Here's something to consider: At around 24:40 into the video when you stated that those who have a background in mathematics that they wouldn't like the idea of 1/0 being infinity. This all comes from the years of people being taught that division by 0 is undefined. I have a very good background in Math, Physics, and other fields of the sciences. Yet, I will argue that division by 0 is NOT undefined. I would claim that may be ambiguous, but not undefined. We are taught that division by 0 is an error, that it cannot be done. I beg to differ. I have experience in C/C++, a little bit of C#, Python and some Assembly, I have no experience within Julia, yet I think the way that Julia is treating 1/0 as infinity is quite accurate. Why? Division is defined as the inverse of multiplication. Multiplication is defined as repeated addition. This implicitly implies that Division isn't just the inverse of multiplication but is also the same as repeated Subtraction. This is one of the key points towards understanding this. Subtraction is defined as either the inverse of Addition or it's simply Addition with the multiplication of the additive inverse. For example: A - B = A + (-B) which is the same as: A + (-1 * B). Here we have an implicit multiplication of (-1). And again, multiplication is repeated addition. Thus, Subtraction is also Repeated Addition in some sense. The only difference (pun intended here) is that the repeated addition in the case of subtraction only occurs Once. This is because multiplication by 1 or (-1) is a single operation or transformation. The +/- determines the direction or orientation. Consider these two expressions or equations. (1 * -1) AND (1-2). Here we have an expression based on the multiplication operator and we have one that is based on the subtraction operator. Yet these two expressions are Equivalent classes as they yield the same result. The transformations being applied might be different, but the destination point is the same. How does this constitute division by 0 not being undefined? We need to also consider some of the basic properties of Arithmetic such as the Additive Identity and the Commutative properties. Additive Identity: 1 + 0 = 1 0 + 1 = 1 1 - 0 = 1 These three equations satisfy the additive identity. The only exception is the expression 0-1 = -1. This does not directly satisfy the Additive Identity because 1 != (-1). Yet it does satisfy the Additive Inverse Property: A * (-1) = (-A) OR (-A) * 1 = (-A). We'll come back to this in a moment as we need to show how this relates to and preserves the Commutative property of A+B = B+A For Addition: 1 + 0 == 0 + 1 is True For Subtraction: 1 - 0 == 0 - 1 is False What's going on here? When evaluating simple arithmetic expressions, we are typically taught to only be concerned with the Output or Result of the operation. We are blindly or ignorantly taught to ignore what happens or takes place when we attempt or try to apply a given operation. This is one of the points within mathematics that is overlooked. With these four base cases I will show a table to demonstrate the importance of what we tend to overlook or bypass. This table will show the basic Additive Identities of both Addition and Subtraction, but what we need to ask here is if the operation being applied is Generative or Not. What does it mean to be Generative? Within the binary operators of both addition and subtraction, we have two operands, a LHS (Left Hand Side) and a RHS (Right Hand Side) respectively. We typically read or evaluate expression from Left to Right by generally accepted conventions. Within the context of this accepted convention and framework, we can easily ask: When applying said operator between LHS and RHS, the question is Does RHS transform or change LHS? If it does, this expression is Generative, if not, it is Non-Generative analogous to a No Op within computer science meaning that a transformation or translation did not occur. Here's the basic table: (1+0) = 1 Non Generative : 0 does not change 1 via addition (0+1) = 1 Generative : 1 does change 0 via addition (1-0) = 1 Non Generative : 0 does not change 1 via subtraction (0-1) = -1 Generative : 1 or -1 does change 0 via subtraction or addition by -1. What does this have to do with division by 0? Here we need to look at the values of 0 and 1 as points on a plane or a vector within a field. When we do this, we can easily see a bunch of patterns that emerge that are all connected through basic Algebra from Linear Equations and their Slopes to the actual Trigonometric Functions, the Dot Product, and more. We can consider the value of 1 as being the vector (1,0) The line segment from the origin (0,0) to this point (1,0) has a slope of 0. From the slope-intercept form of the line: y = mx+b we know that the y-intercept is b and that the slope defined as rise/run is m and can be calculated by any two points (x1,y1) and (x2,y2) on the line from the formula (y2-y1)/(x2-x1) = deltaY/deltaX. This will be important to keep in mind. We can take the two points (0,0) and (1,0) and plug it into the slope formula and we end up with a slope of 