What are the rules for adding, subtracting, multiplication, and division with negative and positive numbers?

Question

Does it matter for addition and subtraction if the first digit in the problem is larger or smaller??

Answer 1

neg + neg = neg
neg + pos = {the sign of the number with the greater absolute value}
pos + neg = {the sign of the number with the greater absolute value}
pos + pos = pos

neg - neg = {the sign of the number with the greater absolute value}
neg - pos = neg
pos - neg = pos
pos - pos = {the sign of the number with the greater absolute value}

pos × pos = pos
pos × neg = neg
neg × pos = neg
neg × neg = pos

pos ÷ pos = pos
pos ÷ neg = neg
neg ÷ pos = neg
neg ÷ neg = pos

By:

{the sign of the number with the greater absolute value}

I mean that the sign of the answer will be the sign of the number whose value would be larger if you don't care about signs.

4 + -9 produces a negative result, because 9 > 4, and the 9 is negative.
-4 + 9 produces a positive result, because 9 > 4, and the 9 is positive.

Answer 2

Addition:

A1: Where both numbers are positive, use the standard algorithm

A2: Where one number is positive and the other negative, subtract the absolute value of the negative number from the positive one (the absolute value is basically the number you get by disregarding sign -- e.g. |1| = |-1| = 1 (the |x| denotes the absolute value of x). The absolute value is always a nonnegative number).

A3: Where both numbers are negative, sum their absolute values and then stick a negative sign in front of the sum.

Subtraction:

S1: Where both numbers are positive and the first number is greater than the second -- just use the standard subtraction algorithm.

S2: Where both numbers are positive and the second number is greater than the first: subtract the first number from the second, and then stick a negative sign in from of the answer (e.g. given a-b, compute -(b-a)).

S3: Where the first number is negative and the second positive: sum the absolute values of the two numbers, and then stick a negative sign in front of it

S4: Where the first number is positive and the second negative: sum the absolute values of the two numbers. Your result will be positive.

S5: Where both numbers are negative: Take the difference of the absolute values of the two numbers, and then flip the sign (i.e. make the result positive if the difference between the absolute values is negative, and negative if the difference of the absolute values is positive).

Multiplication and division:

Multiply (or divide) the absolute values of the two numbers, and then set the sign as follows:

M1: If the original numbers had the same sign (i.e. both positive or both negative), the result is positive.

M2: If the original numbers had opposing signs (i.e. one positive and one negative), the result is negative.

It is generally more useful to know the derivation of these rules. Throughout, a and b are assumed to be nonnegative numbers (so a negative number will take the form -a or -b). Note that these algebraic identities are true even when a and b are allowed to be negative (and if a is negative, then -a is positive), but their usefulness for computation derives from the case where a and b are nonnegative.

A1: a+b = a+b
A2: a+(-b) = a-b, (-a)+b = b+(-a) = b-a
A3: (-a) + (-b) = (-1)a + (-1)b = (-1)(a+b) = -(a+b)

S1: a-b = a-b
S2: a-b = a+(-b) = (-1)²a+(-1)b = (-1)((-1)a+b) = -((-a)+b) = -(b-a)
S3: -a-b = (-1)a+(-1)b = (-1)(a+b) = -(a+b)
S4: a-(-b) = a+(-1)(-b) = a+(-1)²b = a+b
S5: (-a)-(-b) = (-1)a-(-1)b = (-1)(a-b) = -(a-b) (note that since a-b might be negative, -(a-b) is not necessarily negative).

M1: ab = ab, (-a)(-b) = (-1)a(-1)b = (-1)²ab = ab
M1 (division): a/b = a/b, (-a)/(-b) = (-1)a/((-1)b) = (-1)/(-1) * a/b = a/b
M2: (-a)b = (-1)ab = -ab, a(-b) = a(-1)b = (-1)ab = -ab
M2 (division): (-a)/b = (-1)a/b = -a/b, a/(-b) = a/((-1)b) = 1/(-1) * a/b = (-1) * a/b = -a/b

Of course, there is the all important identity (-1)²=1, which gives many students trouble. So we derive this as follows:

0*(-1)=0, but:
0*(-1) = (1+ (-1))*(-1) = 1*(-1) + (-1)² = -1 + (-1)²
So:
(-1)² + (-1) = 0
(-1)² - 1 = 0
Adding 1 to both sides:
(-1)² = 1

Of course we are assuming such things as the associative, commutative, and distributive properties here, and some students might wonder why we make such assumptions. The key to remember here is that the reason negative numbers were invented was not to model the real world (the fact that many situations in physics, and to a lesser extent in finance, lent themselves to modeling with negative numbers was a happy accident. The same thing would later happen again with the imaginary numbers), but rather to simplify algebraic manipulations (think about all the special cases we had to deal with for computations. Now imagine having to do that with symbols of unknown quantities, where you don't have the luxury of knowing ahead of time which rule to use. Before negative numbers became commonplace, you had to have seperate equations to describe, for instance, motion under acceleration parallel to the direction of motion, and motion under acceleration opposite to it). The idea behind negative numbers was to simply ignore the fact that you would get answers that were less than zero and simply treat these "false solutions" as though they obeyed the same laws we already knew positive numbers to obey -- that is, associative and commutative properties of addition, and distributivity of multiplication over addition. By using this assumption, the rules for computing with negative numbers were derived much as you see above (in particular, the identity (-1)²=1). The discovery that using these rules gave you a consistent system of numbers was one of the happier discoveries of mathematics, and was what eventually led to the construction of what we now call the negative numbers. It is because the negative numbers were originally concieved of as a way to extend the properties of positive numbers past zero that we can assume them when trying to figure out the sums and products of negative numbers.