Some Implications and Considerations
Point 1-A.- Case of multiplications of the same type as 20x10
The regular procedure is such as it's shown here:
20
x10
00
20
200
which is an abbreviation of this:
(20 + 0)
x(10 + 0)
(0x0)_ _ _ _ _ _ 0
(20x0)_ _ _ _ _ 00
(0x10)_ _ _ _ _ _ 0
(20x10)_ _ _ _ _ 200
200
To consider that zero multiplied by zero results in zero doesn't hinder us from getting the searched result ALTHOGH THAT CONSIDERATION IS FALSE. This last thing takes us to the issue of the consideration of exceptions where there weren't considered to be.
The axiom (a + b)x(c + d) = ac + bc + ad + bd, AXIOM WHICH MULTIPLICATION PROCEDURES ARE USUALLY RELATED TO, is an example. The axiom is true EXCEPT for (a = 0 or b = 0) and (c = 0 or d = 0).
For the multiplication 20x10, we should proceed just like this: 20x10 = 200, equality which comes from (20 + 0)x(10 + 0) = 20x10 + 0x10 + 20x0 = 200 + 0 + 0 = 200.
In general, (a + 0)x(c + 0) is equal to axc + 0xc + ax0 (or axc + 0xc + ax0 + 0) BUT IS NOT EQUAL TO axc + 0xc + ax0 + 0x0 (for a different from zero and c different from zero).
Point 1-B.- Another important case I'd like to mention here is ax0 = a-a. In this axiom, a is every number except zero for 0x0 results in every number except zero whereas 0-0 results in zero.
Point 2.- MATHEMATICS IS NOT CONDITIONED BY WHAT OCCURES IN THE FIELD OF FACTUAL SCIENCES. IT IS AN INSTRUMENT OF THEM BUT IT IS NOT CONDITIONED BY THEM. Let's imagine three magnitudes A, B and C related between them, ALMOST always, like this: A = a, B = b, C = c, and axb = c. Note A is not exactly the same as a, B is not exactly the same as b, and C is not exactly the same as c. A, B and C are particular magnitudes whereas a is a multiplicand, b is a multiplier and c is a product. NOTHING should say that if A, B and C are equal to such a special number as zero (A = 0, B = 0 and C = 0) then a, b and c have to be equal to zero too.
To consider that zero multiplied by zero results in zero may usually be very useful IN SPITE OF BEING THE CONSIDERATION OF A FALSITY.
Point 1-A.- Case of multiplications of the same type as 20x10
The regular procedure is such as it's shown here:
20
x10
00
20
200
which is an abbreviation of this:
(20 + 0)
x(10 + 0)
(0x0)_ _ _ _ _ _ 0
(20x0)_ _ _ _ _ 00
(0x10)_ _ _ _ _ _ 0
(20x10)_ _ _ _ _ 200
200
To consider that zero multiplied by zero results in zero doesn't hinder us from getting the searched result ALTHOGH THAT CONSIDERATION IS FALSE. This last thing takes us to the issue of the consideration of exceptions where there weren't considered to be.
The axiom (a + b)x(c + d) = ac + bc + ad + bd, AXIOM WHICH MULTIPLICATION PROCEDURES ARE USUALLY RELATED TO, is an example. The axiom is true EXCEPT for (a = 0 or b = 0) and (c = 0 or d = 0).
For the multiplication 20x10, we should proceed just like this: 20x10 = 200, equality which comes from (20 + 0)x(10 + 0) = 20x10 + 0x10 + 20x0 = 200 + 0 + 0 = 200.
In general, (a + 0)x(c + 0) is equal to axc + 0xc + ax0 (or axc + 0xc + ax0 + 0) BUT IS NOT EQUAL TO axc + 0xc + ax0 + 0x0 (for a different from zero and c different from zero).
Point 1-B.- Another important case I'd like to mention here is ax0 = a-a. In this axiom, a is every number except zero for 0x0 results in every number except zero whereas 0-0 results in zero.
Point 2.- MATHEMATICS IS NOT CONDITIONED BY WHAT OCCURES IN THE FIELD OF FACTUAL SCIENCES. IT IS AN INSTRUMENT OF THEM BUT IT IS NOT CONDITIONED BY THEM. Let's imagine three magnitudes A, B and C related between them, ALMOST always, like this: A = a, B = b, C = c, and axb = c. Note A is not exactly the same as a, B is not exactly the same as b, and C is not exactly the same as c. A, B and C are particular magnitudes whereas a is a multiplicand, b is a multiplier and c is a product. NOTHING should say that if A, B and C are equal to such a special number as zero (A = 0, B = 0 and C = 0) then a, b and c have to be equal to zero too.
To consider that zero multiplied by zero results in zero may usually be very useful IN SPITE OF BEING THE CONSIDERATION OF A FALSITY.
Correction: In Point 1-B. I wrote "In this axiom, a is every number except zero for...". I should have written "In this equality, a is every number except zero for...".
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