Infinity

Few concepts in mathematics are more fascinating or confounding than infinity. While mathematicians have a longstanding disagreement over its very definition, one can start with the notion that infinity (denoted by the symbol ∞) is an unbounded number greater than all real numbers.

Writing about infinity dates back to at least the Greek philosopher Aristotle (384 B.C.E.–322 B.C.E.). He stated that infinities come in two varieties; actual infinities (of which he could find no examples) and potential infinities, which he taught were legitimate only as thought. Indeed, the German Karl Gauss (1777–1855) once scolded a fellow mathematician for using the concept, stating that use of infinity "is never permitted in mathematics."

The French mathematician and philosopher René Descartes (1595–1650) proposed that because "finite humans" are incapable of producing the concept of infinity, it must come to us by way of an infinite being; that is, Descartes saw the existence of the idea of infinity as an argument for the existence of God. English mathematician John Wallis (1616–1703) suggested the use of ∞ as the symbol for infinity in 1655. Before that time, ∞ had sometimes been used in place of M (1000) in Roman numerals.

Defining Infinity

Although students are typically taught that "one cannot divide by 0," it can be argued that = 0 (read as "one divided by infinity"). How is this possible? Observe the following progression.

Note that as the denominator, or the divisor, becomes larger, the value of the fraction (or the "quotient") becomes smaller. What happens if the denominators become very large?

One can see that as the denominator becomes extremely large, the fraction values approach 0. Indeed, if one thinks of infinity as "ultimately large," one can see that the value of the fraction will likewise be "ultimately small," or 0. Hence, one informal (but useful) way to define infinity is "the number that 1 can be divided by to get 0." Actually, there is no need to use the number 1 as the numerator here; any number divided by infinity will produce 0.

Using algebra, one can come up with another definition of infinity. By transforming the following equation we see that infinity is what results if 1 is divided by 0.

Then 1 = ∞ × 0

And

Notice that this approach to informally defining infinity produces an equation (the middle equation of the three above) in which something times 0 does not give 0! Because of this difficulty, and because the rules of algebra used to write and transform the equations apply to numbers, some mathematicians claim that division by 0 should not be allowed because ∞ may not be a defined number. They argue that dividing by 0 does not give infinity, but rather that infinity is undefined.

Another method of attempting to define infinity is to examine sets and their elements. If in counting the elements of a set one-by-one the counting never ends, the set can be said to be infinite.

Infinity as a Slope. Infinity is also sometimes defined as "the slope of a vertical line on the coordinate plane." In coordinate geometry, it is accepted that the slope of any straight line is defined as the change in vertical height divided by the change in horizontal distance between any two points on the line. The slope is often shown as a fraction in lowest terms, and sometimes called "rise over run."

In the figure, the slope of line (a) is ½. If a line is very steep, the rise will be very large compared to the run, giving a very large numerical slope. The slope of line (b) is as . A much steeper line will result in a fraction suchall . Such a line would appear to be vertical, even though it would not be quite vertical if viewed in greater detail. Thus, the slope of extremely steep lines approaches infinity, and the slope of a "completely steep" line, that is, a vertical line, can be thought of as equal to infinity.

Yet on a "completely steep" or vertical line, any two points give a run of 0. This means that one could define the slope of the line as any number over 0. This again allows the conclusion that division by 0 results in infinity, unless one maintains that the slope of a vertical line is undefined.

The Nature of Infinity

Although several definitions of infinity were provided, note that none of them state that infinity is the highest possible number. Consider this: On a number line, how many points are between points 4 and 5? An infinite number, of course, because actual points have no dimension, even though their two-dimensional representations have a very small dimension on the paper, blackboard, or computer screen. But consider further: How many points are between points 4 and 6? Also an infinite number, certainly, but this set appears to be twice as large as the one between points 4 and 5. This use of set theory as an approach to understanding infinity forces one to look at several curious possibilities.

There are different sizes of infinity.
A set with an infinite number of elements is the same size as one of its "smaller" subsets.
Elements can be added to a set that already has an infinite number of elements.

Which of these possibly contradictory statements is true? It may be impossible to answer the question. Galileo (1564–1642) felt that the second statement was true. The great German mathematician and founder of set theory Georg Cantor (1845–1918) added to our understanding of infinity by choosing not to see the statements as contradictions at all, but to accept them as simultaneous truths. Cantor defined orders of infinity. An infinite set that can be put into one-to-one correspondence with the counting numbers is the smallest infinite set, called aleph null. Other larger infinite sets are called aleph one, aleph two, and so on. One can see that working with infinity produces various counterintuitive and even paradoxical results; this is why it is such an interesting concept.

There are numerous examples of infinity in pre-college mathematics. One case: it is accepted that 0.999… is exactly equal to 1.0. Yet how can a number which has a 0 in the units place be exactly equal to a number with a one in that place? The idea that there are an infinite number of nines in the first number allows us to make sense of the proposition. The number 0.999… is said to "converge on 1," meaning that 0.999… becomes 1 when the infinite number of nines is considered.

Another example of how infinity comes into play in common mathematics is in the decimal representation of π (pi), or 3.14159…. The digits making up π go on forever without any pattern, even though the size of never π gets even as large as 3.15.

No one has ever come across an infinite number of real things. Infinity remains a concept, brought to life only by the imagination.

Infinity

Defining Infinity

The Nature of Infinity

Bibliography

DEVELOPMENT OF SET THEORY