How is subtracting a negative number becoming addition? [duplicate]

Solution 1:

Let's see if a geometry approach helps.

Draw a number line, on the number line mark down the location of $-4$ and the location of $2$.

We would want the distance between them to be equal to $2-(-4)$.

The distance between $2$ and $0$ is $2-0$, the distance between $-4$ and $0$ is $4$. Hence the distance between $2$ and $-4$ would be $2+4$.

Hence we would want $2-(-4)=2+4$.

Solution 2:

Notation

It's a bit unfortunate that we use the same notation for subtraction as negation, because the two $-$ symbols in something like $2--4$ mean different things. The first one means "subtract" and the second means "negate". To better distinguish the two, I'll use the notation $\ominus$ for subtraction and $-$ for negation. So, just for this answer, I'll rewrite the equation $2--4$ as $2\ominus -4$ so that we can immediately and unambiguously see that this means "subtract negative four from two".

Negation

A number can be seen geometrically a point on the number line.

In this sense, negation means "go the opposite direction" as seen from $0$. For instance, $4$ is to the right of $0$, so $-4$ means go the same distance ($4$ units) but to the left of $0$:

But we can chain negations, too. For instance, $--4 = -(-4)$ means go the same distance as you would for $-4$ ($4$ units) but in the opposite direction of $-4$. We know that $-4$ is to the left of $0$ so $-(-4)$ is $4$ units to the right. But that just ends up at $4$.

So $-(-4)$ and $4$ are the same point on the number line. Hence they are the same number.

Your turn: what number is $---4$ the same as?

Addition

Before we can talk about subtraction, we need to know what does adding mean? Adding involves a few steps that are usually all done together, but let's be a bit more explicit. Here's the algorithm (meaning the steps we take) for adding $a+b$:

1. Start at the point $a$ on the number line.
2. Figure out if $b$ is to the right or left of $0$.
3. Go $|b|$ (the absolute value of $b$) units in that direction from $a$.

The point you end up at will be the value of $a+b$.

Let's look at an explicit example. Let's calculate $2+4$ in this geometric way.

1. First we start off at $2$ on the number line:

2. Next we consider the number $4$. It is to the right of $0$ on the number line.
3. So we go $4$ units to the right of $2$ and we end up at $6$:

Hence we see that $2+4=6$.

Your turn: Try to show geometrically that $2+(-4) = -2$.

Subtraction

Subtraction is a geometric process that's almost the same as addition. The difference comes in step three of our addition algorithm above. For addition we move $|b|$ units in the direction that $b$ is from $0$ away from $a$. For subtraction, we move in the opposite direction.

Example. Let's calculate $2\ominus 4$ geometrically. The first two steps will be the same as when calculating $2+4$ above:

1. First we start off at $2$ on the number line:

2. Next we consider the number $4$. It is to the right of $0$ on the number line.

But in step $3$ we change directions:

3. So we go $4$ units to the left (the opposite direction as found in step two) of $2$ and we end up at $-2$:

So $2\ominus 4 = -2$.

Your turn: Now try to show that $2\ominus -4$ gives exactly the same number as $2+4$. Then try to explain why. Feel free to comment below once you've come up with an explanation.

Bonus Question

Once you understand the geometric way of talking about negation, addition, and subtraction, try putting them all together. See if you can figure out how to calculate the following using our geometric approach:

$$-(-3\ominus 4)+---2$$