0. This is because this line is horizontal, or it is parallel with the horizontal. To keep things simple and throughout the rest of this we are going to fix the y-intercept to be 0. This will simplify the slope-intercept form of the line y =mx+b to simply y =mx. Where does the trigonometric functions come in? How does this show that division by 0 isn't undefined? If we take the point (1,0) which is also the value of 1. We know that it's slope m from the origin and its magnitude points is 0/1 which evaluates to 0. Let's either multiply this by -1 or subtract it by 2. What point do we end up at? In both cases, we end up at the point (-1,0). This also has a slope of 0. This subtraction of 1 by 2 to get to -1 is a horizontal translation where the multiplication by -1 is also a horizontal translation but is also a rotation. Multiplying by -1 is the same as rotating this vector about the origin (0,0) by 180 degrees or PI radians. Where does division by 0 come into play? What about the Trig functions? Okay what happens when we take this vector (1,0) and we rotate it by say 90 degrees or PI/2 radians; 1/2 the rotation of multiplying by -1? This has the same exact effect by multiplying by i where the head of the vector that extends from the origin (0,0) and points at (1,0) is now rotated with its tail fixed at (0,0) and it now points at (0,1). The original slope-intercept form of the equation with b equal to 0 for the line segment defined by (0,0) and (1,0) is simply y = 0 since y = 0*x + 0 = 0. The slope m is 0 as it has the fraction 0/1. The new slope of this line when it is rotated to (0,1) then has the y-intercept form of y = infinity simply because the slope has the form 1/0. This is Vertical Slope. It is tangent, perpendicular, orthogonal, normal to the x-axis. This is a 90 Degree or PI/2 rotation. It's the same thing as multiplying it by i. (continued...)
(...continued) Within mathematics we are typically taught about sets of numbers. We start off with counting, whole, integers, rational, irrationals, reals, imaginaries, and the complex numbers, etc. Instead of thinking of numbers within these different types of sets. Here I'm going to stick with Integer Arithmetic for simplicity and for the base cases, but this can be extended to floating point as well. Instead of referring to the 2D Cartesian Real plane or referring to the Complex plane. Instead, I propose to look at the x-axis as being the Horizontals (similar to the Real numbers within the complex plane), and the y-axis or i-axis as being the Verticals, or Perpendiculars, Orthogonals or Normals. And the entire field instead of being Complex simply referred as to being the Set of Rotational Numbers. Thinking of it in these terms will end up making things more intuitive as well as resolving many common issues or misconceptions within mathematics. I wouldn't change the common notation that is used within complex arithmetic such as 3 + 5i. This notation is still fine. However, instead of calling them complex or calling the i-values imaginary, I'd prefer to call them Rotational and Perpendiculars respectively. What does this have to do with division by 0 and what about the Trig functions? If we take this rotation of 90 degrees or PI/2 radians and divide it by 2, we end up having a rotation by 45 degrees or PI/2 radians. This arbitrary unit vector will now be at the point (sqrt(2)/2, sqrt(2)/2) and it has a slope of 1. This simply the line y = x. Here we can easily see that there is a direct correlation and relationship between the slope or gradient of a line and the tangent function. We can easily see that within the simplified slope equation: dy/dx is simply sin(t)/cos(t) = tan(t). The reason I state that the numbers are rotational is because they are not scalar as we are taught. They are actually multidimensional because they are rotational. Having this type of perspective and understanding becomes clearer when we understand this primarily due to Modulus Arithmetic: (Integer - Remainder Division). This is where we can see that Division by 0 is not undefined. It may be the case that it can be ambiguous, and it is purely ambiguous when we are only concerned with the Quotient and completely ignore the Remainder. Most operators within mathematics specifically binary operators typically have 2 operands and yields a single result, however when it comes to division, it doesn't yield one result, it in fact yields two results. We have the Quotient and the Remainder. So, what is division by 0? It is repeated subtraction, or repeated addition with the additive inverse or the Multiplicative Inverse Property implicitly applied. When we evaluate and or perform division to find the Quotient, we are asking how many times we can successfully subtract the Divisor or Denominator from the Dividend or the Numerator. Then depending on the temporary results, we have conditional checks that have to be applied. We check to see if the temporary result is less than the divisor and still greater than 0. If this is the case, then we can count up how many times we subtracted, and this becomes our Quotient, and the last temporary becomes our Remainder. In the case that the subtraction results to or is equal to 0, then we terminate, count up the number of times we subtracted where this is our Quotient, and our Remainder is 0 because we have an even or perfect division. There are some other cases to where the subtraction performed becomes negative, but that is beyond the scope of this. For the simple case of division by 0 we can look at 1/0 and try to perform long division:
