Why is subtracting a negative number the same as adding a positive number?

Answer 1

The best way to understand this is probably by thinking of the real number line:

<------negative numbers--------0--------positive numbers------->

Subtracting can be thought of as switching directions as you move along the number line. For instance, 5 + 3 means you start at position 5 on the positive side of the number line and move to the right, the positive direction, 3 integral ticks. However, 5 - 3 means you switch directions. Now, you start at position 5 and move left, the negative direction, 3 integral ticks.

Negation can be thought of in the same manner as subtraction; i.e. as a directional change on the number line. Using the above example, that would mean that for 5 + (-3) you would once again start at position 5 on the positive side of the number line and start to move right, since that what addition means. However, the negative sign in front of the 3 forces you to change direction, so instead you end up moving left 3 integral ticks.

Now, to finally answer your question. 5 - (-3), using the number line, would again mean that you would start at position 5 on the positive side. The subtraction operation, as explained above, means that you would move left 3 ticks instead of right. However, the negation of 3, also explained above, means that you would once again switch directions, this time from left to right, and ultimately move 3 ticks that way, the positive direction. Hopefully, you can see that the end result of both 5 + 3 and 5 - (-3) has you moving 3 ticks in the same direction, that being the positive direction, which is why they can be thought of as the same operation.