phony math/algebra solution

no matter how they manipulate the numbers, the two values
of x are different, even though the difference is negligible.

the resultant value of x before the decimal point is not the same
value as the x after the decimal point.

as you add more nines, it gets closer and closer but will never be equal.

the only time it seems to work is if the original x is 'point nine' and there is no resultant decimal value...
'seems to work'...
by coincidence, not procedure.

... after the decimal point"?

The video never puts x after the decimal point. The equations state:

x = 0.999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9

Do you see something invalid in that sequence of equations?

You can actually view x as the sum of an infinite geometric series: x = 9/10 + 9/100 + 9/1000 + ...

And the sum of an infinite geometric series is the first term, in this case 9/10, divided by (1 - the ratio), in this case 1/10. So:

x = (9/10) / (1 - (1/10))

For 9x =9, x=1

No matter how many 9s you put after the decimal point, that part is almost one.

It's rounded up.

It never becomes one.

Not in a practical sense, because we can never actually do anything an infinite number of times -- such as writing a 9 an infinite number of times.

As a concept, however, they're the same.

Just like 1/3 = 0.333333...

Pi equals 3.14159... -- with decimals that never infinitely repeat since it's an irrational number that can't be expressed as a fraction. (Pi is also the sum of a converging infinite series.)

If the decimals infinitely repeat, the corresponding fraction equals the repeating numbers divided by 10 to the power of the number of repeating numbers minus 1 -- e.g., 0.3 (repeating) = 3/(10-1) = 3/9 = 1/3.

And 0.127 (infinitely repeating thereafter) = 127/(1000-1) = 127/999.

If the decimal value is instead 0.38127 (with 127 repeating thereafter), it equals 38/100 + (127/999)/100 = [38x999 + 127]/99900 = 38089/99900.

... involves asking if there's some tiny value that separates 1 and 0.999...

Since ANY such tiny value can never exist (as 0.999... keeps converging closer to 1), they are equal.

Below is a video about the concept of infinity that I think is hilarious, but I want to first clarify that I'm NOT comparing you to Karl Pilkington!

Karl's mind is EXTREMELY practical during this conversation with Ricky Gervais! (The concept of infinity is actually very practical for mathematics in many instances, though.)

They get mighty close though.

Any incredibly tiny value that you can imagine will be larger than what separates them as the infinite series 9/10 + 9/100 + 9/1000 + 9/10000 + ... continues FOREVER.

If there can never be a value that separates their difference, they're equal.

Just like 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... equals 1, as an INFINITE series.

With each iteration of that particular series, the difference between the summation and 1 equals the last term added. But no matter what incredibly small number we consider, we will eventually reach a term 1 / (2^n) in the series which is even smaller. Thus, no such difference can be found and they're equal.

Without this kind of reasoning, simple formulas for what integrals equal in calculus wouldn't exist. Nobody says, "This algebraic formula ALMOST equals this integral."

It can be accepted as equal for practical purposes but they will never be equal.

If x and y are distinct real numbers, then there are in fact infinitely many real numbers t with x < t < y. What are some numbers between 0.9999... and 1?

It's 'infinitely' small but it's still there.

Can you give me or describe a number that lies between?

"Infinitely small" seems like a number that would be close to zero.
Can you give me or describe a number between 1.9999... and 2, or would those be equal?

Love your play.

Let us know how many decimal places you go to get 0.000000000...1 equal zero.

That collection of symbols, 0.000000...1, can only represent ten to a large negative integral power, which would not be zero.

You'd have to create a new system of representation to have this be equal to zero. (Like the surreal or hyperreal numbers.)