#Infinity times 0 how to

They also represent all real numbers that would round to something smaller than the smallest subnormal. The floating point zeros not only represent the real number zero.

#Infinity times 0 how to

In the next post, I’ll talk about how to carefully tell which set of things is bigger, and then show that some surprisingly different things are the same size.It is easier to understand the behavior of IEEE 754 floating point zeros and infinities if you do not think of them as being literally zero or infinite. After our lucky guest came, we had one more guest and the same number of rooms, but each guest still had their own room. When we started, we had infinitely many guests in infinitely many rooms. Each guest did so and the hotel worker put the new guest up in room number one. They called each room and told them to pack up and move to the room with the next higher number on it. Fortunately for her, the hotel has a policy for such circumstances. One night, every room was filled, when an unfortunate traveler came in begging for a room for the night. The Hitchhiker’s Hotel is the largest hotel in the galaxy. But while is after, is it bigger than ? How about or ? We also have numbers that are after infinity. Using this idea, we’ve come up with a number representing infinity. If he is traveling at 10 m/s (Olympic sprinter pace), the first cycle―100 meters―takes ten seconds, the first two cycles―110 meters total―takes 11 seconds, while the first cycles takes 11.1111… seconds (or 100/9, to be exact).Īnd so, Achilles can quickly catch the tortoise. How long does it take for Achilles to catch the tortoise? And since he’s getting infinitely close during the first finitely many cycles, at the -th cycle, Achilles has exactly caught up to the tortoise. Source: Grandjean, Martin (2014) Henri Bergson et les paradoxes de Zénon : Achille battu par la tortue ?Īfter 20 cycles of this, Achilles has still not caught up, though he is closer than the diameter of an electron.Īchilles is still behind the tortoise, though.īut the -th cycle is, by definition, the cycle after all of those cycles. And after Achilles has gone that additional meter (111 meters total), the dang tortoise is already ahead by another tenth of a meter. After Achilles has gone that next 10 meters (110 meters) total), the tortoise is yet another meter ahead. After Achilles has gone 100 meters, the tortoise is still 10 meters ahead. Let’s say Achilles runs 10 times faster than the tortoise 2, but starts 100 meters behind him. Zeno tried to use this to show that all movement is an illusion, despite our experience to the contrary.īut let’s talk about this using our new idea. Many of these philosophical paradoxes do. Since Achilles must repeatedly reach the spot where the tortoise was, he can never catch the tortoise. Thus, Achilles has to run to that spot, but, again, the tortoise has moved on.

But by the time he arrives, the tortoise has moved farther away. In order to catch the tortoise, Achilles must first reach the place where the tortoise began. However, he can’t ever catch the tortoise. (Actually, Zeno has six surviving paradoxes, but let’s talk about the most famous one.) Achillles, the fastest of the Greek warriors, is chasing a tortoise. While isn’t useful in a lot of places in math, it does provide one resolution of the famous Zeno’s paradox. You have the first thing, the second thing and the -th thing. Numbers like 1, 2 or are called ordinals since they define the order of things. Each turn of the spiral represents one power of

Representation of the ordinal numbers up to. You can go on for as many layers as you wish. This clearly has to be infinity!īut what’s the number after ? Well, of course. Then, we define a number, the Greek letter omega, to be the number just after all of the counting numbers 1. Each counting number (also called natural number) is followed by another. Three is the number after two, and so on. Think about it this way: two is the number after one. What number occurs after all the other numbers? Well, infinity, of course. The first is to ask, “What is the number infinity?” The second is to ask, “What does it mean to have infinitely many?” The first we’ll answer in this post, the second we’ll answer next week.Īll of this depends on what precise definitions you choose to use, but the school child’s idea is still about right. To really understand infinity, we need to think about about it in two different ways. Two kids keep trying to one-up each other with bigger and bigger numbers, until one inevitably pulls out the trump number, infinity. I think most of us have had this conversation.