To compute the sum of signed numbers, for example -8 + (+5), we first use the fact that a negative number such as -8 can be expressed as the sum of two negative numbers (appealing to the adjective/noun theme) and rewrite -8 as -3 + -5. Then we can make use of the fact that the sum of a number and its opposite has been defined to equal 0 to do the following:
-8 + (+5)
= ( -3 + -5) + (+5)
= -3 + ( -5 + (+5) )
= -3 + 0
= -3.
To see this result more concretely, we can use the profit/loss model.
---------------------------------------------- Profit----------Loss
First Transaction ............................................................ 8
Second Transaction ...................................5
___________________________________________________
Net
___________________________________________________
Figure 1: -8 + (+5) = ?
We know that if the situation had been a $5 loss and a $5 profit, the net would have been $0. This is simply the profit/loss model for saying that -5 + (+5) = 0. That is:
---------------------------------------------- Profit----------Loss
First Transaction ............................................................ 5
Second Transaction ...................................5
___________________________________________________
Net..............................................................0
___________________________________________________
Figure 2: -5 + (+5) = 0
Combining the idea of rewriting -8 as -3 + -5 with the profit/loss model, we can view the $8 loss as the sum of a $5 loss and a $3 loss. Then we can rewrite Figure 1 in the form:
---------------------------------------------- Profit----------Loss
First Transaction ............................................................ 5
Second Transaction ...................................5
..........................................................................................3
___________________________________________________
Net ....................................................................................3
___________________________________________________
Figure 3: -8 + (+5) = -3
The $5 profit and the $5 loss then "cancel" one another and we see that the net result is a $3 loss. This result is the same result that is gotten by applying the usual rule for adding two numbers that have opposite signs.
Rule: Subtract the smaller magnitude from the greater magnitude and keep the sign of the number that has the grater magnitude.
In terms of the profit/loss model, the smaller magnitude (5) is in the "profit" column so we subtract it from the greater magnitude (8), which is in the "loss" column. We then place the difference (3) in the "loss" column.
---------------------------------------------- Profit----------Loss
First Transaction ............................................................ 8
Second Transaction ...................................5
___________________________________________________
Net ....................................................................................3
___________________________________________________
However, since $5 is modifying profit and $8 is modifying loss, it appears that we cannot apply the adjective/noun theme that says that we cannot subtract $5 from $8 and obtain $3 as the answer because $5 and $8 are modifying different nouns (Footnote 1). We are faced with a dilemma.
Resolving The Dilemma
First, let’s remove the problem from the profit/loss model but retain the visual aid. We can do this by replacing the headings “Profit” by “Positive”, “Loss” by “Negative” and “Transaction” by “Number”. The display now becomes:------------------------------------------ Positive----------Negative
First Number ....................................................................8
Second Number ...................................5
___________________________________________________
Net ....................................................................................3
___________________________________________________
The subtle point is this: even though we subtracted 5 from 8 we added the two signed numbers. In order to add -8 and +5 we found we had to subtract one adjective (5) from another (8), but we did not subtract -8 and +5 . In other words, when we add two signed numbers that have different signs, we subtract the lesser adjective from the greater adjective and keep the noun that is modified by the greater adjective. To show that this rule is consistent with the adjective/noun theme, we can put the rule we have just discovered into the following form known as the “subtract the opposite” rule.
Subtract the Opposite Rule: a + b = a – (-b) Footnote 2
We have just illustrated that the subtract the opposite rule we discovered is consistent with the rule that says to add two signed numbers, subtract the smaller adjective (in this case, 5) from the greater adjective (8) and keep the noun (sign) of the term with the greater adjective (negative). But, and this brings us to the “punch line” of this note, we have also illustrated that the adjective/noun theme does hold for signed numbers. To make use of the adjective/noun theme to add two numbers that have different signs, just do the following:
Step 1:
Write the sum with the number having greater magnitude placed first.
In this example: -8 + (+5).
Step 2:
The two adjectives are now modifying the same noun so we can do the “usual”
arithmetic. In this example:
-8 – (-5) = -3.
Footnote 1: Using the example that 3 dimes + 2 nickels = 40 cents, we motivated the rule that we add adjectives only if they modify the same noun. Similarly, the fact that 3 dimes – 2 nickels = 20 cents motivates the rule that we subtract adjectives only if they modify the same noun.
Footnote 2: This rule is analogous to dividing two fractions. Namely, to divide by a fraction we multiply by the reciprocal of the fraction. Our rule says to add two signed numbers, subtract the opposite of the second number (formally known as the subtrahend). Also, when dividing by a fraction we are NOT multiplying by the original fraction. Similarly, when adding two signed numbers we are NOT subtracting the original ‘subtrahend”.
To apply the “subtract the opposite” rule to our original problem, replace a by - 8 and replace b by +5 . Since b is equal to +5 , the opposite of b (in symbols (-b) ) is -5. Making those substitutions in the rule a + b = a – (-b), the expression a + b becomes -8 + (+5 ) and a – (-b) becomes -8 – (-5). Therefore, our original problem now takes the form -8 – (-5). In this form both 8 and 5 are modifying “negative”. Since both adjectives (8 and 5) modify the same noun (negative), by the adjective/noun theme the answer is -3. Note that this is the same result we got by using the profit/loss model.
We have just illustrated that the subtract the opposite rule we discovered is consistent with the rule that says to add two signed numbers, subtract the smaller adjective (in this case, 5) from the greater adjective (8) and keep the noun (sign) of the term with the greater adjective (negative). But, and this brings us to the “punch line” of this note, we have also illustrated that the adjective/noun theme does hold for signed numbers. To make use of the adjective/noun theme to add two numbers that have different signs, just do the following:
Step 1:
Write the sum with the number having greater magnitude placed first.
In this example: -8 + (+5).
Step 2:
Change the addition sign to a subtraction sign and replace the second number with its opposite (i.e., change its sign). In this example: -8 – (-5).
Step 3:The two adjectives are now modifying the same noun so we can do the “usual”
arithmetic. In this example:
-8 – (-5) = -3.
Footnote 1: Using the example that 3 dimes + 2 nickels = 40 cents, we motivated the rule that we add adjectives only if they modify the same noun. Similarly, the fact that 3 dimes – 2 nickels = 20 cents motivates the rule that we subtract adjectives only if they modify the same noun.
Footnote 2: This rule is analogous to dividing two fractions. Namely, to divide by a fraction we multiply by the reciprocal of the fraction. Our rule says to add two signed numbers, subtract the opposite of the second number (formally known as the subtrahend). Also, when dividing by a fraction we are NOT multiplying by the original fraction. Similarly, when adding two signed numbers we are NOT subtracting the original ‘subtrahend”.
