Different Interpretations for the Number Zero

Zero graduated from a placeholder for absentee digits to its own concept. Here's a few more interpretations of zero:

  • The void: Utter emptiness, the absence of any activity
  • The neutral zone: A cancellation between positive and negative influences

Mathematically, we can write:

\displaystyle{ 0 = 1 - 1}

And to a calculator, these are the same. Are they? There's a suspicion nothing (0) and complete cancellation (1 - 1) aren't quite identical.


In physics, there's the notion of a stable and unstable equilibrium. Take two pencils. Lay one on the table, balance the other on its tip.

They're both 'balanced'. There's zero motion. Yet one is a precarious position, carefully opposing the pull of gravity, while the other lays peacefully.

Lie on the floor for 10 minutes. Hold the plank pose for 10 minutes. From a physics perspective, no work was done (nothing moved), but your quivering arms tell a different story.


In algebra, we constantly factor equations to find roots.

Why? In short, we want to find the "neutral zones" where all forces balance.


\displaystyle{f(x) = x^2 + 2x + 3}

means "Is there a value where x^2, 2x, and 3 cancel each other out?". We arrange the scenario so the neutral zone is where we want to be (such as having no error, or having competing goals align).

There is often a "trivial solution", where we can plug in x=0 and all inputs disappear (lying the pencil on the table... or just taking it away!). However, we're more interested in finding a "neutral zone", where multiple, existing forces balance.


Programming languages distinguish "void/undefined/null" (a value is not set) and "having a value of emptiness".

var i; // i is undefined
i = 0; // i is now set to 4 bytes of "nothingness"

If we imagine data storage as a light switch, we have

  • 1: switch is on
  • 0: switch is off
  • null: there is no switch (or spoon)

By itself, var i is just a name or pointer, but it's not yet referring to anything (not even nothingness). It's not that Gazasdasrb means "nonsense", it's that Gazasdasrb has no meaning at all.

Division By Zero

Many math explanations say you "can't divide by zero". It's not that you can't, it's that it's undefined. What does division by zero mean? What does Gazasdasrb mean?

If we pick a specific value for the result of a division by zero (let's say 3/0 = 15) then we immediately have contradictions (this means 15 * 0 = 3).

We avoid this trouble by saying division by zero is "undefined", or "we haven't got around to picking a value, nyah". In some games, the only winning move is not to play.

(Sometimes we define a value for strange expressions (such as $0^0 = 1$), if it's useful and doesn't lead to contradictions.)


Calculus dances with the concept of zero. Beyond the study of limits and infinitesimals, we are curious about the meaning of "zero change".

When I say a function isn't changing ("the derivative is zero"), it's usually not enough information. Are we not changing because we're at a minimum, a maximum, or precariously balanced between a hill and ravine?


There are tricks, like the second-derivative test, to see what type of "zero change" we have.

In Society

Society sets many goals for itself. Here's one: reduce littering. Given our "multiple zero" interpretation, we could accomplish this with:

  • A robot army that picks up thousands of pieces of litter each day (1000 drops - 1000 cleanups)
  • Teaching people not to litter (0 drops)

It's the same result -- clean streets -- but what strategy do we prefer?

In general, any negative influence (unemployment, crime, pollution, etc.) can be seen through the lens of prevention or cure, an absence vs. meticulous cancellation. The reading is 0 in both cases, and it's up to us to make the distinction. (Sir, unfrozen Caveman Og is asking about Wooly Mammoth attacks again. Should we sell him more repellent?)


In Eastern philosophy there's the notion of non-doing or Wu Wei. Our brains think of "non doing" as sitting lazily on the couch. But maybe it's another type of zero. (Again, hold a plank for 10 minutes and tell me nothing happened.)

Exploring Ideas with Math

This essay is quick armchair philosophy from an equation. The words "something comes from nothing" aren't convincing. But if I write $0 = 1 - 1$, boom, an idea snaps into place. How did 5 symbols convince you in seconds? Isn't that amazing?

Calculations are nice, but not the end goal of math education. Intuition means you're comfortable thinking, daydreaming, and exploring a concept with math as a guidepost. Now imagine having this comfort with the notions of shape, change, and chance (geometry, calculus, statistics).

Let's learn to sing with math, baby.

Other Posts In This Series

  1. Techniques for Adding the Numbers 1 to 100
  2. Rethinking Arithmetic: A Visual Guide
  3. Quick Insight: Intuitive Meaning of Division
  4. Quick Insight: Subtracting Negative Numbers
  5. Surprising Patterns in the Square Numbers (1, 4, 9, 16…)
  6. Fun With Modular Arithmetic
  7. Learning How to Count (Avoiding The Fencepost Problem)
  8. A Quirky Introduction To Number Systems
  9. Another Look at Prime Numbers
  10. Intuition For The Golden Ratio
  11. Different Interpretations for the Number Zero

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