Is this because the negative negates the negative?
Printable View
Is this because the negative negates the negative?
That is an easy way to think of it, so yes.
It doesn't really mean the same thing as 2+2 because it represents a different math operation but it gives the same value.
(There are more complicated reasons but you don't need to understand them unless you are dealing with field theory or noncommutating operators in Quantum Mechanics.)
Ah, those damnable noncommutating operators. Never one there when you want one, then (4-1) turn up all at once!
I have no idea what these 2 replies are about, so let's try to do this properly.
First I assume we are talking the integers. This means that
1. There exists an operation conventionally called "addition" and written
2. There exists and operation conventionally called "multiplication" and wrtten
3, There exists integer and
4. For every non-zero integer there exists an integer such that
Note that "subtraction" is NOT defined. So when one writes , say, one really means
I offer a very general proof....
Consider the sum of products .
By the associative law write this as and by the right distributive law write this as
from the multiplicative property of zero
On the other hand, using the exact same rules
(although you use the left distributive law here)
Which implies that as desired.
So returning to the OP, write or better
Nice question
Subtracting a minus number is the same as adding a positive one.
Had it been 2+(-2) or 2-(2) it would have been zero. So what would -2-(-2) be?
Now how about -2*-2 or 2*-2?
Imagine a line of numbers with zero in the exact middle, reading to the left gives negative numbers counting down and reading to the right gives positive numbers counting up:
(-5)(-4)(-3)(-2)(-1)(0)(+1)(+2)(+3)(+4)(+5)
To subtract a number you move along the line in the opposite direction (from zero) to the number you're subtracting.
In this case, the number you're subtracting is a negative number (one that reads to the left starting at zero) so you'll need to go two steps in the opposite direction, which means moving two steps to the right. You arrive at +4 which is the positive number "4".
(To add a number, you move in the same direction away from zero as the number you're adding, so if you were to add (+2) + (-2) you'd need to move two steps in the same direction as the number you're adding - in this case two steps to the left because you're adding a negative number - and arrive at 0.)
Never looked at like this but it seems accurate enough. If you add 2 negatives they will produce a larger negative number, just as two positives if added would be a larger positive number. If you subtract a negative, the result will be more in the positive direction and if you add a negative it will be more in the negative direction.
My husband used to teach/reinforce this for year 8 or 9 students who didn't see it clearly enough to work with it. He'd do one of Daecon's number lines large scale on the floor or the ground outside and get the kids to be robots with very limited capacities - a) to take one step at a time b) to turn 180 degrees, neither less nor more.
It's very straightforward once you get it "into your bones" by stepping it out. (Though if you take it too far you can make yourself dizzy if you try to do multiplication of negative numbers, you finish up whirling like a dervish.)
Another teacher used to use the idea of trains getting to stations (points on the number line) and either reversing direction or going straight ahead, but I never fully got how he taught it so I've never been clear how that goes.
My school taught is as such:
- Plus - = more negative
- Minus - = The last two negatives switch signs and turn into negative plus a positive
+ Plus + = more positive
and multiplication is as such:
- Times + = -
- Times - = +
+ Times + = +
- divided by - = +
+ divided by - = -
+ divided by - = -
In this case:
2-(-2)=4
You are seeing an invisible one! you can look at it like this:
2-1(-2)=4
Now if you remember PEMDAS you then multiply the -1 and -2 (negative times a negative is a positive) you then get
2+2=4
It is 2+2 because the two is positive. If it was the other way it would look like this
2-(2)=0
Once again a invisible one:
2-1(2)=0
Then multiply -1 by 2 (negative times a positive is positive) and you would then get:
2-2=0
This is a bit a more complicated way of looking at it because you are using multiplication instead of the normal rules of addition
i remember having a math class in university many years ago. it was after calculus 1,2,3 and diffeq. the class went backwards to the very basic 'things' and set about rigourously proving them. i remember pages of a proof of why an even number multiplied by any number (odd or even) will always yield an even number. multiplication and addition with negative and positive numbers was part of it. it was a very difficult class. it was all the things we do and take for granted doing calcualtions.
That sounds a bit like the real analysis class I took. It was useful, but the main thing I remember about it was the unfortunate abbreviation in the list of courses.