0.999999 . . . . = 1

Dr Huff, You've been exposed to gas station price signs too long. LOL

You can use the same technique with any repeating decimal (it's a sign of a rational number)

Try
x=1/7=0.142857142857142857..........
multiply this by 1,000,000 (10 to the power of the number of repeating digits---in this case 6 or 10^6)
1,000,000x=142857.142857.......
subtract off 1
999,999x=142857
the x=142857/999,999=142857/(7*142857)=1/7

Rap

I usually show the .99999 to be an anomaly by using some other repeating single integer such as .22222, so as to retain the form of the brain teaser.

1. define X as 0.222222 . . . .

2. multiply X by 10: ( 10X )

3. We know that 10 times 0.222222 . . . . = 2.22222 . . . . So this means 10X = 2.22222 . . . .

4. Now subtract X from 10X : 2.222222 . . . . -- 0.222222. . . . = 2. (the parts to the right of the decimal point disappear because they are the same)

5. So 9X =2

6. and if 9X =2, then X must =0 .22222 . . . .

7. Substituting for X from 1., 0.22222 . . . . = 0.22222 . . . .

What goes on in you guy's heads?

Evolution didn't prepare us for that stuff surely?

Yes it did, and don't call me Shirley.

The number 0.9999.... will approach 1 to any desired degree of closeness, but it will never equal 1.

contrex is spot on! .9999 factored makes a big difference in science. It can make the difference between reaching Mars or missing it.

It's obvious, surely, that an arbitrarily small number, no matter how small you may make it, can never be the same as zero?

It's these amateur scientific types contrex. They are forever on the brink of the asymptote. How life began. The secret of everything. Cure for all illnesses.

Getting there does God in you see. Their need to do that causes them to confuse the brink with the real thing. A literary conceit you might call it.

If you define X as 0.999999 . . . . you have an indeterminate number and you cannot subject indeterminate numbers to arithmetical manipulations. That's why, I presume, such things were heresy in Ancient Greece.

Aside from not being able to do it such a multiplication (by 10) multiplies the indeterminancy (by 10) and the gap from the brink to the real thing yawns wider and they are 10 times further away than they had previously thought. If they repeat their manipulation, which infinity invites them to do, a chasm appears and so on and so on until they end up the stupidest people on earth. Possibly in the universe.

They can't grasp that they can't ever grasp infinity. The concept eludes human understanding.

Maybe, but maybe not. The concept of an infinitesimal was widely used in calculus before it was made more rigorous. Mathematicians approximated the infinitesimal by using the smallest numbers they could bother to use when evaluating limits they couldn't solve using other techniques.
Other food for thought, spendi talks about how humans can have no understanding of the infinite, but he's strictly wrong about that. We know things about sets with an infinite number of objects in them. For example, back to your question about arbitrarily small numbers, if we accept the well ordering theorem, we can say that there is a number just barely larger than zero. The well ordering theorem says that every set can be ordered in such a way that there is a first element, which means that there is a number that comes first on the interval (0,inf), and that number is just larger than zero.

Thank goodness for that.

As long as it's "just" larger than zero. (Places thumb and forefinger as close as is possible without them touching after 4 pints of John Smith's Extra Smooth. [Only £2.35 in high class establishments] )

The apparent weirdness only crops up when one mixes numbers like 0, 1, with infinities say 0.9 repeater (that repeater means to infinity, so one is mixing numbers with infinities) = danger Will Robinson.

Even things like 10X = 9.99999 etc required constrained definitions to the terms you are using because times and equals are operations that imply numbers in a field interacting with other numbers in the field. An infinity is not a number in the field rather its an idea about the lack of boundaries in a field. So multiplying a number by the idea of a lack of boundary is a subtle slight of hand really!