# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

It's a simple "glitch" in our decimal representation of fractions.

A cool example is 1/7 = .(142857)
6/7 = .(857142)

1/7 + 6/7 = .(142857) + .(857142)
7/7 = .(9)

You round off 1/7 to .142857, and 6/7 to .857142, and then add them together a don't get exactly 1 and you call it a "glitch" in the decimal representation of fractions?

When I open my Windows calculator and enter 1 divided by 7 and click the equal sign I get an answer. I store that answer in memory. I then enter 6 divided by 7 and click the equal sign and get an answer. I press the plus sign, and then press memory recall, and finally the equal sign, and it adds the two together and the answer is.......1!

You round off 1/7 to .142857
Look again.
1/7 = .(142857)

The parentheses mean that it is a repeating decimal.

...then add them together a don't get exactly 1
You do get exactly 1. That's the point.

Look again.
1/7 = .(142857)

The parentheses mean that it is a repeating decimal.

So you are trying to add two repeating decimals together? How is a repeating number added to anything while it's repeating?

You do get exactly 1. That's the point.

If it was exactly one the number "1" would be on the left side of the decimal and a zero would be on the right side of the decimal. In this case there is a zero on the left side of the decimal and a 9 on the right side of the decimal, followed by an infinite amount of 9's. The two are not the same, Pete, they are different! One is a whole, and the other is a fraction of a whole.

So you are trying to add two repeating decimals together? How is a repeating number added to anything while it's repeating?

It's not that difficult.
Try this:
Code:
``````  0.111...
+ 0.111...
________

0.222...``````
No problem!

Now this:
Code:
``````  0.142857142857142857...
+ 0.857142857142857142...
_______________________

0.999999999999999999...``````
Easy!

If it was exactly one the number "1" would be on the left side of the decimal and a zero would be on the right side of the decimal. In this case there is a zero on the left side of the decimal and a 9 on the right side of the decimal, followed by an infinite amount of 9's. The two are not the same, Pete, they are different! One is a whole, and the other is a fraction of a whole.
Welcome to the party.
Your skepticism of the notion is understandable, and common. See Wikipedia: 0.999...: Skepticism in education

It's not that difficult.

You are just rounding off, adding them together, and then adding the ... at the end. That's like claiming you are correct by giving me a finite number as to how many Jelly Beans are in the pile while I continue to throw more on the pile. Your guess is a finite number with the note that I was still throwing Jelly Beans on the pile when you gave your finite answer.

You can't see that 0.111... + 0.111... = 0.222...?
Do you think that somewhere in the infinite line of 1's, a 3 might pop up?

You can't see that 0.111... + 0.111... = 0.222...?

How many is .111...? Is that exactly .111, or more than .111? If it's more than .111 then how much more?

How many equal pieces of the pie are there if I have one of those equal pieces that is .111... of the pie? How many pieces of pie are there, Pete?

How many is .111...? Is that exactly .111, or more than .111? If it's more than .111 then how much more?

It's more than .111, MD. It's 0.000111... more.
Have you seen repeating decimals before? This link might help: Recurring Decimals

It's more than .111, MD. It's 0.000111... more.

So the more it is the less pieces of pie there are? For instance, if all the pieces are of equal size, and I have 1/10 of the pie, then I have 1 of 10 pieces. If I have .111... of the pie how many pieces are there?

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So the more it is the less pieces of pie there are? For instance, if all the pieces are of equal size, and I have 1/10 of the pie, then I have 1 of 10 pieces. If I have .111... percent of the pie how many pieces are there?

We've had this discussion before, MD. Time to stop this sidetrack.

So you are trying to add two repeating decimals together? How is a repeating number added to anything while it's repeating?
You are welcome to wait until the decimal representation stops repeating if you want...

You are welcome to wait until the decimal representation stops repeating if you want...

That's not the point! The point is that it is finite, not repeating. Take a pie and divide it into 3 pieces. Each piece is an exact percentage of the pie, and the sum total of the three pieces adds to 100% of the pie, not 99.999999...% of the pie. Each piece is an exact percentage of the pie, it is not 33.3333....% of a pie, each piece being a repeating decimal piece of pie. Sound ridiculous? It is! Who ever heard of a repeating decimal piece of pie?

There is one pie and it is divided into three pieces, and those pieces make up ONE WHOLE PIE, not 99.99999...% of a whole pie!

The point is that it is finite, not repeating.

Basic arithmetic says the opposite to your claim.

Take a pie and divide it into 3 pieces. Each piece is an exact percentage of the pie, and the sum total of the three pieces adds to 100% of the pie, not 99.999999...% of the pie. Each piece is an exact percentage of the pie, it is not 33.3333....% of a pie, each piece being a repeating decimal piece of pie. Sound ridiculous? It is! Who ever heard of a repeating decimal piece of pie?

Tough, 0.(3) is the decimal REPRESENTATION of 1/3. They BOTH mean the SAME thing. They explain this in 5-th grade, when they explain fractions and decimals.

You are welcome to wait until the decimal representation stops repeating if you want...
LOL, good one!

Tough, 0.(3) is the decimal REPRESENTATION of 1/3. They BOTH mean the SAME thing.

So you would be paid in full if I gave you 0.(9) dollar payment for the \$1.00 I borrowed from you?

Common sense says you're wrong! 1.0 is greater than 0.(9) all day long! 1.0>0.(9)

Who ever heard of a repeating decimal piece of pie?
It's an anomaly in representation due to us having 10 fingers. Transpose it into base 9 and the repetition goes away. (Although I am sure others will appear.)

So you would be paid in full if I gave you 0.(9) cents payment for the \$1.00 I borrowed from you?

Yep. Live with it.

Common sense says you're wrong! 1.0 is greater than 0.(9) all day long! 1.0>0.(9)
It is refreshing to see that your fringe ideas extend to basic arithmetic, they are not restricted to physics.

Yep. Live with it.

Your math is the problem. When you split a pile of 100 pennies into 3 piles, you can have two piles of 33 and one pile of 34, but you can't have 3 piles of 33.(3), that's just reality, deal with it! If you split a PENNY into three pieces instead of a pile of 100 pennies into three piles then you are back to square one, having at the most 2 equal and one larger piece. Reality is a cold slap in the face, eh Tach?