3 ways to prove that .999... = 1

.999... = 1, demonstrated in three different ways
Proof by fractions, series, and algebra.

Category:

Science & Technology

Tags:

• math
• repeating
• decimals
• proof
• series
• fractions
• algebra

[AxxeelR8]

they will never be equal, cus .999... will always lack . ...1. i know that u will think i'm a brat, but think of it for a second. they are just very close to one another, but will never be equal cus .(9) will always lack . ...1

[jehan60188]

@AxxeelR8 there is no such thing as ". ...1"

think about it if ". ...1" exists then ". ...1000001" exists and also ". ...1...154" exists

and unity minus any of those numbers will be less than .999...

[getonmylevel1on1]

thank you, my friends are non believers, but do understand fundamental algebra and can't refute the facts.

[Cabernal]

here's another neat "proof" that actually convinces a lot of people

suppose that .99...is not equal to one, well then take the average of the two, it's strictly greater than .99... and strictly less than 1, right?

how would you write that number?

[AxxeelR8]

@AxxeelR8 ok ok. i give up/ if 9/10 = 1 then i give up on math

[GozillaFireschocks]

So you are basically saying that if you eat 1 apple, you haven't eat 1 apple, but 0,99999 apple, leaving a tiny piece of it on the table. Seriously, WTF... Guys, one is one.. You got one birthday a year right? Not 0,9999.

[Rubenite22mg]

Well, 1-0.999... = 0.000...1  0.000...1 being infinitely small is equal to 0 therefore 1=0.9999... but infinity isn't real, point at it! :)

[xanakoj]

In addition, you would also have to conclude that 1.000...1=.999... (1.000...1 has infinitely many 0's followed by a 1). I would treat them the same mathematically due to them approaching 1, but I do not agree that they are identical. Practically, yes. Actually, no. I think 1.000...1>.999... By how much? An unimaginably small number.

The problem is that you can't actually use infinities because it is a concept, not a number. You have to use tricks like limits, which are approximations.

[brissydhiller]

@AxxeelR8 another way to show that 0.9999999...=1 is that there is no real numbers between 0.9999.....9 and 1 as they are INFINITELY close to each other. by there not being a real number between 0.9999....9 and 1 using the definition of real numbers we can see that 0.99999....9 equals 1.

also i would have done the sum of an infinite series by breaking down 0.999999999....99 into

0.99999...999= 0.9+0.09+0.009+...

then summarise it into terms of a geometric series.

[brissydhiller]

@AxxeelR8 another way to show that 0.9999999...=1 is that there is no real numbers between 0.9999.....9 and 1 as they are INFINITELY close to each other. by there not being a real number between 0.9999....9 and 1 using the definition of real numbers we can see that 0.99999....9 equals 1.

[elborrador]

0.333... is not equal to 1/3. 0.333.. approaches 1/3 as the number of decimal places approaches infinity, but is never equal to 1/3. That's why, in reality, 0.333...≈1/3. These "proofs" are important in understanding why we may use the approximation 0.999 for 1 etc but they are not proofs for 0.999... being equal to 1. You are trying to provide a limit to an indefinite process (i.e. 0.9999.... repeating "infinite" times) is all.