Видео

Thanks for your comment! As for Julia, a lot of progress has been made on "TTFX" (Time to first X - i.e. first plot takes a while due to JIT). I haven't measured it recently but it feels like it's less than 10-20 seconds now and it's only paid once. This is better due to better package precompilation. Adding new packages still causes a lot of precompilation, but this usually happens rarely. If you are using a REPL based workflow (instead of running scripts) then you usually pay this only once/twice per day so it doesn't feel that bad at all. In the next few videos, I'll talk about an optimal workflow to try and minimise the precompilation cost so TTFX doesn't feel like too much of an issue.

Definitely agreed that these "micro benchmarks" are limited in what they can teach us. Correctly (and fairly) benchmarking is difficult, and next week I'll be releasing a video on that topic. In this example, the main take away is the difference between interpreted Python and compiled code, which are orders of magnitude different. You're right to be skeptical about performance comparisons - statements like language X is faster than language Y aren't very useful without looking at a specific example and trying to explain why differences arise.

Many people get a bad impression of Julia due to the JIT compilation - but many issues have improved (and are still improving) with latency from JIT. A lot of the issue can also be avoided by using a good workflow or at least mitigated.

(...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!

I think what he meant is that, due to the inevitable rounding errors when representing irrational numbers with a finite number of bits, associating numbers in different ways may lead to different results. I can't come up with an example off the top of my head, though.

Here is a classic example of the challenges with floating point rounding. a = 10^(15). # large number b = 10^(-15) # small number. (a+b)-a == b # evaluates to false. (a+b) is rounded to a and then subtracting a gives 0.0 which is not equal b. (a+b) == a # evaluates to true since a+b is rounded to a.

rounding error of floating point numbers