Escape Velocity

I went to bed early, around midnight. But I awoke at 1:30AM and knew my sleep was over for the night. Anyway, I was thinking about mathematics.

Suppose you could write a number like so, 0.999… (the dots denote an infinite number of 9s). Mathematicians also write this as 0.9. Note the bar above the 9; it’s called a macron. It’s surprising how difficult it is to make a computer do something so simple. But it’s actually easy if you use CSS… Cascading Style Sheets. (At this point I know I’m talking to myself, so I’ll get on.)

Now imagine this number: 1.0. My question is this: “Is the number 0.999… with 9s to infinity exactly the same as the number 1.0?”

It is obvious that each succeeding 9 in the series 0.999… gets you a tad closer to 1.0, but none of them will get you exactly to 1. There is never a 9 where you can say, “That’s it, we’re at 1.0 now.” Nope, there’s always going to be an infinite number of nines remaining. You can get closer and closer and ever so closer to the number 1, but you can never quite get there. So the two numbers are not the same. Right?

Wrong. The number 1.0 and the number 0.999… with it’s infinite string of 9s are equivalent ways to write the same quantity. They are equal.

You might ask, “Who cares about infinity? It doesn’t exist, it’s just a math trick.” It’s true that infinity is a math concept, but it’s a very useful one that is often used in “the real world” by scientists and engineers, as well as mathematicians.

In a university physics class many years ago, I was asked to compute Earth’s escape velocity. Some of you might ask, “What is escape velocity?”

If you throw a ball into the air, straight up, it will fall back down. If you throw it harder, it will go higher but still fall back down. But suppose you had a cannon that could shoot that ball straight up as fast as you wanted. How fast would the ball have to be going up for it to never fall back to Earth? The ball would have to keep going forever. It would go slower and slower, of course, but because Earth’s gravity gets weaker with distance from Earth, the ball’s deceleration would decrease as distance from our planet increased, so that the ball will never stop. Mathematically, we can say the ball will stop at infinity, after traveling an infinite distance on a journey that required infinite time. The key insight is that if we reverse that impossible journey, so that the ball starts at zero speed at infinity and falls to Earth, the speed of the ball when it hits Earth will be Earth’s escape velocity. The two journeys are mirror images.

To calculate the speed the ball is going when it hits Earth requires using a branch of mathematics called calculus. If you make a career of engineering, it will probably be the first math you’re taught. I spent a year learning integral calculus. I spent another year learning differential calculus. Then—I got to the hard stuff. But just because you have to learn it doesn’t mean you’ll ever need to use it. Maybe you won’t. But if you’re a science-type person, you might find it so intriguing that you keep going.

We have sent probes into space traveling faster than escape velocity. Those probes will never return to Earth. They’ll travel through space and time for as long as our Universe exists, barring an encounter with a planet or a star or a black hole. Or—aliens.

Now it’s after 6AM and it’s still dark. Maybe I’ll fix myself some breakfast. It’s a tossup between raisin bran cereal and sausage biscuits. So many decisions.