1 -0 --- 1 -0 --- 1 ... Hmm we can successfully subtract 0 from 1 forever. This becomes an infinite loop where the counter of the loop, the Quotient Ramps Up to Infinity. It is also a No-Operation. Remember above: 1+0 and 1-0 Are Non-Generative? When we see 1/0 we can treat this as simply being either of the following: 1 + 0 + 0 + 0 + 0 ... where we continuously add 0 never changing 1. 1 - 0 - 0 - 0 - 0 - 0 ... where we continuously subtract 0 never changing 1. This is well behaved and well defined. What makes it ambiguous? When only considering the Quotient and completely ignore the Remainder we can see the following: 1/0 = Q = Infinity 2/0 = Q = Infinity 3/0 = Q = Infinity ... N/0 = Q = Infinity This is where the ambiguity comes in. We cannot undo this operation without more information. There's an infinite family of expressions that all tend to infinity. How can we resolve this ambiguity? It's quite simple! We Can Not Ignore the Remainder! Okay but how does division by 0 produce a remainder? Consider 1/0 from the long division. Our Quotient is ramping up to Infinity. However, for every single iteration of the subtraction loop, the Numerator or Dividend is left Unchanged since there is no transformation being applied. So, in the event of division by 0 we can see that the remainder is the Numerator. We can observe this new table where we do not ignore the Remainders: 1/0 = Q = Inf, R = 1 2/0 = Q = Inf, R = 2 3/0 = Q = Inf, R = 3 ... N/0 = Q = Inf, R = N Within Division by 0, other than the only exception being when the numerator is 0 as this does give us the indeterminate form 0/0 where this is simply the Origin (0,0) and that it is the Zero Vector a 0D Point where it is only relative to itself. This form of 0/0 is a point of rotation, a point of reflection a point of symmetry and so on. With this special case being considered, I then propose the convention that within division by 0 when the numerator or dividend is NOT zero that the Quotient itself will always be Infinity. The Remainder will always be the Absolute Value or Magnitude of the Numerator or Dividend and that the Quotient will take on the (Sign) of the Numerator or Dividend. For example: -42/0 = Q = -Inf, R = 42 With this type of convention, we can now easily distinguish division by 0 based on the remainder. Here the Quotient of being +/- Infinity strictly implies that this is Vertical Slope which is Perpendicular to the Horizontal. The Sign of the Infinity implies the direction we are heading Up or Down. The Remainder itself the absolute value of the Numerator or Dividend is simply where we are at on the vertical. How many steps or rungs up the ladder we are. This is division by 0. It is Not Undefined! Ambiguous sure it can be, but not always. Undefined? No! It's well defined and well structured. One of the misleading things to fully understanding this is to better understand what 0 really is. Zero itself is both even and an integer. Zero is Not Positive Nor is it Negative. And zero is also Not a Number. It is the Opposite of a Number. It is the Antithesis of a Number. It is Empty, Void, Null. In some sense Zero is a Type of Infinity but it's the Opposite of Infinity. Consider the Tangent Function at 90 Degrees. There's a vertical asymptote that tends off to both + and - infinity. This section of the tangent function (within this particular period) is a smooth and continuous curve that goes on forever. And this pattern repeats itself horizontally. So here, when we have values or functions that tend towards +/-infinity away from zero these are Divergent. When we have things that tend towards 0 or some other discrete value which is pointing to that value which happens to be a 0D Arbitrary Point are Convergent. We just have to consider the way we've been looking at perceiving and treating 0, and how it associates with all other non-zero values based on some operation that may or may not result in a transformation. Every non-zero value is unequivocally without a doubt, orthogonal, perpendicular, normal, tangent to zero, they are 90 degrees separated. This is why we see a state change of the vector (1,0) which has a slope of 0/1 to becoming (0,1) with a slope of 1/0 when we rotate it by 90 degrees or PI/2 radians. This is the exact difference in horizontal translation between the Sine and Cosine functions in which the Tangent function is defined as Sine/Cosine. And there you have it! Division by 0 is NOT UNDEFINED! I Just Now Defined It and Demonstrated It! All Just Food For Thought! Challenge Everything! Question Everything, Put It To The Test!
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Great Video: Keep up the good work!
