How can you get from number 1 to number 2, if there are an infinite amount of numbers in between?

Question

If you can always cut a number in half, and cut that in half, and cut THAT in half, doesnt that mean there are an infinite amount of numbers (or fractions of a number) between the number 1 and 2?

Also, how can you ever stand on one side of the room, and then walk out the door across the room, if there is an infinite amount of numbers in the distance between you and the door? I am so confused. Am I thinking about it the wrong way?

Answer 1

The numbers between 1 and 2 only become infinite if you count the smallest, tiniest most minute of numbers like, 1.0000000000000000000000001, 1.0000000000000000000000002
1.0000000000000000000000003

Even if you did that above, it still would be finite, but you actually already counted to "2" if you just consider the very last number in the decimal place.

Try this. Say I am a whole person. If I eat some steak, you could say I am now 0.000000298 more body than I was, if you were viewing me as just cells. When you view me as a body, I could always keep eating and gaining almost an infinite number of cells and nutrients until I was the size of the earth, but still be one body. (theoretically heheh).

If two people were side by side, you could count both of their amount cells to realize there's enough for two people, but you'd never get to that conclusion, because there are so many. However, because the cells are separated and aggregated into two separate bodies, you can step back and just count " 1. 2."

The trap and paradox of achieving the smallest of counting is BECAUSE of our ability to think about the problem. Words, thinking and logic themselves generally work in sequence and in small pieces. "Feeling" generally involves aggregates and "wholes". You'll feel less confused when you get too tired to think about it. You'll just start to "feel" ok that your door is in reach, the frig is down stairs, and there's a definite amount of money in your wallet. The mind works in mysterious ways...

Answer 2

This is related to the the paradox of Zeno of Elea, where in basic you are constantly traveling half the distance to your destination, so it takes an infinite number of steps to do so. The problem with that is that you can apply that to any distance, thus it takes an infinite number of steps to move anywhere. By this logic, motion is impossible, but try telling that to the pie heading towards your face. :-)

The way to think about it is that the infinite numbers between 1 and 2 are infinitely small. (The term for that is infintesimals.) It's a matter of applying an infinite number of infintesimals. Using calculus and converging series, this may result in a finite number. In your case, that would be 1, the difference between 1 and 2.

Same logic sort of applies to the journey to the door. However, keep in mind that when you are walking, you are stepping in halves, but in a roughly even stride.That is, at a certain point, you are covering multiple halves at one time, and that leads you to the door.

Wikipedia has information on this and some other of Zeno's paradoxes at http://en.wikipedia.org/wiki/Zeno%27s_paradoxes. Hope that helps.

Answer 3

You can get from 1 to 2 by jumping over an infinite amount of infinitely small numbers in a single finite bound.

The longer you spend adding this series 1/2 + 1/4 + 1/8 + 1/16 ... the closer you will get to 1. But that's because you are taking a finite amount of time on each addition. Instead of doing it that way, math geeks have found ways to add an infinitely large number of numbers in a finit amount of time, which is exactly what the basic principles of calculus are meant to do.

Answer 4

Yes, there are infinite numbers between 1 and 2

Answer 5

Even though there is an infinite amount of numbers in between, that means that all the fractions that you come up with have an infinitely small difference between one another. Infinite (the number of distances possible) x infinite small (the difference between the closest two fractions) = finite (distance). Same with the floor. However, if the distance is infinite. Then you just can't reach the end. It depends on distance, not the number of distances or numbers possible.

Answer 6

But the space in between the 'infinite amount of numbers' are also also decreasing. Yes, there can be said to be infinte, but they also get closer and closer in proportion as they approach infinity, yet still maintain the distance of '1'

For example, if you cut the space between 1 and 2 in half, you get:

2 .5s ('two' point fives)

and if you continue you get the following:

4 .25
8 .125
16 .0625

Remember, the distance is still One, no matter how many times you divide it. You divide it more, the spaces get smaller, you divide it less, the spaces get larger, but the finite distance remains constant. The extension of space between the numbers approaches zero as the amount of numbers approach infinity: this leaves us with a finite extension.

Another example would be, consider a finite 2 by 2 inch square. The square is finite, however, you can divide the square into smaller and smaller parts to infinity, however, of all those infinite parts, it still makes up the finite square. So also with the number line, and the ability to move from one location to another.

Answer 7

You're thinking actually 100% correctly, but that's why they call them "counting numbers"....1 2 3 4 5 6 7 8.....and so on. If you didn't use counting numbers, you could never really even get to 1. ......for example.

0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001. I gave up holinding the zero....you got the idea though.

Answer 8

Theoretically yes. But the logic only works that way for an infinitesimally small object that can move an infinitesimally small distance. A person has physical limitations on how small a distance he or she can move. Once the 1/2 distance you are trying to move falls below the human tolerance, you would only have two options: 1) not moving; 2) move, and complete the distance. The same is true for any other object that's not infinitesimally small - anything of any size has a finite tolerance on it's movements.

Answer 9

There are an infinite amount of numbers between 1 and 2, but why do you have to count in those units, you can just count the who numbers, or just count in the tenth digit. You skip numbers as you calculate, you don't have to hit everyone of them.

Answer 10

Yes you are thinking of it in the concrete versus the abstract. Numebrs are Abstract ideas. Space is finite. You could divide the space down into the quantum level but you would reach the limits and there find the end but numbers are abstract ideas and as these ideas you can always add one more.