Integers
The integers are a whole new ballgame from the counting numbers.
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We know integers to be numbers like 1, 2, 3, and so on. So doesn't that make them counting numbers -- i.e., 1, 2, 3, and so on?...Well, yes, it does. And no, it doesn't.
It does, because the set of integers includes numbers that look and act like the counting numbers. The symbol 2, for example, is used both for the natural (counting) number and the positive integer. And all other counting numbers have the same look as their counterparts in the integers. Furthermore, 2 + 3 in the counting number domain equals 2 + 3 in the integer domain. In addition, the number of these integers is exactly the same as the number of counting numbers. So it's perfectly reasonable to think that's what they are. After all, if it walks like a duck and quacks like a duck, ... Well, you get the picture.
However, it is not a duck, because there's more to "integers" than counting numbers, and that breaks them apart -- they're not the same animal, though they may have similar stripes. Integers form a different class. The reason is that the integers also include negative numbers, which aren't at all the same as counting numbers, which in fact don't have a sign attached to them. There is no negative counting number!
Another difference is the presence of that funny symbol shaped like a donut, namely 0, the number zero. There is no zero among the counting numbers. This is a new number and was added as an integer as a kind of facilitator, a go between, to pacify the "transformed" counting numbers and the negative whole numbers.
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Integers have a lot of class -- that's for sure! Infinitely so! The natural numbers more or less grew like topsy, unlike the integers, which had to have been practically a divine inspiration. Maybe that's just the god-like quality of the great mathematicians.
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No. of Integers = No. of Naturals
What? The number of counting numbers equals the number of integers? Are you mad? Don't the integers include both negative and positive numbers? And doesn't that make the integer class twice as big as the counting number class, not including the donut, which only makes the integer class one number bigger?
Fair enough -- I said that. And I stand by what I said. So let me try to make sense of the words. Intuition here is very slippery!
Would you agree that two sets are the same size (have the same number of members) if they can be put in one-one correspondence with each other? Like showing that John and Mary are the same height by having them stand back to back for comparison? If so, maybe you'll agree that the integers and naturals are the same size if I can put them in one-one correspondence. But how in the world could that possibly be done, considering we have to compare one of something with twice that something?
Here's what we'll do, ignoring the zero, for the moment. The main thing is first to re-order the integers. Normally we think of them as ranging along a horizontal line from large negatives on the left, going to zero in the middle, and starting the positives to extend them to the right.
... -3, -2, -1, 0, +1, +2, +3,...
Viewed this way, the integers are hard to compare with the naturals, because there's no starting point. So, instead of this, let's begin the new sequence with 1 and alternate between positive and negative integers. Better yet, let's start with 0, and then go with the plus and minus integers. This gives us:
0, +1, -1, +2, -2, +3, -3, ...and so on.
With this trick out in the open, we now have a real starting point, so we can count them. That is, we can number them from left to right, using the counting numbers, 1, 2, 3, 4, 5, and so on. Now, amazingly, but surely, numbering this way puts the integers in one-one correspondence with the naturals, and we can conclude that the integers equal the naturals in size.
To see this more precisely, we can write a formula for the correspondence. Defining the counting number, n, in terms of the integer, I, we write:
nI = 2I, if I is positive and greater than 0 (1)
nI = - 2I + 1, if I is negative and greater than 0 (2)
Check it out!
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