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The Instigator
Pro (for)
Anonymous
Winning
38 Points

# .9999...Repeating exactly equals 1

Post Voting Period
The voting period for this debate has ended.
after 9 votes the winner is...
Anonymous
 Voting Style: Open Point System: 7 Point Started: 5/15/2014 Category: Science Updated: 7 years ago Status: Post Voting Period Viewed: 1,366 times Debate No: 54726
Debate Rounds (3)

 Pro I am confident as well as many mathematicians that .9 repeating is equal to 1. Not an estimation but an equivalency.For characters sake I define .99... (with an ellipse) or .9¯ or as being .9 repeating.Report this Argument I accept, noting two points. First, BOP is entirely upon upon my opponent, he must prove his resolution, while.me replenish simply to refute his arguments. Second, my opponent has taken the position that 0.9r is exactly equal to one. My opponent must prove this resolution, proving that the values are too close to differentiate does not prove that they are exactly the same.Report this Argument Pro Thank you for debating. I know the resolution is a concept that on intuition seems false to many people. However, it is a perfectly valid statement to make mathematically. Mathematics is like science and logic. There is an agreed upon process of performing operations. Without this agreed upon process 2 different people would get 2 different unequal answers. For example something as simple as 2+4x3. Thanks to the Order of Operations we know this is 4x3=12 then 2+12=14. Multiplication is done before addition. With this in mind I move to my first proof.1. 3 thirds1/3 = .3333... repeating That is the 3's go on forever. This is not an aproximation. The sign is not ≈ it is =.2/3 = .6666... This is becuase the fraction is a type of equation itself. It says 2 divided by 3.P1: 2/3 + 1/3 If we see this as an equation with no fractions it is (2 divided by 3) add (1 divided by 3)..6666... + .3333... = .999... Order of Operations PEMDAS , we do the division first. Then add together equaling .9¯P2:2/3 + 1/3 = 3/3 = 1 If we see this as adding fractions we have 3 over 3 or a whole. Which equals 1.Again all the rules of Math were followed. So we can not have 2 different people come up with 2 different "Quantities". Sure one person may right an answer as 8 and another write it as 2^3 (2 with exponent 3 or power of 3). 2 different ways of writing the same answer. But the answers are equal, they are the same quantity. Same equation, different but equal answers.C1: So 2/3 + 1/3 = .999... OR 2/3 + 1/3 = 1 Therefore .999... = 1My Opponent will have to show where I broke a rule of Math. Notice that I did not mix fractions with decimals. I added fractions to fractions, and I added decimal to decimal. It's important to see that they are not approximations of each other. But that the decimals are the result of a mathematical operation, division.2. Transitive Propertyx = .999... We start with an equation. As long as we perform the same operation to both sides, the equation stays equal.10x = 10 times .999...10x = 9.999...10x -x = 9.999... - x (which we know from earlier x = .999...)Becuase x and .9¯ are equal we can exchange them based on the Axioms of Equality of math. This allows us to calculate "like" terms. This more accurately is called the Symmetric Property.9x = 99x/9 = 9/9x = 1C2: So again we have x = .999... and we have x=1 therefore .999... equals 1. Or 1 = x = .999... This is called a transitive property of equality. 3. Number line explanantionWhen we have 2 numbers that are "not equal" and we place them on a number line, like 3 and 1. We know there is a quantity between them. We figure out this quantity by subtraction ie: 3 -1 =2. 2 is the quantity between 3 and 1.0-----1-----2-----3-----4 There are 2 "-----" between 3 and 1.P1: IF 1 does not equal .999... Then there should be a quantity between them.What is 1 - .999... = ?Well it is not .0001 Becuase the 9's go on forever. It is not .00000000001 for the same reason. This implies that the answer is a decimal point followed by an infinite amount of 0's and at the end 1. There is no end to infinity. This 1 never comes. So the answer is an infinite amount of 0's. Which is 0P2: The quantity between 1 and .999... is 0C3: There is no quantity between 1 and .999... therefore they are equal.Closing:This can be a hard truth to swallow. Maybe even harder than accepting quantum entanglement or negative energy."All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident" -Arthur Schopenhauer, German philosopherThis is not a mathematical riddle to trick someone, like the bellhop and missing dollar. This resolution is a fact. I obseved all mathematical conventions and await my opponent's rebuttals.http://www.purplemath.com...http://www.wausau.k12.wi.us...Report this Argument After extensive research and consulting with some of my professors, I no longer believe that I can contend my opponents arguments. I apologise to my opponent for being unable to present a debate. I concede to my opponent.Report this Argument Pro I am sorry to hear my opponent concedes.In the mean time here is another example:1/3 + 1/6 = 1/2 if solved using fractions..33(33)... + .166(66).. = .499(99)... If solved using repeating decimalsSo .49999999.... equals 1/2 or .5It's 2 different ways of saying the exact same thing.Some interesting video on infinityVote ProReport this Argument This debate is conceded. Vote pro.Report this Argument 8 comments have been posted on this debate. Showing 1 through 8 records.
Posted by DeletedUser 7 years ago
I was hoping you would have used zeno. approximations in my opinion would be weak. But If we walk away learning something, we are better for the experience.
Posted by PotBelliedGeek 7 years ago
For the record, I still could have argued this, bit after the research I no longer felt the confidence in my arguments. I still could argue Zeno's paradox and approximations.
Posted by DeletedUser 7 years ago
@WilliamP where were my spelling mistakes? I've been trying to improve on S+G so any help would be great.
Posted by PotBelliedGeek 7 years ago
@ n7, I have no intention of debating semantics. I will debate this mathematically and logically.
Posted by DeletedUser 7 years ago
Repeating in the title is notation not a number. Notations are for understanding not calculation. Math problems can have 2 parts Syntactic, which is the numbers and operators. And is can have the Semantic, which is words and symbols to describe the numbers.
Posted by n7 7 years ago
Pro didn't put "no semantics". Con could win this by saying "repeating" isn't a valid number. So .9999 repeating doesn't equal anything.
9 votes have been placed for this debate. Showing 1 through 9 records. 