Here are rules you can use to operate on positive and negative numbers.

Operation	Case	Algebraic	English	Example
a + b	Both positive	a + b	Add in the usual way.	3+4 = 7
	Both negative	−(|a| + |b|)	Add the absolute values, then make the result negative.	-3 + (-4) = -(|-3| + |-4|) = -(3 + 4) = -7
	One positive, one negative	(Assuming |a| ≥ |b|.) If a is positive, then |a| − |b|, otherwise −(|a| − |b|)	Take the difference of the absolute values; if the one with the greater absolute value is positive then the result is positive; otherwise the result is negative.	3 + (-4): First compute the difference of the absolute values: |-4| - |3| = 4 - 3 = 1. Then, since |-4| >|3|, the sign of the result is negative so the result is -1.
a − b		a + (−b)	Subtraction is the same as adding the opposite, sometimes called "keep change change" (keep the sign of the first term; change the subtraction operator to an addition operator, and change the sign of the second term.)	3 - (-4) = 3 + (+4). Then use addition rules above to compute the result.
a × b	Same sign	|a| × |b|	Multiply the absolute values.	-3 × (-4) = |-3| × |-4| = 3 × 4 = 12
	Opposite signs	−(|a| × |b|)	Multiply the absolute values, then make the result negative.	3 × (-4) = -(|3| × |-4|) = -(3 × 4) = -12
a ÷ b	Just like multiplication but divide instead of multiply.			12 ÷ (-4) = -(|12| ÷ |-4|) = -(12 ÷ 4) = -3

If you are comfortable with the algebraic column in the table above, you may be ready for the proof challenge!