Richard asked in Science & MathematicsPhysics · 7 months ago

# If .999... is equal to 1, then infinitely small must be 0 right?

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• 7 months ago

0.9999999........ is an actual number you can perform mathematical operations on that shows its equal to 1. Infinitely small is a subjective phrase that has no basis to have mathematical operations performed on it to say it equals anything.

• 7 months ago

0.999 no matter how many nines will never be one, BUT

0.999... is actually EQUAL to one because the three dots mean "unto infinity",

it has actually reached infinity.

While you can never write out how many nines that is,

it can be represented mathematically with the three dots.

Consequently, the opposite end is also true: 0.000...1 is actually equal to zero.

Infinitely small, is actually having reached infinity, possible only in the rarified world of mathematics.

Do not get this confused with an "Infinitesimal", which is negligibly small, but not actually zero.

• 7 months ago

Let x=0.999999....

10x=9.99999...

10x-x=9

9x=9

x=1.

Infinity small doesn't mean 0.

• goring
Lv 6
7 months ago

99.99999.........9999 is a mathematical operation of a percent of an entity. The number of decimal places has a real meaning if and only it the entity is a very large definite volume. The same reality also applies to never ending irrational  number.

Mathematics does not always follows logic or reality.note infinity has no end.hence all infinities are equivalent.

For example if you divide a pie by zero you are  not dividing the pie by anything ,hence you still have the same whole pie.

• 7 months ago

Limit as x approaches infinity of (1/x) is zero.

• 7 months ago

In terms of simplicity, the reason why we would round off so many digits is to make it easier for scientists to replicate and to write down on paper. Making a single stroke versus writing out 9 9s is for the sake of time and ease. Sure, .99999999999999999999 is VERY close to 1, it is, in fact, NOT 1. No matter how many more nines you would add to that, it still wouldn't be 1. The same concept for .0000000000000000000000001. Yes, it is small, and yes it's pretty insignificant, but it's still SOMETHING. That is something we can't round off. It's still there, just, very, very small. Hope this was helpful.

• 7 months ago

in essence. .999999... does not actually equal 1 though. It is just effectively 1. At some point you can't tell the difference, yet it is never truly there. Same with small. eventually, the tiny little 1 (0.000...1) is too small to even see, so there is no effective difference. But it is still not truly 0. any more than 0.999... is truly 1.

• oubaas
Lv 7
7 months ago

it tends to zero , but > 0

• geezer
Lv 7
7 months ago

I can see what you are asking ?

If 0.9999999999999999999 ... etc ... forever .. is ''equal to 1''

is 0.0000000000 .. forever .. but with a 9 at the end .. ''equal to 0'' ?

Yes .. if one is true then the other one is as well !

• Petter
Lv 7
7 months ago

No, no matter how inifinetly small, it's still something.