Another interesting topic is

N = {1,2,3,4,5,6,...}

This set has an infinite number of elements.

Now consider the set of Integers, where we throw in Zero and all the Negative Numbers:

I = {...,-6,-5,-4,-3,-2,-1,0,1, 2, 3, 4, 5, 6,...}

This set also has an infinite number of elements.

At first look it appears this set should be twice as big, but actually you can do what's called a "mapping" that is "one-to-one and onto" where you pair up all the Natural numbers with all of the Integer numbers. One way of doing this is as follows:

N = {... 13 11 9 7 5 3 1 2 4 6 8 10 12 ...}

I = {...,-6,-5,-4,-3,-2,-1,0, 1, 2, 3, 4, 5, 6,...}

We start with the 0 in set I and then bounce back and fourth between the positive and negative numbers.

With this mapping you can choose any integer number from set "I" and I can tell you a number from set "N" that matches up with it. Thus both sets in a peculiar way have the same infinite number of elements in them. We can say that set I is a "countable order of infinity" because we can count all the elements, that is, we can match up our set of Natural counting numbers N with every element in set I.

We can go further with this and examine the set of all positive fractions

Q+ = (a/b)

where a is a Positive Integer and b is a Positive Integer.

Our set Q+ now is the set of all positive fractions, what we call the set of all Rational Numbers (I think they choose the letter Q for this set to stand for Quotient, and add a plus sign to indicate we're doing only the positive numbers for now).

At first look it appears our set Q+ of all fractions should be much bigger than our set N of all Natural Counting Numbers. But when we examine it closely we discover we can make a "mapping" that is "one-to-one and onto" from set N to set Q+. That is, you can choose any fractional number from set Q+ and I can give you a number from set N that matches up with it. So in a peculiar sense our sets Q+ and N have the same infinite number of elements.

This is usually proved by making a table of all fractions as is done below. We place the numbers 1, 2, 3,... down the side to represent all possible values of a, and again the numbers 1, 2, 3,... across the top to represent all possible values of b:

1 | 2 | 3 | 4 | 5 | 6 | ... | ||

1 | | | 1/11 | 1/23 | 1/36 | 1/410 | 1/515 | 1/621 | ... |

2 | | | 2/12 | 2/25 | 2/39 | 2/414 | 2/520 | 2/6 | ... |

3 | | | 3/14 | 3/28 | 3/313 | 3/419 | 3/5 | 3/6 | ... |

4 | | | 4/17 | 4/212 | 4/318 | 4/4 | 4/5 | 4/6 | ... |

5 | | | 5/111 | 5/217 | 5/3 | 5/4 | 5/5 | 5/6 | ... |

6 | | | 6/116 | 6/2 | 6/3 | 6/4 | 6/5 | 6/6 | ... |

. | . | . | . | . | . | . | ||

. | . | . | . | . | . | . | ||

. | . | . | . | . | . | . |

Note in this table that some numbers are represented more than once. For example, 1/1 is the same as 2/2, 3/3, 4/4, 5/5, and 6/6. We don't care about the duplicates. If we can count this set then we can count Q+.

The trick is to count this set by going diagonally up and across. Take a look at the blue numbers and see how we're going diagonally upwards. You can select any fraction from set Q+ and I can give you at least one counting number from set N that corresponds to it.

With a little ingenuity you can expand the argument to show that the set Q of all fractions, both positive and negative, can be counted.

Q = (a/b)

where a is an Integer and b is an Integer.

So our set Q of all fractions has a "countable infinity" of elements in it, the same as our set I of all Integers, and our set N of all Natural counting numbers.

We can take one more step here and consider all the "irrational numbers" (or "crazy" numbers) like

Pi = 3.14159265358...

or

Square root of 2 = 1.41421356237...

or

natural logarithm e = 2.718281828459045...

These are numbers which can not be expressed as a fraction a/b and so are called "irrational" (as in "crazy") numbers. This is the set of all Real numbers R.

R = {Set of Real Numbers, all those decimal numbers}

It can be proven (the proof is actually rather simple) that it is impossible to create a "mapping" that is "one-to-one and onto" from the set of Natural Numbers N to the set of all Real numbers R. No matter what scheme you come up with to count all the elements (Real numbers) of set R, I can come up with an element (a Real number) that you missed. Hence the set of Real Numbers R is said to be of a "higher order of infinity", an "uncountable infinity" of elements.

So there are at least two distinct "infinities", one a "countable infinity", and a higher order of infinity that is an "uncountable infinity". Math gets interesting.

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