How to Do the Sub Steps of Neutral Operations
Enter the following into a new worksheet in a New Workbook or just use pencil and paper. , To be discussed are the 5 basic Neutral Operations: 1) a+b = a*b = c; 2) a-b = a*b = c; 3) a+b = a/b = c; 4) a-b = a/b = c; and 5) a*b = a^b = c. In previous...
Step-by-Step Guide
-
Step 1: Enter the following into a new worksheet in a New Workbook or just use pencil and paper.
a+b+c+...+z = abc...z and abcdefghi = a^(bcdefghi).
Those will be referred to at the bottom in Related LifeGuide Hubs. , for the same two numbers a and b, a+b = a*b.
The algebraic sub-steps to write down and learn are: 1) a+b = a*b = c; 2) a+b-b = ab
- b; 3) a = b*(a-1) as b is factored out on the right side while the left side is simplified; 4) a/(a-1) = b as both sides are divided by (a-1) and the right side's (a-1)/(a-1) reduces to simply
1.
Now b has been isolated and defined in terms of a and
1.
Since it was also possible to begin by subtracting a instead of b, and because both addition and multiplication are commutative, it's also true that b/(b-1) = a.
And since ab = c, it is true also that a^2 / (a-1) = c and b^2 / (b-1) = c, which is very symmetrical.
It is well worth noting that the formula works wherever in Math or Physics one finds the relationship of c = ab, such as F = ma, or E = mc^2 or for a+b = c as in the Pythagorean Theorem, as a^2 + b^2 = c^2 can be set equal to a^2 * b^2 (or even a^2/b^2).
As a = b/(b-1), a is set to 5 and b then = 5/4. a+b = ab becomes 5+ 5/4 = 5 * 5/4 which both = 25/4.
Check: the hypothesis holds.
Neither a nor b may equal 1 lest 0 result in the denominator, for division by 0 is undefined. , The algebraic steps to write down and learn are: 1) a-b = a*b; 2) a-b+b = ab + b; 3) a = b*(a+1) as b is again factored out on the right while the left is simplified; 4) a/(a+1) = b after both sides are divided by (a+1 and simplified.
Now b has again been isolated and defined in terms of a and 1 but subtraction is not commutative so there is not a symmetrical relationship.
If a is subtracted to start with instead of adding b, one obtains 2) a-a-b =ab-a; 3)
-b = a*(b-1); 4)
-b/(b-1) = a and a has been isolated and defined in terms of b and
1. 5)
-b/(b-1) *
-1/-1 = b/(1-b) = a.
Letting b equal 5 again then with b/(1-b) = a, a =
-5/4 and
-5/4
- 5 =
-5/4 *5 because
-5/4
- 20/4 =
-25/4, check. , We need to be able to factor out a from the numerator, so we lose discretion.
The steps to write down and learn are as follows: 1) a+b = a/b and since we must factor out a on the right, we must subtract it on the left; 2) a-a+b = a/b
-a; 3) b = a*(1/b
-1) factoring out a on the right and simplifying the left; 4) b/(1/b
-1) = a as both sides have been divided by (1/b
-1) and a has been isolated and defined in terms of b and
1.
There is no symmetry.
Letting b = 5, for b/(1/b
-1) = a then, 5/(1/5
-1) = a = 5/(-4/5) =
-25/4 = a.
However, since a/b = c, it's possible to set c equal to /b so that c = 1/(1/b
- 1).
Letting b = 5 and for c = 1/(1/b
- 1), we have c = 1/(1/5
- 1) or 1/((-4/5) =
-5/4 = c.
Testing the hypothesis of a+b = a/b = c,
-25/4 + 5 = (-25/4)/5 becomes
-25/4 + 20/4 =
-25/20 which is
-5/4 = c, check..
And c = 1/(1/b
- 1) = 1/(1/5
-1) = 1/(-4/5) =
-5/4, check also. , The steps to write down and learn are as follows: 1) a-b = a/b and since a must be factored out on the right, it must be subtracted on the left; 2) s-s-b = a/b
- a; 3)
-b
- a*(1/b
- 1) factoring out a on the right and simplifying the left; 4)
-b//(1/b
- 1) = a can become, by multiplying the left by
-1/-1, b/(1
- 1/b) = a and a has been isolated and defined in terms of b and 1 by dividing through by (1/b
-1).
There is no symmetry.
However, since a/b = c, it's possible to set c equal to /b = c so that c = 1/(1
- 1/b) ... sort of the opposite of the above.
