Infinity is an abstract concept used to describe something that is endless or boundless. It is important in mathematics, cosmology, physics, computing, and the arts.
The Infinity Symbol
Infinity has its own special symbol: ∞. The symbol, sometimes called the lemniscate, was introduced by clergyman and mathematician John Wallis in 1655. The word “lemniscate” comes from the Latin word lemniscus, which means “ribbon,” while the word “infinity” comes from the Latin word infinitas, which means “boundless.”
Wallis may have based the symbol on the Roman numeral for 1000, which the Romans used to indicate “countless” in addition to the number. It’s also possible the symbol is based on omega (Ω or ω), the last letter in the Greek alphabet.
The concept of infinity was understood long before Wallis gave it the symbol we use today. Around the 4th or 3rd century B.C.E., the Jain mathematical text Surya Prajnapti assigned numbers as either enumerable, innumerable, or infinite. The Greek philosopher Anaximander used the work apeiron to refer to the infinite. Zeno of Elea (born circa 490 B.C.E.) was known for paradoxes involving infinity.
Of all Zeno’s paradoxes, the most famous is his paradox of the Tortoise and Achilles. In the paradox, a tortoise challenges the Greek hero Achilles to a race, providing the tortoise is given a small head start. The tortoise argues he will win the race because as Achilles catches up to him, the tortoise will have gone a bit further, adding to the distance.
In simpler terms, consider crossing a room by going half the distance with each stride. First, you cover half the distance, with half remaining. The next step is half of one-half, or a quarter. Three quarters of the distance is covered, yet a quarter remains. Next is 1/8th, then 1/16th, and so on. Although each step brings you closer, you never actually reach the other side of the room. Or rather, you would after taking an infinite number of steps.
Pi as an Example of Infinity
Another good example of infinity is the number π or pi. Mathematicians use a symbol for pi because it’s impossible to write the number down. Pi consists of an infinite number of digits. It’s often rounded to 3.14 or even 3.14159, yet no matter how many digits you write, it’s impossible to get to the end.
The Monkey Theorem
One way to think about infinity is in terms of the monkey theorem. According to the theorem, if you give a monkey a typewriter and an infinite amount of time, eventually it will write Shakespeare’s Hamlet. While some people take the theorem to suggest anything is possible, mathematicians see it as evidence of just how improbable certain events are.
Fractals and Infinity
A fractal is an abstract mathematical object, used in art and to simulate natural phenomena. Written as a mathematical equation, most fractals are nowhere differentiable. When viewing an image of a fractal, this means you could zoom in and see new detail. In other words, a fractal is infinitely magnifiable.
The Koch snowflake is an interesting example of a fractal. The snowflake starts as an equilateral triangle. For each iteration of the fractal:
- Each line segment is divided into three equal segments.
- An equilateral triangle is drawn using the middle segment as its base, pointing outward.
- The line segment serving as the base of the triangle is removed.
The process may be repeated an infinite number of times. The resulting snowflake has a finite area, yet it is bounded by an infinitely long line.
Different Sizes of Infinity
Infinity is boundless, yet it comes in different sizes. The positive numbers (those greater than 0) and the negative numbers (those smaller than 0) may be considered to be infinite sets of equal sizes. Yet, what happens if you combine both sets? You get a set twice as large. As another example, consider all of the even numbers (an infinite set). This represents an infinity half the size of all of the whole numbers.
Another example is simply adding 1 to infinity. The number ∞ + 1 > ∞.
Cosmology and Infinity
Cosmologists study the universe and ponder infinity. Does space go on and on without end? This remains an open question. Even if the physical universe as we know it has a boundary, there is still the multiverse theory to consider. That is, our universe may be but one in an infinite number of them.
Dividing by Zero
Dividing by zero is a no-no in ordinary mathematics. In the usual scheme of things, the number 1 divided by 0 cannot be defined. It’s infinity. It’s an error code. However, this isn’t always the case. In extended complex number theory, 1/0 is defined to be a form of infinity that doesn’t automatically collapse. In other words, there’s more than one way to do math.
- Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton University Press. p. 616.
- Scott, Joseph Frederick (1981), The mathematical work of John Wallis, D.D., F.R.S., (1616–1703) (2 ed.), American Mathematical Society, p. 24.