Indeterminate: the hidden power of 0 divided by 0

Indeterminate: the hidden power of 0 divided by 0

Welcome to another video by Mathologer Today I am going to introduce these crazy calculations At school the teacher would say If you don't manage to avoid these things Will be dreaded But there is actually a great set of rules that can be taught to the general public To prevent them from thinking about this situation. But if you think that ’s enough Many modern mathematics will not be realized So let me introduce them a little bit OK, so why would they tell you that 3 divided by 0 is not working? Or is 0 divided by 0? Let's take a look at this number that everyone must have no problem Three-eighths Mathematically 3/8 actually represents the only solution to this simple equation: 8x = 3 OK, let ’s replace 8 with 0 and see what happens.

We were in trouble right away Because no solution can satisfy this equation The equation will always become "0 = 3" which must be wrong So … the reason to avoid this number sounds reasonable now, right? What about 0 divided by 0? Here comes another problem All numbers here satisfy this equation This is another kind of trouble Hmm … let's stay away from it For most people it is enough to know the above knowledge But if the mathematics we study ends here Calculus will never appear in the world No one will ever know Isaac Newton It's really sad, right? OK so what exactly is calculus Calculus is basically finding derivatives, integrals I drew a nice function here And we want to find the derivative of this function at a specific value You all know how important this is in calculus But it can be represented in a very simple way geometrically That way of presentation is The derivative of the function at this point is actually the slope of its tangent But we can't directly see from the function what its tangent slope is The more convenient form of calculation is more like the secant here then…

the way is If I move these two points closer Then the closer they are This secant will become more tangent Then the slope of this secant The closer to the value we are interested in OK, how do I calculate the actual slope of the secant? You can directly read the width and height here Then we can calculate the slope by dividing the height by the width very easy Now see what happens when I bring the two points close Height and width are approaching zero So in the end it approaches the taboo number 0/0 But nothing terrible happened this time What terrible problems we should have expected But it didn't happen We are just getting closer The slope of this tangent It feels a little strange But remember the solution of the polynomial corresponding to 0/0 Can be any real number So what really happened here is We built a very narrow range Make 0/0 a specific value in it Then you can see When this range changes We will get different numbers So let's take a look at this special case Try to see the derivative of x ^ 2 at 1/2 Ok we need to find out the width and height Set width to w This will give us a second point (translation: 1/2 + w) Then calculate the function value of these two points Gives (1/2) ^ 2 and (1/2 + w) ^ 2 respectively So high is of course the difference between these two function values So high is of course the difference between these two function values Expand them Note that these two items can be eliminated, so you calculate high After having high We can also calculate the slope You can see the slope …

It … The numerator approaches zero and the denominator approaches zero But as long as it's not really zero We can eliminate w Then it will become this beautiful formula Just observe this formula It should be very obvious that the limit of this formula when w is close to zero It should be very obvious that the limit of this formula when w is close to zero Which 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 Right? Actually it's not just 1/2 We can actually use any point If we do this we will find that the derivative of x ^ 2 is 2x We can see that the derivative of x ^ 2 is 2x.

Once we understand this, With calculus everything is in our hands With calculus everything is in our hands .