How to read this to others, so they can write it down just by listening?
1 + (-2) = -1
1 - 2 = -1
I want to know the differences.
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How you explain this really depends on how much you want to convey, and how much you think will be assumed. For instance, the most specific wording of the first example would go something like this:
One plus open parenthesis negative two close parenthesis equals negative one.
The second example would be as follows:
One minus two equals negative one.
The important difference (pun intended) is how to say the '-' sign. I have found, contrary to the other answers so far, that a '-' when used on only one number (such as on the right side of those equations) is referred to as a 'negative' operator, whereas when used on two numbers it's a 'minus' sign. Saying 'two plus minus one', to me, implies 'plus or minus', which might not apply in most contexts but is definitely worth avoiding. On the other hand, 'two plus negative one' is immediately clear to me.
This even makes the parentheses in your first example unnecessary, and you can just say
One plus negative two equals negative one.
Unless you actually want everyone to write down the parentheses, in which case you should be explicit.
As a maths teacher, I would accept any of the following as correct (listing my most preferred reading first):
One plus negative two equals negative one
One plus minus two equals minus one
One add negative two equals negative one
(Other arrangements of "negative" versus "minus", and "add" versus "plus", are also possible, although I'd be a little bemused if someone said "negative two" and then "minus one"!)
Other readings are, of course, possible, although I would regard something like "the sum of one and minus two is minus one" as an interpretation, or maybe a paraphrase—describing the meaning, rather than reading it as written.
One take two equals negative one
One subtract two equals negative one
One take away two equals minus one
One minus two equals minus one
This last one illustrates the potentially confusing status of "minus" as both "something done to two numbers" and "a type of number" (in grammatical terms, both a verb and an adjective; in mathematical terms, both a binary and a unary operation).
There's no much difference on those examples.
For example the second one is read as one minus two equals minus one. The first one can be read similarly (one plus minus two equals minus one), but having something like 1 - (1 - 2) = 2, that's different, you need to say there's a bracket, and then that's one minus, open bracket, one minus two, close bracket, equals two. You can also use parenthesis.
In the first example, i would personally say
One plus [pause] [quickly] minus two [pause] equals minus one
Depending on my mood, i might say "negative" instead of "minus".
It's shorter than saying aloud "parenthesis/bracket", but still conveys the separation.
I've never had anyone complain about it being hard to understand, at least not with short expressions.
A slightly more complex example to explain:
1 + (2 * 3) = 7
One plus [pause] [quickly] two-times-three [pause] equals seven
(1 + 2) * 3 = 9
[quickly] One-plus-two [pause] times three equals nine
You could simply say :
One plus minus two is equal to minus one.
One minus two is equal to minus one.
Logically, there is no difference between the two expressions. I'm sure you're aware of that. Generally brackets () are used to avoid confusions with the arithmetic operators. If you want to know more about the mathematical aspects of the expression, you should ask the same questions here. The math geniuses there will certainly help you. In these expressions, you need not specify the brackets as they are mostly used only for written purposes and not used verbally. Additionally, you could swap 'is equal to' with 'equals'.