I Learned How to Divide by Zero (Don't Tell Your Teacher)

you have been lied to all

your

life if there is one thing they told you that you cannot do at all it is

divide

by

zero

, but the code has been cracked and by the end of this video you too will know how to

divide

by

zero

ant is a nice strong word, especially in mathematics, not long ago it was that you can't take the square root of a negative number, before you couldn't have irrational numbers, soon mathematicians discovered that wasn't the case, these things just weren't defined, we define the square root of negative one as i and we introduced irrational numbers into our number system.

You may have been told that division by zero is not defined. In fact, the slope of a vertical line is an indefinite slope and we see that it appears in the slope formula. for line equations when we divide by zero now I don't know about u but I think it's time to define what division by zero really means let's look at why dividing by zero has been so difficult in the first place a way to try dividing by zero is dividing one between numbers that are very close to zero, for example one divided by one one divided by half one divided by a quarter one divided by a tenth and so on making the denominator of the fraction always smaller, which What you'll notice if you use a calculator is that this number starts increasing pretty quickly, it gets bigger and bigger, which makes sense because we're dividing it by something smaller and smaller, so a natural assumption might be that 1 divided by 0 is infinite, but there is a problem if we do it from the other side if we start dividing one by small numbers that are negative one divided by negative one one divided by negative half one divided by negative one quarter and so on we will see that the quantity returns to get bigger and bigger in the opposite direction toward negative infinity; in fact, what we are doing here is a calculus problem.

Actually, we're taking the limit of the function one over x from the left and right of zero and we get two different ones. answers on the right we get positive infinity, on the left we get negative infinity, and generally if you were in a calculus class you would say that the limit doesn't exist, so while it may seem like infinity is a good answer for What 1 divided by zero is that we can't have two definitions for a thing and this might be enough to stop many people but not us, maybe what's really wrong is the way we think in our two dimensional Cartesian plane in instead of having a top and a bottom an infinite plus and minus maybe there really shouldn't be a top and bottom introduce a stereographic projection this is a system where we impose a sphere on top of our regular two-dimensional coordinate system with the top bottom of the sphere sitting at the origin, the top of the sphere sitting at one and we map each point in the two-dimensional plane to this sphere by drawing a line across the top of the sphere to a point.

This is a little complicated, it is used in some areas of geometry some areas of complex analysis but what happens is that we are folding the entire two dimensional plane into a sphere, there is a one to one correspondence with all the points and what happens is that all infinities go in any direction on the map towards a singular upper point that represents infinity, so if we think about our problem now, if we take the plus and minus infinity, we are actually joining them together, like joining the ends of a wire to make a circle and this plus and minus infinity we consider infinity, the only true unsigned infinity, so now we have one divided by zero equals infinity and we are not considering plus or minus, this is just the problem of infinity no sign resolved true, not quite, you see that infinity is an idea, it's not actually a real number, but if we are going to define it this way, we would still like to treat infinity the way we intuitively think of it.

You could think of the kids on the playground trying to think of the biggest number one will say ten the other will say one hundred the other will say infinity and then the next person will say infinity plus one and then the person who said infinity will say oh no, no you can say infinity plus 1 because infinity plus 1 is infinity, we want infinity to work like this, it has an absorbing nature, where if you add infinity to something it's still infinity, but this creates a little problem if we say 1 divided by 0. it is equals infinity and we want to keep our algebra rules in place, it would make sense that 1 is equal to 0 times infinity 0 times anything is still 0, so instead of saying zero we could say zero times two, so now that we have one equals zero times two infinity and zero times infinity we just said it equals one so one equals one times two or one equals two now that's not entirely correct and we didn't have to use two, we could have used any number three four five pick

your

favorite and this says that every number is equal to every other number, so that's not a very convincing definition, so now our problem is really the term zero multiplied by infinity and If you've taken a calculus class, you may know this as an indeterminate form zero multiplied by infinity infinity minus infinity zero over zero or infinity over infinity, these are all classifications of indeterminate forms.

Typically we would do limit problems and find out what they are in the context of the function and the limit. Enter wheel algebra, where we define what zero over zero is and all these other indeterminates. forms we define them as nullity or I have heard it called background and the way this new element works is that it is an absorbing element that is even more powerful, in fact, every time we interact with it it simply becomes that element, any element more this. nullity becomes nullity any element times nullity becomes nullity and somehow we have solved our problem with this final absorbing element, so we have solved all our problems with the introduction of this element, well, somehow We lose some things.

Don't we do it if something multiplied by nullity is nullity? Well, that means that zero multiplied by null is the tune and we lose the fact that anything multiplied by zero is zero. What about anything about itself is one? If that is nullity, then nullity upon nullity. is nullity and we lose it too, especially x minus x, which is traditionally zero, but nullity minus nullity is nullity, so we have to settle for the relationship x minus x equals zero equation as convenient as the one that exists. a bunch more little nuances of abstract algebra to go along with the wheel theory and I encourage you to Google them and review them to answer the question: can we divide by zero?

Yes, define one divided by zero as unsigned infinity, define all of those. indeterminate ways to be the melody and apply the rules to work with nullity, the only drawback is that we limit ourselves a little with our algebra taking those restrictions into account, perhaps a better question is if we divide by zero, I will leave it for you to decide