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BitBuster

If 1+1 didn't equal 2

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1 + 1 = 10 in binary.

What now?

 

To clarify...

 

(The subscript denotes the base scale. 2 is binary, 10 is decimal.

 

The number 112 represents:

 

2 + 1

 

1 1

 

2 + 1

 

which is 310 .

 

 

 

To get 1110 you need 10112.

 

10112 represents

 

8 + 4 + 2 + 1

 

1 0 1 1

 

8 + 0 + 2 + 1 .

 

which is 1110 .

 

 

But to respond directly,

 

102 is

 

2 + 1

 

1 0

 

2 + 0

 

= 2

 

TL;DR = 1 and 1 is both 2, 11 and 3. It all depends on what you do with the numbers.

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What then?

 

Most probably it would mean our notation for the number 'two' would be different from what it actually is. It wouldn't need a large event in the past for '2' to actually notate something else, like maybe 'seven'.

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110 + 110 = 111.

 

Ummmm.......NO

 

There is no base 1. (Base 1 field theory would require dividing by zero.)

 

Or, if you prefer, there is no "1" in base 1 -- only zeroes. The base is never one of the digits.

 

However,

 

110 + 110 + 110 = 112

 

Also 1 + 1 + 1 + 1 = 1 (mod 3)

 

12 apples + blender = applesauce

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...in theory, could it exist?

 

That really depends on what your definition of "exist" is. (Sorry Bill Clinton.)

 

The problem with using 1 as a base is that it wouldn't match the rules for any other base. For example, we know that the third place left of the decimal in base 2 is 22 or 4. So, 1012 = 510 . But 1x = 1, which means every place has the same value.

 

Also, 0 would be the only valid digit. So the only number you can really express is zero.

 

Now, in Set Theory (which I only took because I was a Math major, but I loved it), the numbers we know and love (the Natural Numbers), are defined in terms of something very similar to this. Zero is the empty set. One is a set containing zero. Two is a set containing one and zero. And so on. Each number is a set with the right number of elements, and all the elements are defined recursively from NOTHING. This is where most people say "stop, I wanna get off."

 

Anyway, you could theoretically change your definition of base to include stings of zeroes, and the number of zeroes would determine the size of the number. But it's not really the same as other bases.

 

Where I tend to jump off the train is when people start talking about base e or base pi. No thanks -- I like my bases to be positive integers (greater than 1).

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That really depends on what your definition of "exist" is. (Sorry Bill Clinton.)

 

The problem with using 1 as a base is that it wouldn't match the rules for any other base. For example, we know that the third place left of the decimal in base 2 is 22 or 4. So, 1012 = 510 . But 1x = 1, which means every place has the same value.

 

Also, 0 would be the only valid digit. So the only number you can really express is zero.

 

Now, in Set Theory (which I only took because I was a Math major, but I loved it), the numbers we know and love (the Natural Numbers), are defined in terms of something very similar to this. Zero is the empty set. One is a set containing zero. Two is a set containing one and zero. And so on. Each number is a set with the right number of elements, and all the elements are defined recursively from NOTHING. This is where most people say "stop, I wanna get off."

 

Anyway, you could theoretically change your definition of base to include stings of zeroes, and the number of zeroes would determine the size of the number. But it's not really the same as other bases.

 

Where I tend to jump off the train is when people start talking about base e or base pi. No thanks -- I like my bases to be positive integers (greater than 1).

Very interesting. I was thinking that it was more of an arbitrary decision not to include base 1, but I didn't notice the differences between base 1 and the rest of them.

 

Unfortunately. ;)

You started it. :D

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I remember the proof that 1=2. But in that one you divide by zero.

 

It reminds me of the proof of "All billiard balls are the same color" or "all lines are parallel". These are basically improper uses of inductive proofs.

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I'm pretty sure that's not the upper limit. :P :P

 

Oh, I know. I debated on putting it as 1 + 1 could equal OVER 9000!!!!!, but that didn't seem very believable :P

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