When does 1 + 1 not equal 2?

When I do the math many things don’t equal what they should. This would be one of those cases.

In binary arithmetic, 1+1 = 10.
10 base 2 is 2.

@GeorgeGee people like you are the pride and joy of all educators! Thank God for guys like you.

when 1 doesn’t = 1

This many and this many always equal this many. But we use different names and symbols for that many, depending on what number system we are working in. “10” always stands for the base number. In our usual way of counting, “10” means this many.

@Jeruba its the same picture so technically 1 + 0 = 1

Not at all. It’s a quantity.

Isn’t it also when it’s an equation like -1+1= 0 ?
Or am I missing the point here?

@Jeruba one apple + one apple doesn’t necessarily = 2 because all apples are slightly different shapes and sizes…sometimes a half an apple is called an apple…when you ask someone what they are eating sometimes they say an apple they don’t always say 0.5 apples…the same is true for S/O’ers 1 + 1 – 1 + 1 isn’t the same as 1 + 1… you can’t quantify and replace someone in your life equally with anyone else

We’re not adding fingers or apples. We’re adding quantities. I am making a distinction between a symbol (1) and the thing it represents, namely, as many of something as you can indicate with a single finger. Could be a single marble, apple, planet, whatever. I chose fingers because they are a very common way to signify this abstract thing that is quantity.

One and one of something will always equal the quantity symbolized by two fingers. But it will not always equal “2” or a mark named “two” because that is a symbol and not a number. “2” is not a number. It’s a symbol. The quantity is the number, and I showed it instead of naming it because the name is the symbol.

I’m giving the same answer that @GeorgeGee gave, and adding an explanation. It has nothing at all to do with personal relationships or the irregularities of fruit. It is about the abstract idea of number and the different ways it’s symbolized, depending on the base you are counting in.

In base 2 you count like this:

1 10 (this is the base)
11 100 (this is the square of the base)
101 110
111 1000 (this is the cube of the base)

In base 6 it’s like this:

1 2 3 4 5 10
11 12 13 14 15 20
21 22 23 24 25 30
31 32 33 34 35 40
41 42 43 44 45 50
51 52 53 54 55 100
etc.

Here “100”—the square of the base—is equal to 36 in base 10, which is 6×6.

Mathematical operations are not physical actions, nor is counting the sole use of numbers. For example, 1+1 = 0 (mod 2), and the numbers available in this system suffice only to count up to 1. When dealing with an algebraic structure similar to the familiar number systems, one customarily uses “1” to denote the multiplicative identity in that structure. Outside of everyday arithmetic, the expression “1+1” is context dependent.

I remember an infamous mathematical equation or proof that proves 0 = 1, and others. These invalid proofs are known as mathematical fallacy. Here’s a good example:

No Study = Fail
+ Study = No Fail
No Study + Study = Fail + No Fail
(No + 1) Study = (No + 1) Fail
Study = Fail

@RomanExpert I’m no mathematician but I’m guessing that the error is in the simplification stage:
(No + 1) Study = (No + 1) Fail

When you put 1 and 1 together on paper. It becomes 11 not 2. :D

@downtide The (No + 1)s cross each other out…Lol!