Letting b = 5 and for a = b/(1
- 1/b), a = 5/(1
-1/5) = 5/(4/5) = 25/4 (whereas above a equaled
-25/4).
Testing the hypothesis of a-b = a/b = c, 25/4
- 5 = (25/4)/5 = 25/4
- 20/4 = 25/20 = 5/4, check.
And c = 1/(1
- 1/b) or 1/(1
- 1/5) = 1/(4/5) = 5/4, check again. , Letting b=9, and for b^(1/(b-1)) = a then, 9^(1/(9-1)) = a and the left side evaluates in Excel to the value
1.31607401295249 = a.
Testing the hypothesis of a*b = a^b gives
1.31607401295249 * 9 =
1.31607401295249 ^ 9 and the first answer =
11.8446661165724 and the second value =
11.8446661165722, which is off just a smidgen, but acceptably so because of how Excel cuts off digits when calculating.
Because a*b = c and a = b^(1/(b-1)), then it is also true that b^(1/(b-1))*b or b^(1+(1/(b-1))) = c. 9^(1+(1/(9-1))) = 9^(9/8) and this answer =
11.8446661165724 = c, check. , Perhaps another time.
Or perhaps you will want to work those out.
The thrust of this article has been to simply cover the basics and now you have them.
Good going!, It means the point of first tangence when acceleration first touches mass, or light speed squared does.
It is the instantaneous addition and multiplication state and does nothing to invalidate the basic law involved., For more art charts and graphs, you might also want to click on Category:
Algebra, Category:
Mathematics, Category:
Spreadsheets or Category:
Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page. -
Step 2: To be discussed are the 5 basic Neutral Operations: 1) a+b = a*b = c; 2) a-b = a*b = c; 3) a+b = a/b = c; 4) a-b = a/b = c; and 5) a*b = a^b = c. In previous LifeGuide Hubs
-
Step 3: more complicated relations have been solved
-
Step 4: The first equation neutralizes Addition versus Multiplication
-
Step 5: The second equation neutralizes Subtraction versus Multiplication and is handled in much the same way as the first.
-
Step 6: The third equation is handled somewhat differently when neutralizing Addition versus Division.
-
Step 7: The fourth principle equation
-
Step 8: neutralizing Subtraction versus Division
-
Step 9: operates much like the third where we lose some discretion.
-
Step 10: Lastly
-
Step 11: we have the principle neutral operation of a*b = a^b
-
Step 12: which may be written down and learned from as follows: 1) a*b = a^b; 2) ab/a = (a^b)/a; 3) b = a^(b-1); 4) The root of (b-1) is taken of both sides so that b^(1/(b-1)) = a and a has been isolated and defined in terms of b and 1.
-
Step 13: Many other side-excursions are possible
-
Step 14: a+b = -a/b
-
Step 15: What does f = m+a or E = m+c^2 mean?
-
Step 16: Make use of helper articles when proceeding through this tutorial: See the Related LifeGuide Hubs below and the article How to Do the Sub Steps of Neutral Operations for a list of articles related to Excel
-
Step 17: Geometric and/or Trigonometric Art
-
Step 18: Charting/Diagramming and Algebraic Formulation relating to Neutral Operations.
Detailed Guide
a+b+c+...+z = abc...z and abcdefghi = a^(bcdefghi).
Those will be referred to at the bottom in Related LifeGuide Hubs. , for the same two numbers a and b, a+b = a*b.
The algebraic sub-steps to write down and learn are: 1) a+b = a*b = c; 2) a+b-b = ab
- b; 3) a = b*(a-1) as b is factored out on the right side while the left side is simplified; 4) a/(a-1) = b as both sides are divided by (a-1) and the right side's (a-1)/(a-1) reduces to simply
1.
Now b has been isolated and defined in terms of a and
1.
Since it was also possible to begin by subtracting a instead of b, and because both addition and multiplication are commutative, it's also true that b/(b-1) = a.
And since ab = c, it is true also that a^2 / (a-1) = c and b^2 / (b-1) = c, which is very symmetrical.
It is well worth noting that the formula works wherever in Math or Physics one finds the relationship of c = ab, such as F = ma, or E = mc^2 or for a+b = c as in the Pythagorean Theorem, as a^2 + b^2 = c^2 can be set equal to a^2 * b^2 (or even a^2/b^2).
As a = b/(b-1), a is set to 5 and b then = 5/4. a+b = ab becomes 5+ 5/4 = 5 * 5/4 which both = 25/4.
