliers, that is, 90, 100 and 100. But this is a red herring because it covers the stupidity of having calculated the arithmetic mean in the first place. One should not have to re-examine the data if it makes sense to redistribute for equality, that is, the suggestion of outliers is not an excuse or valid reason, but a faux pas, confirming the ignorance of those responsible for the construction of such an arithmetic mean.2. One might say it is useful to get an overall impression when comparing two different classes. Suppose we have class A and B with 10 students each, and the following scores/grades:Class A:    30,  30,  30,  30,  30,  30,  30,  90,  100, 100       (Mean = 50)Class B:    50,   50,  50,  50,  50,  50,  50,  50,   50,   50       (Mean = 50)It is very tempting to infer from the arithmetic means, that each class has students of roughly the same test taking ability. Of course this is completely untrue, as 70% of class A students are below average ability if a score of 50 is considered to be average.There are many other ignorant uses and those responsible are unfortunately "educators" who are perceived as highly educated.Learn more about me at: http://thenewcalculus.weebly.com
Also follow me on Space Time and The Universe:


http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html#post21409



What 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 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.
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