I am going to make a claim. Here it is:

…000000.0000000… = 0

…111111111.1111111111… = 0

…222222222.22222222… = 0

…3333333333.3333333… = 0

(and so on for every possible digit)

So!

The lesson to take away

If a single digit repeats before and after the decimal place infinitely, that number is equal to 0

Do you want proof, my inquisitive friend?

Well, you’ve come to the right place!

Watson! My calculator!

Imagine a number.

0.111111111… (1s repeating to infinity)

This number can be called x

10x = 1.11111111…

1.1111111… is just x + 1

10x = x + 1

9x = 1

x = 1/9

So, 1 repeating infinitely to the right of the decimal is equal to 1/9

But… imagine a different number.

Call the number y.

y = …11111111111 (1s extending higher and higher to the left of the decimal point)

Where x had a finite value, y is infinitely large.

But it too has a finite value!

“Great Scott!” - Watson

We’ll get there.

…111111111111111 multiplied by ten is simply …11111111111110, or y - 1

10y = y - 1

y = -1/9 = -x

 

x + y = …11111111111.11111111111…

x + y = x - x = 0

…111111111111.1111111111… = 0

y is what we call a 10-adic

An infinitely large number that is equivalent to another number

Its called a 10 adic because it is written in base-10.

But there’s a problem with 10-adics which I will explain elsewhere (not today, though) which makes mathematicians prefer what is called a p-adic, or prime-adic

This is any infinite number in a prime number system

Most common are 3-adics and 5-adics

These are cool!

 

But let’s think about 17-adics

”Great Scott!” - Watson

It’s prime out of the goodness of my heart

With 10-adics, it’s easy to think that the only adics that can zero out with rational pairs (y being the adic and x being the rational) are made of the numbers 1-10

But thats not the case

G is the highest digit in base 17, representing the value of 16

Set …GGGGGGGG to θ

And 0.GGGGG… to φ

θ * 10 (since we are in base 17, this is actually times 17) is

…GGGGGGG0, or θ - G

10θ = θ - G

Gθ = -G

θ = -1

”Great Scott!” - Watson

Lets think about φ now

φ * 10 = G.GGGGGGGG…

or

10φ=φ+G

Gφ=G

φ = 1

So

just like before

θ = -φ

and

…GGGGGGGGGG.GGGGGGGGG… = 0

So if we increase the size of our number system, we can fit more non-zero zeros into consideration.

But we can increase as much as our heart desires! So

I make a rule, in base-10, to describe all rational-adic sums that are equivalento 0

Take a function:

f(x) = b(a^x)

Sum the value of the function at every single integer for x, and it will always be 0

a and b can be any value at all.