Check: the hypothesis holds.
Neither a nor b may equal 1 lest 0 result in the denominator, for division by 0 is undefined. , The algebraic steps to write down and learn are: 1) a-b = a*b; 2) a-b+b = ab + b; 3) a = b*(a+1) as b is again factored out on the right while the left is simplified; 4) a/(a+1) = b after both sides are divided by (a+1 and simplified.
Now b has again been isolated and defined in terms of a and 1 but subtraction is not commutative so there is not a symmetrical relationship.
If a is subtracted to start with instead of adding b, one obtains 2) a-a-b =ab-a; 3)
-b = a*(b-1); 4)
-b/(b-1) = a and a has been isolated and defined in terms of b and
1. 5)
-b/(b-1) *
-1/-1 = b/(1-b) = a.
Letting b equal 5 again then with b/(1-b) = a, a =
-5/4 and
-5/4
- 5 =
-5/4 *5 because
-5/4
- 20/4 =
-25/4, check. , We need to be able to factor out a from the numerator, so we lose discretion.
The steps to write down and learn are as follows: 1) a+b = a/b and since we must factor out a on the right, we must subtract it on the left; 2) a-a+b = a/b
-a; 3) b = a*(1/b
-1) factoring out a on the right and simplifying the left; 4) b/(1/b
-1) = a as both sides have been divided by (1/b
-1) and a has been isolated and defined in terms of b and
1.
There is no symmetry.
Letting b = 5, for b/(1/b
-1) = a then, 5/(1/5
-1) = a = 5/(-4/5) =
-25/4 = a.
However, since a/b = c, it's possible to set c equal to /b so that c = 1/(1/b
- 1).
Letting b = 5 and for c = 1/(1/b
- 1), we have c = 1/(1/5
- 1) or 1/((-4/5) =
-5/4 = c.
Testing the hypothesis of a+b = a/b = c,
-25/4 + 5 = (-25/4)/5 becomes
-25/4 + 20/4 =
-25/20 which is
-5/4 = c, check..
And c = 1/(1/b
- 1) = 1/(1/5
-1) = 1/(-4/5) =
-5/4, check also. , The steps to write down and learn are as follows: 1) a-b = a/b and since a must be factored out on the right, it must be subtracted on the left; 2) s-s-b = a/b
- a; 3)
-b
- a*(1/b
- 1) factoring out a on the right and simplifying the left; 4)
-b//(1/b
- 1) = a can become, by multiplying the left by
-1/-1, b/(1
- 1/b) = a and a has been isolated and defined in terms of b and 1 by dividing through by (1/b
-1).
There is no symmetry.
However, since a/b = c, it's possible to set c equal to /b = c so that c = 1/(1
- 1/b) ... sort of the opposite of the above.
Letting b = 5 and for a = b/(1
- 1/b), a = 5/(1
-1/5) = 5/(4/5) = 25/4 (whereas above a equaled
-25/4).
Testing the hypothesis of a-b = a/b = c, 25/4
- 5 = (25/4)/5 = 25/4
- 20/4 = 25/20 = 5/4, check.
And c = 1/(1
- 1/b) or 1/(1
- 1/5) = 1/(4/5) = 5/4, check again. , Letting b=9, and for b^(1/(b-1)) = a then, 9^(1/(9-1)) = a and the left side evaluates in Excel to the value
1.31607401295249 = a.
Testing the hypothesis of a*b = a^b gives
1.31607401295249 * 9 =
1.31607401295249 ^ 9 and the first answer =
11.8446661165724 and the second value =
11.8446661165722, which is off just a smidgen, but acceptably so because of how Excel cuts off digits when calculating.
Because a*b = c and a = b^(1/(b-1)), then it is also true that b^(1/(b-1))*b or b^(1+(1/(b-1))) = c. 9^(1+(1/(9-1))) = 9^(9/8) and this answer =
11.8446661165724 = c, check. , Perhaps another time.
Or perhaps you will want to work those out.
The thrust of this article has been to simply cover the basics and now you have them.
Good going!, It means the point of first tangence when acceleration first touches mass, or light speed squared does.
It is the instantaneous addition and multiplication state and does nothing to invalidate the basic law involved., For more art charts and graphs, you might also want to click on Category:
Algebra, Category:
Mathematics, Category:
Spreadsheets or Category:
Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.
About the Author
Rachel Rogers
Writer and educator with a focus on practical lifestyle knowledge.
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