Here's something to consider:
At around 24:40 into the video when you stated that those who have a background in mathematics that they wouldn't like the idea of 1/0 being infinity. This all comes from the years of people being taught that division by 0 is undefined. I have a very good background in Math, Physics, and other fields of the sciences. Yet, I will argue that division by 0 is NOT undefined. I would claim that may be ambiguous, but not undefined. We are taught that division by 0 is an error, that it cannot be done. I beg to differ.
I have experience in C/C++, a little bit of C#, Python and some Assembly, I have no experience within Julia, yet I think the way that Julia is treating 1/0 as infinity is quite accurate.
Why?
Division is defined as the inverse of multiplication. Multiplication is defined as repeated addition. This implicitly implies that Division isn't just the inverse of multiplication but is also the same as repeated Subtraction. This is one of the key points towards understanding this. Subtraction is defined as either the inverse of Addition or it's simply Addition with the multiplication of the additive inverse.
For example: A - B = A + (-B) which is the same as: A + (-1 * B). Here we have an implicit multiplication of (-1). And again, multiplication is repeated addition. Thus, Subtraction is also Repeated Addition in some sense. The only difference (pun intended here) is that the repeated addition in the case of subtraction only occurs Once. This is because multiplication by 1 or (-1) is a single operation or transformation. The +/- determines the direction or orientation.
Consider these two expressions or equations. (1 * -1) AND (1-2). Here we have an expression based on the multiplication operator and we have one that is based on the subtraction operator. Yet these two expressions are Equivalent classes as they yield the same result. The transformations being applied might be different, but the destination point is the same.
How does this constitute division by 0 not being undefined?
We need to also consider some of the basic properties of Arithmetic such as the Additive Identity and the Commutative properties.
Additive Identity:
1 + 0 = 1
0 + 1 = 1
1 - 0 = 1
These three equations satisfy the additive identity. The only exception is the expression 0-1 = -1. This does not directly satisfy the Additive Identity because 1 != (-1). Yet it does satisfy the Additive Inverse Property: A * (-1) = (-A) OR (-A) * 1 = (-A).
We'll come back to this in a moment as we need to show how this relates to and preserves the Commutative property of A+B = B+A
For Addition:
1 + 0 == 0 + 1 is True
For Subtraction:
1 - 0 == 0 - 1 is False
What's going on here?
When evaluating simple arithmetic expressions, we are typically taught to only be concerned with the Output or Result of the operation. We are blindly or ignorantly taught to ignore what happens or takes place when we attempt or try to apply a given operation. This is one of the points within mathematics that is overlooked.
With these four base cases I will show a table to demonstrate the importance of what we tend to overlook or bypass. This table will show the basic Additive Identities of both Addition and Subtraction, but what we need to ask here is if the operation being applied is Generative or Not. What does it mean to be Generative? Within the binary operators of both addition and subtraction, we have two operands, a LHS (Left Hand Side) and a RHS (Right Hand Side) respectively. We typically read or evaluate expression from Left to Right by generally accepted conventions. Within the context of this accepted convention and framework, we can easily ask: When applying said operator between LHS and RHS, the question is Does RHS transform or change LHS? If it does, this expression is Generative, if not, it is Non-Generative analogous to a No Op within computer science meaning that a transformation or translation did not occur.
Here's the basic table:
(1+0) = 1 Non Generative : 0 does not change 1 via addition
(0+1) = 1 Generative : 1 does change 0 via addition
(1-0) = 1 Non Generative : 0 does not change 1 via subtraction
(0-1) = -1 Generative : 1 or -1 does change 0 via subtraction or addition by -1.
What does this have to do with division by 0? Here we need to look at the values of 0 and 1 as points on a plane or a vector within a field. When we do this, we can easily see a bunch of patterns that emerge that are all connected through basic Algebra from Linear Equations and their Slopes to the actual Trigonometric Functions, the Dot Product, and more.
We can consider the value of 1 as being the vector (1,0)
The line segment from the origin (0,0) to this point (1,0) has a slope of 0.
From the slope-intercept form of the line: y = mx+b we know that the y-intercept is b and that the slope defined as rise/run is m and can be calculated by any two points (x1,y1) and (x2,y2) on the line from the formula (y2-y1)/(x2-x1) = deltaY/deltaX. This will be important to keep in mind.
We can take the two points (0,0) and (1,0) and plug it into the slope formula and we end up with a slope of 0. This is because this line is horizontal, or it is parallel with the horizontal. To keep things simple and throughout the rest of this we are going to fix the y-intercept to be 0. This will simplify the slope-intercept form of the line y =mx+b to simply y =mx.
