Indeterminate: the hidden power of 0 divided by 0

Welcome to another Mathologer video. Today I will present to you these crazy calculations. At school, the teachers said that if they didn't try to avoid these things, they would be severely cursed. set of rules that can be taught to the general public to prevent them from getting confused when they encounter this situation, but if you think this is enough, a lot of modern mathematics will not be implemented, so let me introduce them a little. So why do they tell you that it is not possible to divide 3 by 0? Or is there no definition of dividing 0 by 0? Let's first take a look at this number which is actually no problem for everyone. the only solution to this simple equation: 8x = 3 OK So let's replace 8 with 0 and see what happens.

We immediately get into trouble because there is no solution that satisfies this equation. The equation will always be "0 = 3". It must be wrong. So... avoid this. The ratio of the numbers now makes sense, right? Well... here's another problem. Stay away from that. For most people, it is enough to know the above knowledge, but if the mathematics we study only stops here, calculus will not appear in the world, and no one will know Isaac Newton. This is really surprising. Well, what exactly is calculus? Calculus basically consists of finding derivatives and integrals. I drew a nice function here and we want to find the derivative of this function at a specific value.

Everyone knows that this is very important. but it can be presented in a very simple way in geometry. The way it is presented is that the derivative of the function at this point is actually the slope of its tangent line. But we can't see directly in the function at a glance what the slope is. of its tangent line is The more or less convenient way of calculation is more like the secant line here and then... The method is that if I bring these two intersection points closer together, the closer they are, the more the secant line will become like a tangent line, then the slope of this secant line will be closer to the value we are interested in.

Okay, so how should I calculate the actual slope of the secant line? You can directly read the width and height here, and then. we can calculate the slope. We simply divide the height by the width. Now look what happens when I bring the two points together. Both the height and width approach zero. This time nothing terrible happens. We expected something terrible to happen, but it didn't. We simply get closer and closer to the slope of the tangent line. This seemed a little strange, but remember that the polynomial solution corresponds to. 0/0 can be any real number, so what's really happening is that we've created a very narrow range within which 0/0 can be a specific value.

And then you can see that when this range changes we get different numbers. So let's look at this special case. Let's look at the derivative of x^2 in 1/2. Okay, we need to figure out what the width and height are here. This will give us the second. point (Annotation: 1/2 + w) Then calculate the function values ​​of these two points to obtain (1/2)^2 and (1/2 + w)^2 respectively. Of course, the difference between them is so high. these two function values. Of course, the difference between these two function values ​​is high. Note that these two terms can be eliminated, so that we can calculate the height, we can also calculate the slope.

You can see that the slope... it... the numerator approaches zero and the denominator also approaches zero is nothing, but as long as it's not actually zero, we can eliminate w, and it will become this beautiful formula. By looking at this formula, it should be very obvious that the limit of this formula should be when w is close to zero. It is very obvious that the limit of this formula when w is close to zero is the slope we want to find. This means that the derivative of x^2 at 1/2 is equal to 1. This means that the derivative of x^2 at 1/2 is equal to 1.

Not bad, right? In fact, it's not just 1/2. , we can use any point. If we do this, we can find that the derivative of x^2 is 2x. Once we understand this point, in terms of calculation, everything is under our control. In terms of calculation, everything is under our control.