The Instigator
Mhykiel
Pro (for)
Winning
38 Points
The Contender
PotBelliedGeek
Con (against)
Losing
2 Points

.9999...Repeating exactly equals 1

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Post Voting Period
The voting period for this debate has ended.
after 9 votes the winner is...
Mhykiel
Voting Style: Open Point System: 7 Point
Started: 5/15/2014 Category: Science
Updated: 2 years ago Status: Post Voting Period
Viewed: 896 times Debate No: 54726
Debate Rounds (3)
Comments (8)
Votes (9)

 

Mhykiel

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.
PotBelliedGeek

Con

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.
Debate Round No. 1
Mhykiel

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 thirds

1/3 = .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 [1], 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... = 1

My 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 Property

x = .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 = 9
9x/9 = 9/9

x = 1

C2: 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. [2]


3. Number line explanantion

When 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 0

P2: The quantity between 1 and .999... is 0

C3: 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 philosopher

This 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.


[1]http://www.purplemath.com...
[2]http://www.wausau.k12.wi.us...
PotBelliedGeek

Con

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.
Debate Round No. 2
Mhykiel

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 decimals


So .49999999.... equals 1/2 or .5

It's 2 different ways of saying the exact same thing.

Some interesting video on infinity

https://www.youtube.com...

Vote Pro
PotBelliedGeek

Con

This debate is conceded. Vote pro.
Debate Round No. 3
8 comments have been posted on this debate. Showing 1 through 8 records.
Posted by Mhykiel 2 years ago
Mhykiel
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 2 years ago
PotBelliedGeek
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 Mhykiel 2 years ago
Mhykiel
@WilliamP where were my spelling mistakes? I've been trying to improve on S+G so any help would be great.
Posted by Christian_Debater 2 years ago
Christian_Debater
Man, nice argument Mhykiel
Posted by PotBelliedGeek 2 years ago
PotBelliedGeek
@ n7, I have no intention of debating semantics. I will debate this mathematically and logically.
Posted by Mhykiel 2 years ago
Mhykiel
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 2 years ago
n7
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.
Posted by Enji 2 years ago
Enji
but i was going to debate this :(
9 votes have been placed for this debate. Showing 1 through 9 records.
Vote Placed by Zarroette 2 years ago
Zarroette
MhykielPotBelliedGeekTied
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Total points awarded:30 
Reasons for voting decision: Con gracefully concedes.
Vote Placed by creedhunt 2 years ago
creedhunt
MhykielPotBelliedGeekTied
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Reasons for voting decision: Con conceded, and made a spelling mistake
Vote Placed by whiteflame 2 years ago
whiteflame
MhykielPotBelliedGeekTied
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Reasons for voting decision: Conceded. I felt the arguments were good, so it would likely have been a tough hill to climb anyway.
Vote Placed by Jonbonbon 2 years ago
Jonbonbon
MhykielPotBelliedGeekTied
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Reasons for voting decision: Concession of arguments and implied concession as to the reliability of sources.
Vote Placed by Wylted 2 years ago
Wylted
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Reasons for voting decision: Con concedes
Vote Placed by TN05 2 years ago
TN05
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Reasons for voting decision: Con conceded.
Vote Placed by SeventhProfessor 2 years ago
SeventhProfessor
MhykielPotBelliedGeekTied
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Reasons for voting decision: CC
Vote Placed by Stephen_Hawkins 2 years ago
Stephen_Hawkins
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Reasons for voting decision: This is how to do a debate on this issue properly, well done to PRO.
Vote Placed by WilliamsP 2 years ago
WilliamsP
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Reasons for voting decision: Conduct is awarded to Con due to his concession. Spelling and grammar also goes to Con because Pro happened to misspell a few very basic words. Convincing arguments goes to Pro because he actually made an argument. Finally, reliable sources goes to Pro again because he actually presented sources.