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Zero is one of the most important Mathematical breakthroughs. The first zero concept seems to be forged by al-Khwārizmī, who lived in the 9th century, but it ain’t reached Europe until the 13th.

Despite zero’s importance, it’s not a natural number, given the description.

atural numbers are countable, this is their definition. You can count one thing, you can count two things, but you can’t count zero things, as well as you can’t count negative two things or π things.

The Fibonacci Numbers

eonardo Bigollo Pisano, a.k.a. Fibonacci, was an Italian Mathematician of the first half of the 13th century and is credited by the introduction of zero into European Mathematics. He is known by the Fibonacci Numbers, a very important numerical sequence leading to the golden reason and several natural discoveries.

The Fibonacci Numbers are a natural sequence, but in a phenomenal way prior to Mathematical.

Let me explain the Fibonacci’s discovery process:

He literally had a couple of rabbits. He noted that in an almost fixed time span – let’s talk about a month for simplicity – the couple reproduced, giving birth to a new couple.

Every couple reproduced every month, taking the double of this time to grow to the spawning age.

This means that in the 1st month he had a couple of rabbits. In the 2nd he still had a couple. In the 3rd month a new couple was born, so he had two couples. In the 4th month the first couple give birth to its second litter, but the first was not prepared to copulate yet, so he had 3 couples. In the 5th month both the initial couple and the first descendants had litters, bringing a total of five coupes.

Recapping:

  • 1st month: 1 couple;
  • 2nd month: 1 couple;
  • 3rd month: 2 couples;
  • 4th month: 3 couples;
  • 5th month: 5 couples;
  • 8th month: 8 couples;
  • and so on.

But why cannot the sequence start on zero? 0, 1, 1, 2, 3…

Because out of nothing, may come nothing. If you have no couple of rabbits (zero) in the 1st month, you’ll get no couple in the 2nd. Still no couple in the 3rd, 4th, 5th… yet zero in the 10th month (or in the nth month).

This means that a real natural sequence starting from zero should only be 0, 0, 0, 0, ad eterno.

Zero Is Not a Natural Number… Until It Is

owever it’s very convenient taking zero as a natural number.

We gain a addition identity element, ’cause anything plus zero is the same thing, just like one is the multiplication identity element.

Interesting things start to work, like zero multiplication patterns.

So, even if zero isn’t a natural number, you can use it as one.

λ-Calculus

ere’s an example where zero is helpful as natural number: λ-calculus.

In λ-calculus, natural numbers are functions. For example:

  • 1 is λsz.sz
  • 2 is λsz.s(sz)
  • 3 is λsz.s(s(sz))
  • and so one

As you can see, each natural number is a 2-argument function that applies the second argument as parameter to the first each times of the very number. Following the same foundation, the zero is when the second argument is returned with no appliance of the first: λsz.z.

It’s very suitable, ’cause it’s the same false definition:

  • True: λsz.s
  • False: λsz.z

So zero is the same as false.

Zero Is a Convenient Natural Number… Until It Isn’t

ut not everything is like la vie en rose. We got some tricky issues on zero.

Like zero division: the zero division is very annoying, ’cause undesirable things start to happen.

For example, 1×0 = 0, as well as 2×0 = 0, so 1×0 = 2×0. Dividing each side by zero we got 1 = 2.

In order to work around this misbehavior, the Math deals with zero division as indeterminate – Computation calls it “not a number” (NaN).

So any number divided by zero (even zero itself) is indeterminate – or NaN.

The Curious Case of Zero Raised to Zero

bone of contention is zero raised to zero.

Let’s found out.

The power operation means how many times a number is multiplied by itself after one.

Follow the pattern:

  • 2⁴ = 1×2×2×2×2 = 16
  • 2³ = 1×2×2×2 = 8
  • 2² = 1×2×2 = 4
  • 2¹ = 1×2 = 2
  • 2⁰ = 1 = 1

The same procedure can applied to zero:

  • 0⁴ = 1×0×0×0×0 = 0
  • 0³ = 1×0×0×0 = 0
  • 0² = 1×0×0 = 0
  • 0¹ = 1×0 = 0
  • 0⁰ = 1 = 1

And that’s how computer works!

But there’s another way. You can consider each power as the higher power divided by the base, so:

  • 2⁰ = 2÷2 = 1
  • 0⁰ = 0÷0 = NaN

This is how Mathematicians do. But… why computers don’t act like that?

Because it would be inconsistent. Realise:

  • 2² = 2³(=8)÷2 = 4
  • 0² = 0³(=0)÷0 = NaN

Through that path, no zero power could ever exist.

Written by

Musician, senior software engineer, autistic, and autistic parent (not necessarily in this order)

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