Does 0.999~ = 1? [oldskool debate]

0.999999999999999999999999999999999999999999999999999999999999...

...is not the same as, 1.

Maybe, via manipulation of certain mathematical formulas, it is possible to make 0.999 equal 1, but they are ultimately two different things.

 

Well considering .9 repeating and 1 are both numbers, "certain mathmatical formulas" are the only things we should be using to analyse these numbers, as they are indeed part of math. It has been widely proven that .9 repeating equals 1. At this point you'll need a counterproof to disprove it, so please provide one.

 

The definition of "equal" differs from that of a physicist and a mathematician.
This is the reason why we have no reached a consensus and this debate will never end.

 

Using the previous mentioned algebra, it does under a caveat; the 9's extend to an infinite length. If they do not, then it doesn't work.

I'll do three:

x = 0.9
10x = 9 (0.9*10)

10x - x = 9-0.9
9x = 8.91
x = 0.9

Next

x = 0.99
10x = 9.9 (0.99*10)

10x - x = 9.9 - 0.99
9x = 8.91
x = 0.99

Next

x = 0.999
10x = 9.99 (0.999*10)

10x - x = 9.99-0.999
9x = 8.991
x = 0.999

Only when the 9's extend to an infinite length and are at parity will 0.999 recurring = 1. If you notice the pattern, the number of decimal points for the aforementioned finite numbers are never equal; reiterating, they are only equal for an infinite number.

As for the 1/3 example, that extends to infinite, and, that is why it works.

 

^ This. If infinite was a number, then x = 0.999...

x1 = 3/3
10x = 30/3
10x - x = 27/3
9x = 27/3
x2 = 3/3

x1 = x2

 

Looking at the number .999... and 1 they are not equal. But if used in an equation/proof then it can be concluded they are equal.

Hard to say really. All depends on how you want to look at it.

 

Incorrect.

The mathematical definition of equal is more stringent than its definition in physics, as some physics is based on theories, or widely agreed upon and tested yet not completely proven hypotheses. In both physics and math, .999 repeating equals 1 as both assume limits are true (which use the principle).

 

nope as it never actually goes to one but continuously keep adding 9 so 0.999999 and keeps going on, take limit laws in consideration.

 

This doesn't make sense. The manipulation of mathematical equations never breaks the rule of equality. When you have an equation and you're trying to simplify it or solve for a variable, never in any of the steps you take does the two components on either side of the equal sign not equal each other.

 

How did you get x2 = 3/3?

 

.999~ = 1.

So yes. It does, a never ending number rounds up, so you can end it. That's what I was taught.

 

This thread made me laugh;

To partially answer the question I say; "What is the meaning of life?". This is completely relevant because this question can be interpreted in many ways. Technically, 1/3 * 3 = 1, yes. But 1/3 = .33333 and 2/3 = .6666666 then 3/3 should be equal to .99999999, however, you forget about your sigfigs. Significant figures (sigfigs), state that an answer is only accurate to the least accurate point in an equation. 3/3 could technically be equal to .9999 but with only 1 sigfig, anything about .5 rounds up.

The second part of my answer consists of some relatively basic physics laws, so if you don't know some basic physics don't answer this question and say I am incorrect. The speed of light (c) is NOT 3x10^8 m/s. It is far from it. The index of refraction of air is NOT 1. Therefore, the speed of light on earth is not equal to C which is not equal to 3x10^8. Therefore 3/3 does not equal .9999, it equals 1. Saying 3/3 = .9999 is the exact same thing as saying to a 6-year-old, "Mommy and daddy have 3 apples, you get an apple but without a stem, and mommy and daddy each get an apple without the stem, but the stem no longer counts as the apple so all of the apple has been used up!" when in theory, the step is still present because matter cannot be created or destroyed under ordinary circumstances. THEREFORE; .99999 = 1

tl;dr
1.78 * 1 = 2.

 

This would only be possible in an ambiguous case which would result in a wrong answer.
/false
/prove me wrong and I can prove to you that 5.1 = 10.

 

I can't really understand what you are trying to say in the second paragraph. But 1.78 * 1 does not equal 2.

As for the first paragraph:
Sig figs are really only used in low level physical physics (Not theoretical), with real life measurements. As soon as you get to something that hasn't been designed, such as making the first rocket, or conceptual physics, (Quantum mechanics, Superstring theory), sig figs vanish. Why? because one just cannot afford the small amount of error that may result from the significant figures. A 1% error on the rocket, for example, may lead you to far outshoot the moon. On String theory, it may lead you to wrongly deduce the amount of dimensions.

 

Rounding plays no part in this. 1 is exactly equal to 0.(9).

That's an approximation. The speed of light is exactly

They're approximations. 1 is not an approximation for 0.(9); it is the exact same value.

The difference between 0.(9) and 1 is 0. Your simplistic analogy is flawed.

 

To people who don't think it equals 1, what does 1/(.9) equal then?

 

Nononono. Maybe your conclusion is similar to the OP's, but your reasoning is completely off. This concept has nothing to do with rounding, as .999... is exactly equal to 1. I don't really understand your "matter cannot be created or destroyed" example with respect to this topic, but .999... = 1 is a definitive, logical proof, not just some pragmatic approximation.

And 1.78 = 2 makes no sense whatsoever, unless you're implying that all numbers are exactly the same and that numbers mean nothing. If you add/subtract/divide/multiply from each side of 1.78 = 2 you can deduce that all numbers all equivalent, which is a patently ridiculous notion.

*Your post made me laugh~

 

i don't get it. will someone explain to me how 0.999...=1 without rounding

they are both different numbers and if .999 is infinite then it will never = 1 but always be a decimal. to me it looks like you're trying to say orange = yellow. because both are clearly different numbers so how are they equal?

 

0.999 is 0.999.

~0.999=1

 

They aren't. Rounding plays no part in this. 1/3 is .333~ therefore, 2/3 = .666~ and 3/3 = .999~ = 1

I don't get what you're trying to say.