How to Subtract Positive and Negative Integers

This is the continuation of the series of Civil Service review in mathematics particularly on operations of integers. In this post, we are going to discuss the most complicated operation on integers. I have taught people of all ages about this topic and it seems that for many, this is the most difficult among the four operations. In this post, we are going to learn how to subtract positive and negative integers or signed numbers. Note that in subtracting integers, there are only  four forms. If a and b are positive, the subtraction are of the following forms.

Case 1: positive minus positive (a – b)
Case 2: negative minus positive (^–a – b)
Case 3: positive minus negative (a – ^–b)
Case 4: negative minus negative (^–a – ^–b)

How to Subtract Positive and Negative Integers

What most people don’t know that a – b is the same as a + ^–b, or subtracting a number is the same as adding its negative. That means that you only have to memorize the steps in addition of integers. Given a subtraction sentence, you then transform it  into addition. Here are a few examples.

Case 1 Exampe 1: 5 – 8

Subtracting is the same as adding its negative, so 5 – 8 = 5 + ^–8. Note that 5 + ^–8 is already addition and 5 + ^–8 = ^–3.

Case 2 Example: ^–10 – 4

The expression ^–10 – 4 is the same as ^–10 + ^–4 = ^–14.

Remember also that if you see two consecutive – signs or a minus and a negative sign, you can transform it to +. That is, -(^–a) = + a and -(^–a) + a. In most exam, the negative signs are not usually superscript, so you will likely -(-a).

Case 3 Example: 5 – ^–6

The above expression might be written in 5- -6 or 5-(-6). In any case, two negative signs, a minus and a negative sign can be transformed into a plus sign so, 5 – (–6) = 5 + 6 = 11. Notice that the last equation is also an addition sentence.

Case 4 Example: ^–8 – ^–6

The expression ^–8 – ^–6 = ^–8 + 6 = ^–2.

Observe that the four forms are already completed in the examles. From the strategy above, we only remember two strategies: (1) transform any subtraction sentence to addition sentence and (2) replace two consecutive negatives or a minus and a negative with + sign.