# What do you think of this explanation of why subtracting a negative is the same as adding a positive?

The key observation is that if we confine ourselves to addition and subtraction (no multiplying), there is no difference between positive and negative numbers. Think of them as forces pulling in opposite directions. We tend to think of north and east as being positive and south and west as being negative. So think of the forces as being northwest and southeast, arbitrarily assigning positive to one and negative to the other.

For any equation involving addition and subtraction, we can reverse the signs of all the numbers and still have a true statement. Note that the + or – denoting addition or subtraction will remain. Part of the problem is the confusion of having these symbols to denote both operations and sign.

(+6) + (+3) = (+9) and (-6) + (-3) = (-9)

(+9) – (+3) = (+6) and (-9) – (-3) = (-6)

(-9) – (+3) = (-12) and (+9) – (-3) = (+12)

The equations on the right are obtained by reversing the signs of the equations on the left. If you accept the equations on the left then you must accept the ones on the right.

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## 17 Answers

I should have added that if you accept that subtracting a positive is the same as adding a negative then it follows that subtracting a negative is the same as adding a positive.

I think I just follow the rule and don’t think about it much. That’s the thing about the basics in math, the people (kids) who just learn the rules and apply them usually do well in beginning math. The kids who keep trying to understand sometimes have a much harder time. Although, the math geniuses I think understand it and apply it and move through the information in amazing way.

For me, the graph showing the positive and negative numbers on each side of zero explained it effectively. Thereafter, whenever I saw a negative number added to a positive, it was the same as a subtraction problem with two positive numbers. If I wasn’t sure, I would envision the graph.

…-3 —- -2 —- -1 —- **0** —- +1 —- +2 —- +3…

Thinking about it as the numbers being on either side of zero, as shown immediately above, is the best way to visualize it. When we subtract we move to the left, but when you subtract a negative it moves you to the right.

However, it’s still easier in my opinion to just know the rule. When you see subtract a negative, just change the symbols to a plus sign, and don’t overthink it.

I imagined it like @Espiritus_Corvus described. We probably went to school around the same time and used similar textbooks. That worked very well for me. Complex numbers added a second dimension for graphing, then a third for vectors, then a fourth for Quaternians for describing spacial mechanisms.

I probably went to school at around the same time as @Espiritus_Corvus and @LuckyGuy, too, and used somewhat similar logic. But I think I was taught to think of it as a logic issue more than a ‘number issue’. That is, a plus-plus was always a plus, and a double-negative minus-minus was also a plus. I’m trying to recall how and when I first came to the realization that a plus-minus or a minus-plus was always going to be a negative, though. (It can’t have been just now!) However, as noted below, I don’t really recall the process by which this became innate to my thought and calculations processes.

But that’s true, isn’t it? Adding a positive number to a value will result in a value shifted rightward along the number line (or higher along the y axis, if working in two dimensions), and subtracting a negative number will do the same thing, since we’re “taking away” a negative amount, which is the same as adding.

In contexts where the process involves subtracting a positive value or adding a negative value, though, the movement is always leftward (or down) on the axis. And the same principle holds through multiplication, too.

I don’t recall the process by which I learned that, it’s so far back in memory. All I know now is that it’s part of my “flow” calculation ability, and I don’t have to think about it at all.

Well, it’s like looking in a mirror. It’s backwards.

When I taught it I don’t go into great depths as to the logic. I just had them (and me) memorize the chant, “A negative minus a negative equals a positive.”

Math is one of my favorite subjects to teach. It’s so straight forward. It’s also hard to be creative and interesting like you can be with all other subjects. Memorization works best. And memorizing can be fun!

I’m pretty sure I’m a generation younger and I learned it the same way as Lucky and Espiritus. Are they not teaching it that way anymore?

@CWOTUS A plus-minus and vice versa isn’t always a “negative” in that it doesn’t always have a negative answer. It depends on which is larger, the negative number or the positive one. I think that the language can get confusing in that I try to stick to using the word negative and positive only describing the number, and minus and plus when talking about the symbol used to dictate the calculation.

I can see that the words and images I introduced in the first paragraph of my response may have indicated a certainty in final result that was not intended, @JLeslie. I thought that I corrected that adequately in my second paragraph, when I referred to “movement among a number line” as being in a positive or negative direction, depending upon the values and operators.

You did. I should have read through more carefully. I think people who are confused by the concept might still be unsure even after all of our explanations. LOL. I think it’s difficult to learn math from reading.

The number line is a very useful tool for picturing real numbers. There is additionally a symmetry between positive and negative numbers with respect to addition and subtraction that people don’t pick up on. What I was trying to say, though not very well, is that you can picture positive and negative numbers as opposing forces. When you add two of the same type, they combine to create a bigger force in the same direction. When you subtract one of them, the direction of the force is reversed. This holds for both positive numbers and negative numbers.

@LuckyGuy, I was just reading that it took a while before complex numbers were represented as points in the plane. It simplifies things conceptually when you can think of multiplying complex numbers as adding angles and multiplying lengths.

I wonder if, in these days of “unfriending” and “unfollowing,” you could just say “unsubtracting” and get it across.

@Jeruba

“Unsubtracting” would be correct Newspeak

@jaytkay, thanks for the confirmation. The point is, wouldn’t that make it easy for Newspeakers to understand the concept? “Unsubtract” = “add.” No operators and parentheses involved.

I like it, especially the talk about opposing forces. I wonder what people who are just learning arithmetic think about it.

@Jeruba , I like the idea, but I would think of it as saying that subtracting is unadding.

Sure. But I thought the difficulty was with explaining subtracting a negative, which does take a little thought to wrap around when it’s a new idea. I assumed that you were talking about explaining this to one or more students. Anyway, the idea is yours if you want it. Probably somebody else has already tried it out.

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