Math Journey Archive

Math Journey 4, Part 2

To Infinity via Arithmetic

"Uncle Bob"

Sequences and Series. A mathematical sequence is an ordered list of items, and we'll limit our discussion to numbers. Some sequences are calculated, that is, each successive member is determined by some formula that uses either the ordinal of the member or the previous member in the list. For example, the halving sequence that begins {1/2, 1/4, 1/8, ...} can be formulated by halving at each step, or by the formula [(1/2)^ n] (one-half raised to the 1st, 2nd, 3rd, ... powers). This formula can be used to calculate the fourth member.

1/2 x 1/2 x 1/2 x 1/2 = 1/16

A mathematical series is a sequence of numbers that is keeping a running total of some other sequence. For the halving sequence, the series is

{1/2, (1/2) + (1/4), (1/2) + (1/4) + (1/8), ...} = {1/2, 3/4, 7/8, ...}

Halfway to the Wall. If I start at one end of a room and each minute proceed half of the remaining distance to the opposite end, then the sequence above tells me how far, as a fraction of the entire length, I must go at each minute, while the series above tells me the total fraction I have traveled. Do I ever reach the wall?

The answer is no because I never go more than half the remaining distance at any time. If we imagine that an infinite number of minutes can elapse, where would I be? I would be at the other end of the room. The value of the fractions in the series get closer and closer to one, and that is one whole room length. For example, the fraction 1023 / 1024 is reached after only ten minutes, and that is over 99.9% of the way across.

The conclusion here is that even though a sequence has an infinite number of members, its series doesn't necessarily result in an infinite total. Let's see one that does.

The sequence {1/2, 1/2, 1/2, 1/2, ...} is associated with the series {0.5, 1, 1.5, 2, ...} and this series will reach an infinite total, given an infinite number of halves in the sequence. We can show this is true with symbols, but it's just as easy to reason that, with enough terms, the series will get larger than any number you can name, and that's the nature of infinity. [See Part 1 of this journey in UB January]

Infinity via Counting. So far we've allowed our imaginations to entertain the possibility of sequences with an infinite membership. Let's formalize this. If infinity is larger than any number, and the counting sequence

{1, 2, 3, 4, 5, ...}

can be extended to exceed any number, then it can be considered an infinite set. In the 1800's a German mathematician named Georg Cantor made attempts to comprehend infinity; in particular, he attempted to count it. He knew that the mathematical essence of counting was the act of matching objects to the numbers 1, 2, 3, ... until the objects were all counted. We know that our fingers aren't named "one", "two", et cetera, but we recite those names as we match fingers, numbers, and objects. It's called making a one-to-one correspondence. For example, if I visit a first grade class out for recess in January, and I ask them all to hold up their left hands, and I see a mitten on every hand; and then I ask them to hold up the other hand, and again, a mitten covers every one, I know there is a one-to-one correspondence between right and left mittens. I haven't counted them yet, but I know the counts will be equal.

Cantor thought that the counting numbers could count anything, including infinite sets, because they are an infinite set as well. His strategy was to match other infinite sets to the counting numbers, à la mittens. That would establish the equality of one infinite set, counting-wise, with another. And that is when things began to get really strange.

It's Saturday and we are at the training school for volunteer firemen. Imagine an infinitely tall ladder reaching, like Jack's beanstalk, up through the clouds and into the heavens. We see a firefighter on each rung. We imagine it's a bit crowded. A station wagon pulls up with five more firefighters. We make room on the ladder for them by ordering all firefighters to climb up five rungs. Mission accomplished. Later, an Infiniti arrives with an infinite number of volunteers. We can accommodate them as well by ordering all on the ladder to climb up to an odd-numbered rung. The even-numbered rungs, infinite in number, are now vacated.

Cantor thought similarly that parts of infinite sets count up the same. He matched all the even numbers with their halves and saw that they could be counted as equal in number to all the counting numbers:

2 4 6 8 10 12 ...
1 2 3 4  5    6 ...

Cantor also discovered that he could not make a complete list of all the real numbers, not even the real numbers between zero and one. This set includes fractions and the irrational numbers such as half the square root of 2 and one-fourth pi. Numbers like

0.5000000...
0.6666666...
0.7071068...
0.7853982...
...

Every time he imagined a listing he saw how he could construct a number that wasn't in the list. To add to the list above he simply picked a tenths digit different than the first one (5); a hundredths digit different than 6; a thousandths digit different from 7, a fourth digit different than 3, and so on. And he saw that even if his list were infinite, he could form a new number with the same technique. Cantor theorized that there was an infinity greater than the one the counting numbers arrived at. Wow!

In the next leg of this journey we'll relate some of these concepts to geometrical and other types of objects.

Something to think about – would the series

S = 1/2 + 1/3 + 1/4 + 1/5 + ... ever reach infinity? [Answer below]

Math Journey Archive

The series S above does reach infinity, but at a snail's pace. We have shown that
1/2 + 1/2 + 1/2 +...

reaches an infinite total. By taking bunches of terms of S we can show that it eventually reaches that total. For examples, beginning with the 2nd and 3rd terms

1/3 + 1/4 > 1/2; 1/5 + 1/6 + 1/7 + 1/8 > 1/2; and so on.

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