Where does the trigonometric functions come in? How does this show that division by 0 isn't undefined?
If we take the point (1,0) which is also the value of 1. We know that it's slope m from the origin and its magnitude points is 0/1 which evaluates to 0.
Let's either multiply this by -1 or subtract it by 2. What point do we end up at? In both cases, we end up at the point (-1,0). This also has a slope of 0.
This subtraction of 1 by 2 to get to -1 is a horizontal translation where the multiplication by -1 is also a horizontal translation but is also a rotation. Multiplying by -1 is the same as rotating this vector about the origin (0,0) by 180 degrees or PI radians.
Where does division by 0 come into play? What about the Trig functions?
Okay what happens when we take this vector (1,0) and we rotate it by say 90 degrees or PI/2 radians; 1/2 the rotation of multiplying by -1? This has the same exact effect by multiplying by i where the head of the vector that extends from the origin (0,0) and points at (1,0) is now rotated with its tail fixed at (0,0) and it now points at (0,1).
The original slope-intercept form of the equation with b equal to 0 for the line segment defined by (0,0) and (1,0) is simply y = 0 since y = 0*x + 0 = 0. The slope m is 0 as it has the fraction 0/1.
The new slope of this line when it is rotated to (0,1) then has the y-intercept form of y = infinity simply because the slope has the form 1/0. This is Vertical Slope. It is tangent, perpendicular, orthogonal, normal to the x-axis. This is a 90 Degree or PI/2 rotation. It's the same thing as multiplying it by i.
(continued...)
(...continued)
Within mathematics we are typically taught about sets of numbers. We start off with counting, whole, integers, rational, irrationals, reals, imaginaries, and the complex numbers, etc.
Instead of thinking of numbers within these different types of sets. Here I'm going to stick with Integer Arithmetic for simplicity and for the base cases, but this can be extended to floating point as well. Instead of referring to the 2D Cartesian Real plane or referring to the Complex plane. Instead, I propose to look at the x-axis as being the Horizontals (similar to the Real numbers within the complex plane), and the y-axis or i-axis as being the Verticals, or Perpendiculars, Orthogonals or Normals. And the entire field instead of being Complex simply referred as to being the Set of Rotational Numbers. Thinking of it in these terms will end up making things more intuitive as well as resolving many common issues or misconceptions within mathematics. I wouldn't change the common notation that is used within complex arithmetic such as 3 + 5i. This notation is still fine. However, instead of calling them complex or calling the i-values imaginary, I'd prefer to call them Rotational and Perpendiculars respectively.
What does this have to do with division by 0 and what about the Trig functions?
If we take this rotation of 90 degrees or PI/2 radians and divide it by 2, we end up having a rotation by 45 degrees or PI/2 radians. This arbitrary unit vector will now be at the point (sqrt(2)/2, sqrt(2)/2) and it has a slope of 1. This simply the line y = x.
Here we can easily see that there is a direct correlation and relationship between the slope or gradient of a line and the tangent function.
We can easily see that within the simplified slope equation: dy/dx is simply sin(t)/cos(t) = tan(t).
The reason I state that the numbers are rotational is because they are not scalar as we are taught. They are actually multidimensional because they are rotational. Having this type of perspective and understanding becomes clearer when we understand this primarily due to Modulus Arithmetic: (Integer - Remainder Division).
This is where we can see that Division by 0 is not undefined. It may be the case that it can be ambiguous, and it is purely ambiguous when we are only concerned with the Quotient and completely ignore the Remainder.
Most operators within mathematics specifically binary operators typically have 2 operands and yields a single result, however when it comes to division, it doesn't yield one result, it in fact yields two results. We have the Quotient and the Remainder.
So, what is division by 0? It is repeated subtraction, or repeated addition with the additive inverse or the Multiplicative Inverse Property implicitly applied.
When we evaluate and or perform division to find the Quotient, we are asking how many times we can successfully subtract the Divisor or Denominator from the Dividend or the Numerator. Then depending on the temporary results, we have conditional checks that have to be applied.
We check to see if the temporary result is less than the divisor and still greater than 0. If this is the case, then we can count up how many times we subtracted, and this becomes our Quotient, and the last temporary becomes our Remainder.
In the case that the subtraction results to or is equal to 0, then we terminate, count up the number of times we subtracted where this is our Quotient, and our Remainder is 0 because we have an even or perfect division.
