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Why is point nine repeating equal to one?

To answer that question we should first talk about why any non-termination decimal number is equal to whatever it is. And to talk about that, we should first talk about the value of ordinary terminating decimals. Consider a terminating decimal, say 0.314. This decimal represents the sum of the fractions 3/10 + 1/100 + 4/1000; and longer (but still terminating) decimals can be computed in a similar way. But how do we decide what value a non-terminating decimal represents, say 0.314159265458979... and so on with a never-ending sequence of digits? By analogy, it should be equal to 3/10 + 1/100 + 4/1000 + 1/10,000 + ... and so on; but how can we figure out what such a never-ending sum adds up to? Well, one way of looking at it is as follows: Whatever value the decimal has, we know that (say)0.314 is off by no more than 0.001, since 0.314159... - 0.314 = 0.000159..., and 0.000159... is clearly < 0.001. Likewise, 0.3141 is off by no more than 0.0001, and 0.31415 is off by no more than 0.00001, and so on. In other words, the sequence of (terminating) decimals, 0.3, 0.31, 0.314, 0.3141, 0.31415, etc. gives us a list of better and better approximations to the ultimate value of the non-terminating decimal; and in fact by taking enough decimal places, the error in the approximation can be made as small as you like. If you've studied calculus, you may recognize this sort of discussion--it means that the value of the non-terminating decimal acts like the limit of the sequence of terminating decimals. In fact, it just *is* the limit of the sequence. So mathematicians have chosen to define the value of a non-terminating decimal as the limit of the sequence of approximations. Now we can talk about the specific case of 0.9 repeating: It turns out that the limit of the sequence 0.9, 0.99, 0.999, ... is just equal to 1, exactly (which should not be too hard to convince yourself of) and therefore the value of the non-terminating decimal 0.9 repeating is, by definition, equal to 1.


How do you remember how many ft are in a mile?

5279.9894 feet Should you repeat that in your memory would be registered.


Why do three circles show the epicenter of an earthquake?

Each circle is actually the radius around a reporting station. To find an earthquake's epicenter, you need at least three reporting stations. The radius around each station should meet at one point, the epicenter. This point should be about the size of a town, depending on how close or far the reporting stations are. With any less than three reporting stations, the exact point of the epicenter may not be located.


When should a swift code be used?

Not sure, I think when you need to transfer money in over-seas bank


When 2 over 8 is a decimal is the decimal repeating or terminating?

2/8 can be reduced to 1/4, which you should know to be .25 in decimal form. Because the decimal "terminates" after the hundredths place, it is a terminating decimal. An example of a repeating decimal would be 1/3. This would come out to be .333333... (etc.) Most calculators will display something like 2/3 as .6666667 placing the 7 on their last available digit displayer as a means of rounding the repeating decimal.

Related questions

Should i get a job as a registered nurse or cardiovascular?

you should get a job as registered you should get a job as a registered nurse


Where should an assignment of a copyright be registered?

An assignment should be registered where the original copyright was registered.


Identify the complete predicate.He should have registered for composition?

"Should have registered for composition" is the complete predicate. It includes the verb "registered" as well as the auxiliary verbs "should" and "have."


How do you determine the state your vehicle should be registered in?

Your vehicle should be registered in the state you live in.


What should Petroleum and hazardous liquid pipeline structures include?

Pump Stations


Why you should carry on using coal power stations?

We should carry on using coal power stations because we can make them eco-friendly.


How often should work stations should be cleaned?

every day


How is the real estate trust registered as owner?

The legal owner of the property must transfer the property by deed to the trustee of the trust. That should only be done by a professional who is familiar with both trust law and conveyancing in your jurisdiction.


Where can you get a list of serial numbers for firearms registered in your name?

AnswerThis question is a bit confusing. If they are registered in YOUR name, YOU must have registered them, you should know what agency you registered them with, and they should be in YOUR gun safe.


How far should eyewash stations be from chemicals?

Eyewash stations should be no more than 10 seconds walk from where chemicals are used and splashed may occur.


Can you transfer your Sims 4 expansion packs to a new computer?

With all Sims 4 games being registered through Origin, so long as you have access to that account, you should be able to redownload your original game and expansions.


Should a newborn litter of puppies be registered with limited or full registration after being registered with the akc when both parents of litter are akc registered but were sold with limited?

If the parents were sold on limited registration, they should never have been bred and their puppies will be unable to be registered at all