How do you understand subtracting integers?

Subtracting integers can be best understood as adding the opposite. In other words, instead of subtracting a number, you add its additive inverse.

Core Concept: Adding the Opposite

The fundamental idea is to transform a subtraction problem into an addition problem. This is achieved by changing the sign of the integer being subtracted (the subtrahend) and then adding it to the first integer (the minuend).

Mathematically:

a - b = a + (-b)

Where:

• a is the minuend (the number being subtracted from)
• b is the subtrahend (the number being subtracted)
• -b is the additive inverse (opposite) of b

Examples:

Let's illustrate with a few examples:

• Example 1: 5 - 3

  This can be rewritten as 5 + (-3). Adding a positive 5 and a negative 3 results in 2. Therefore, 5 - 3 = 2.

• Example 2: 2 - 7

  This becomes 2 + (-7). Adding a positive 2 and a negative 7 results in -5. Therefore, 2 - 7 = -5.

• Example 3: 4 - (-6)

  This is where it becomes particularly useful. It is rewritten as 4 + (+6). Adding a positive 4 and a positive 6 results in 10. Therefore, 4 - (-6) = 10. Subtracting a negative number is the same as adding a positive number.

Steps for Subtracting Integers:

1. Identify the minuend and the subtrahend.
2. Change the sign of the subtrahend (find its additive inverse).
3. Change the subtraction operation to addition.
4. Add the integers. Follow the rules for adding integers:
   • If the signs are the same, add the absolute values and keep the sign.
   • If the signs are different, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

Why This Works:

This method works because subtraction is the inverse operation of addition. Subtracting a number is equivalent to adding its opposite. This concept is fundamental in understanding how integers interact on the number line. Imagine walking forward 5 steps (positive 5) and then walking backward 3 steps (subtracting 3, or adding -3). You would end up 2 steps forward from your starting point.

By understanding subtraction as the addition of the opposite, you can simplify calculations and avoid common errors.