There are some other cases to where the subtraction performed becomes negative, but that is beyond the scope of this.
For the simple case of division by 0 we can look at 1/0 and try to perform long division:
1
-0
---
1
-0
---
1
... Hmm we can successfully subtract 0 from 1 forever. This becomes an infinite loop where the counter of the loop, the Quotient Ramps Up to Infinity. It is also a No-Operation. Remember above: 1+0 and 1-0 Are Non-Generative?
When we see 1/0 we can treat this as simply being either of the following:
1 + 0 + 0 + 0 + 0 ... where we continuously add 0 never changing 1.
1 - 0 - 0 - 0 - 0 - 0 ... where we continuously subtract 0 never changing 1.
This is well behaved and well defined. What makes it ambiguous?
When only considering the Quotient and completely ignore the Remainder we can see the following:
1/0 = Q = Infinity
2/0 = Q = Infinity
3/0 = Q = Infinity
...
N/0 = Q = Infinity
This is where the ambiguity comes in. We cannot undo this operation without more information. There's an infinite family of expressions that all tend to infinity.
How can we resolve this ambiguity? It's quite simple! We Can Not Ignore the Remainder! Okay but how does division by 0 produce a remainder?
Consider 1/0 from the long division. Our Quotient is ramping up to Infinity. However, for every single iteration of the subtraction loop, the Numerator or Dividend is left Unchanged since there is no transformation being applied. So, in the event of division by 0 we can see that the remainder is the Numerator.
We can observe this new table where we do not ignore the Remainders:
1/0 = Q = Inf, R = 1
2/0 = Q = Inf, R = 2
3/0 = Q = Inf, R = 3
...
N/0 = Q = Inf, R = N
Within Division by 0, other than the only exception being when the numerator is 0 as this does give us the indeterminate form 0/0 where this is simply the Origin (0,0) and that it is the Zero Vector a 0D Point where it is only relative to itself. This form of 0/0 is a point of rotation, a point of reflection a point of symmetry and so on. With this special case being considered, I then propose the convention that within division by 0 when the numerator or dividend is NOT zero that the Quotient itself will always be Infinity. The Remainder will always be the Absolute Value or Magnitude of the Numerator or Dividend and that the Quotient will take on the (Sign) of the Numerator or Dividend.
For example:
-42/0 = Q = -Inf, R = 42
With this type of convention, we can now easily distinguish division by 0 based on the remainder. Here the Quotient of being +/- Infinity strictly implies that this is Vertical Slope which is Perpendicular to the Horizontal. The Sign of the Infinity implies the direction we are heading Up or Down. The Remainder itself the absolute value of the Numerator or Dividend is simply where we are at on the vertical. How many steps or rungs up the ladder we are.
This is division by 0. It is Not Undefined! Ambiguous sure it can be, but not always. Undefined? No! It's well defined and well structured.
One of the misleading things to fully understanding this is to better understand what 0 really is.
Zero itself is both even and an integer. Zero is Not Positive Nor is it Negative. And zero is also Not a Number. It is the Opposite of a Number. It is the Antithesis of a Number. It is Empty, Void, Null.
In some sense Zero is a Type of Infinity but it's the Opposite of Infinity. Consider the Tangent Function at 90 Degrees. There's a vertical asymptote that tends off to both + and - infinity. This section of the tangent function (within this particular period) is a smooth and continuous curve that goes on forever. And this pattern repeats itself horizontally.
So here, when we have values or functions that tend towards +/-infinity away from zero these are Divergent. When we have things that tend towards 0 or some other discrete value which is pointing to that value which happens to be a 0D Arbitrary Point are Convergent.
We just have to consider the way we've been looking at perceiving and treating 0, and how it associates with all other non-zero values based on some operation that may or may not result in a transformation. Every non-zero value is unequivocally without a doubt, orthogonal, perpendicular, normal, tangent to zero, they are 90 degrees separated.
This is why we see a state change of the vector (1,0) which has a slope of 0/1 to becoming (0,1) with a slope of 1/0 when we rotate it by 90 degrees or PI/2 radians. This is the exact difference in horizontal translation between the Sine and Cosine functions in which the Tangent function is defined as Sine/Cosine.
And there you have it! Division by 0 is NOT UNDEFINED! I Just Now Defined It and Demonstrated It!
All Just Food For Thought! Challenge Everything! Question Everything, Put It To The Test!