Confused about negatives!

[CBP]

Hello,

I can understand why 10 - 3 = 7 and why 10 + (-3) = 7, but I can't figure out how to explain why 10 - (-3) = 13. It doesn't seem enough to simply say that a negative and a negative equal a positive. Why exactly do we do this? Every explanation I have found expects me to merely memorize the rule. But if we use number lines to explain the first two problems, then surely we can use a number line to help explain this problem, too, right?

Thank you!

[stapel_eliz]

Subtraction of a negative number has to be a negative or a positive value. If it's a negative value, then you would, for instance, have 3 - (-2) = 3 - 2 = 3 + (-2). Taking out that step in the middle would leave you then with 3 - (-2) = 3 + (-2). Adding 3 to both sides would give you -(-2) = (-2). Would this make sense?

In a sense, the negative of a negative is positive because it has to be; math wouldn't work, otherwise.

[CBP]

Well, what you wrote makes sense mathematically. Meaning, I've learned the rules, so when I apply them, that makes sense. But why can we explain 10 + 3 and 10 + (-3) using a number line or apples, but we can't explain 10 - (-3) using those same things? Or can we/you? My dad, a retired engineer with a Ph.D., says that if adding a negative number yields a smaller number, then subtracting a negative number would do the opposite by increasing that number. He says it's "intuitively obvious." But . . . that's not an explanation. Shouldn't we be able to explain WHY this happens/works?

[stapel_eliz]

...if adding a negative number yields a smaller number, then subtracting a negative number would do the opposite by increasing that number.... But . . . that's not an explanation.

It might not be an explanation you accept, but saying that "doing the opposite ought to lead to an opposite result" is an explanation, and a fairly reasonable one.

The explanation provided here earlier was more logical than mathematical: the result must be "this" or "that", and "that" leads to a logical contradiction, so the result then must be "this". (A mathematical proof would involve graduate-level field theory.) Your father has provided you with another logical explanation; namely, if "this" leads to "the other", then "not this" ought sensibly to lead to "not the other". And there are various other explanations and mental pictures, such as those provided here, here, here, and here.

If none of these "works" for you, then you may just need to accept the rule for now, and give yourself some time to become comfortable with its practice.

[CBP]

I'm comfortable enough with the practice of this rule; I've made it through a second term of calculus several years ago. But now that I'm revisiting math as a graduate student in special education (hoping to be a math specialist), all of my old questions keep popping up. I was happy to learn that there is a mathematical proof for my question and that it involves higher mathematics. I'm intending to get my B.S. in math someday and perhaps beyond, so maybe I'll bookmark this and revisit my question down the road. Thank you!

[CBP]

Ha, that first link you provided did the trick. It's not a proof, as it says, but still . . . the hot cube and cold cube thing gave me what I was looking for. Thank youuuu!!!!!!!!