On Understanding the Void through Dividing by Zero
A lot of the following ideas are built on notes from a great podcast episode called Zeroworld from Radiolab. Listen here.
As you may have remembered from math class, we're taught that you can do almost anything with the medium of numbers, except divide by zero.
That’s because in math, division by zero is undefined. Here’s two simple points as to why:
Division is fundamentally defined in mathematics as the inverse of multiplication. For example, 6/3=2 because 2*3=6. However, when you try to apply this to division by zero, things fall apart. A key concept of math is that things need to be "undone", but there is no number that you can multiply by 0 to get a non-zero number, and therefore division by zero cannot be "undone" through multiplication. Therefore by allowing division by zero, it would create contradictions within mathematics. For example, if you were to define 1/0 = some number, let's say 'Y', then according to multiplication, 0*Y should be 1. But we know that anything multiplied by 0 is 0, so this breaks the basic rules of arithmetic.
When you divide a number by another number that gets closer and closer to zero, the result becomes larger and larger, approaching infinity. The problem with this is infinity is actually not a real number; it's a concept. Dividing by zero doesn't produce a definable number but rather leads towards an infinite limit, which is not a specific, finite value.
For example:
1/1 = 1
1/0.1 = 10
1/0.01 = 100
1/0.001 = 1000
and so forth
Therefore, as we divide something with numbers that approach zero, the result approaches infinity.
This relationship between zero and infinity can give us insights into another reason why division by zero is “not allowed” - when we divide something by zero, we’re actually getting an answer that’s not a number. Remember, infinity is a spatial concept. Maybe, then, division by zero is actually a portal into understanding their inextricably linked qualities.
Zero & Infinity
As Charles Seife states, "Zero and Infinity are two sides of the same coin--equal and opposite, yin and yang, equally powerful adversaries at either end of the realm of numbers". The essence of zero cannot be fully appreciated without its counterpart, infinity, and vice versa. Their interdependence is such that the presence of one suggests the existence of the other.
Zero represents a numerical foundation, a point of origin. In mathematics, it's the central point in the number line, demarcating the positive from the negative. As a number, zero signifies nullity or the absence of quantity, which is essential in arithmetic and algebra for maintaining the integrity of the number system.
Infinity on the other hand, is often perceived as a spatial concept. It represents boundlessness or unending extension. In geometry and calculus, infinity is used to describe endless lines, curves, or surfaces. It's not a number in the conventional sense but rather a state or quality of being limitless. In math, infinity is used to describe the behaviour of functions as they stretch out towards endlessness. In physics, infinity is used to describe concepts like infinitely small scales (quantum mechanics) or infinitely large expanses (cosmology). Infinity's relationship with zero in this sense is clear, where operations involving limits explore how functions behave as they approach zero (infinitesimally small) or stretch towards infinity.
A math concept that discovered an interesting relationship between zero and infinity is the Riemann Sphere. Imagine a sphere sitting in a 3D plane, where