How is subtracting a negative number becoming addition?

Question

For me that subtracting a negative number becomes addition is currently magic. I have watched and read a number of "explanations" and they all basically boils down to:

-- = +

Which for me is magic!

None of these sources have been able to give me a logical and mentally understandable explanation about how exactly subtracting a negative number becomes addition.

As a frame of reference, the below makes sense:

2 + (-4) = -2

For me this is logical. It that tells that the number to add starts at a minus, a deficit.

However how:

2 - (-4)

becomes

2 + 4

is beyond my current understanding of how math works.

The closest I can get is that negative + negative is somehow turning into positive. But that to me is currently just a magical rule.

I think I have missed something very important here, something that makes what I at the moment see as illogical for others clearly is logical and makes perfect sense.

I would be immensely grateful to the person who will take their own time to explain to me how this works, because right now I am very stuck.

Answer 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 $2-(-4)=2+4$.

Answer 2

Here is one way to think: Substraction is the inverse of addition.

To see this, let us ask ourselves, what is $4-2=x$?

Here, the idea is to find a number $x$, such that $4=2+x$, which turns out to have a unique solution of $x=2$.

Now, what is $4-(-2)=x$?

As before, the idea is to find a number $x$, such that $4=-2+x$, which turns out to have a unique solution of $x=6$. Given you understand adding negatives, this last step should make sense to you. ***

We write the last step as $x=4-(-2)=6=4+2$.

***This $x$ must be bigger than $4$ so that even after adding $-2$ it equals $4$! Here is another intuition for why it must be $4+2$, i.e, yielding a bigger answer than $4$.

Answer 3

Real Life

"My friend borrowed $2$ of my pencils."

It sounds like the number of my pencils decreased by two, and the change is negative because there's one negative (borrowing).

"My friend borrowed $-2$ of my pencils."

The change in my pencils must have opposite sign, $+2$.

Subtraction

Think like this:

    4
-   2
------

You should write $2$ as answer, because the bigger one is the first one and their difference is 2.

    4
-  -2
------

Now you should write $6$ as answer, because the bigger one is the first one and their difference is $6$.

Answer 4

Another way you could think about it is with an example.

Lets say we currently have a debt of $5. We could say the amount of money we have is negative 5.

If the person we owe the money to decides to forgive our debt, we can picture this as subtracting the $-5$ as it is no longer valid.

Therefore our current money is $-5 - (-5)$. But them forgiving our debt means we no longer owe money, so our total should be at zero.

Indeed with the magic $--=+$ rule we have $-5--5=-5+5=0$ as we would expect.

Answer 5

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:

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

Answer 6

Try thinking of subtraction as a special form of addition. Instead of addition and subtraction being separate operations, subtraction is really addition in disguise and given a special name. Specifically, subtraction is when you perform addition with a negative number.

For example, you mention that 2 + (-4) = -2 makes sense to you. What about 2 - 4 = -2 ? These are the same thing, it's just written differently so that the first case is addition while the second way of writing it is subtraction. That's why I say that subtraction is really addition in disguise.

Again, 1 - 5 = -4 is the same as 1 + (-5) = (-4). Now apply the same mode of thinking to the subtraction of a negative number: 2 - (-6) = 8 would be the same as 2 + (-(-6)) = 8. What is (-(-6))? Just as a negative in front of a positive number "flips" the number to other side of the number line {as in the number 4 is 4 units above 0 while -4 is 4 units below 0}, the same "flip" happens when applied to a negative in front of a negative number. So instead of -6 which is a number 6 units below 0, we'll have the opposite of that: 6 units above 0, which is positive 6. So 2 - (-6) = 8 really is the same as 2 + 6 = 8.

Answer 7

Let $a$ and $b$ be integers. We define $$ a-b=a+(-b)\tag{1} $$ where $-b$ is the symbol for the integer $c$ such that $$b+c=0\tag{2}.$$ We can show that such an integer is in fact unique and $-b$ is called the additive inverse of $b$.

We claim that $-b=(-1)b$ (where $(-1)b$ is the multiplication of the additive inverse of $1$ and $b$). Indeed, note that $$ \begin{align} b+(-1)b &=(1+(-1))b\\ &=0\times b=0.\tag{3} \end{align} $$ where $(-1)b$ is multiplication, in the first line we used the distributive property, and the fact that $-1$ is the additive inverse of $1$. Thus $-b=(-1)b$ since the additive inverse is unique. Now we return to your example.

Let $x$ and $y$ be integers. In particular $$ x-(-y)=x+[-(-y)]\tag{4} $$ where $-(-y)$ is the unique integer $z$ such that $-y+z=(-1)y+z=0$. Thus $z=y$ (since $(-1)y+y=(-1+1)y=0$) and in (4) we can write $$ x-(-y)=x+[-(-y)]=x+z=x+y.\tag{5} $$ as desired.

Answer 8

NOTE: This would be nicer with a few more images, but I can't find my phone charger right now so just try to draw the relevant pictures for yourself as you go through it.

There are already some geometric and algebraic answers. Now let's try a physical one.

There exist two types of electric charges in the universe. We call them positive and negative charge. Think of a positive number as representing an excess of the positive type of charge on an object. A negative number likewise represents an excess of negative charge on an object.

Addition is adding charge to an object and subtraction is removing charge from an object.

What happens when you have an object with $+3$ charge and you add $+2$ charge to it? Your object will end up with $+5$ charge because $3+2=5$.

Similarly, adding $-4$ charge to an object that already has $-2$ charge means that the object now has $-6$ charge.

But what happens when you add the negative type of charge to a positively charged object? Some of the charges will "neutralize" each other. For example, if the object originally had $+2$ charge and you add $-3$ charge to it, then 2 units of the negative charge that you've adding will go into negating the $+2$ charge and you'll be left with $-1$ charge on the object. Hence $2+(-3) = -1$.

Similarly, if the object originally had $-7$ charge and you add $+4$ charge, then it'll end up with $-3$ charge because all of the added charge gets used up neutralizing $4$ units of positive charge already on the object. Hence $-7+4=-3$.

Now what does subtracting charge mean? It means removing that much charge. So, for example, if you have $+10$ charge on an object and you remove $+2$ of it, then you're left with $+8$. I.e. $10-2=8$. Likewise if you start with $-4$ excess charge and remove $-2$ of it, then you're left with just $-2$ excess charge. I.e. $-4-(-2) = -2$.

Now look at the first picture I have above -- the one labelled Neutral. In the image there is $0$ excess charge (i.e. no excess positive or negative charges). But if I remove 2 blue negative charges, then there will be an excess of 2 red positive charges. Hence $0-(-2) = 2$.

So what if you have $+4$ excess charge on an object and you remove $-3$ charge from it? Remember that the excess charge isn't the only charges in the object -- it's just that the rest of the charge is "neutralized". Just like in the "Neutral" image, you can think of this as a positive and negative charge stuck together and then behaving as it weren't charged at all. So to remove $-3$ charge, you'll have to break up some of these neutralized pairs. But then there'll be $+3$ charge left over + the original $+4$ excess charge. So $4-(-3) = 4+3$. That's what subtracting a negative number means physically.

Answer 9

A simple algebraic demonstration:

$$3 - 4 = -1$$ $$3 - 4 - (-1) = 0$$ $$3 - (-1) = 4$$ $$3 - (-1) = 3 + 1$$

Answer 10

No answer will satisfy you until you have created your own mental picture of what "$-4$" means. Mind me; I am talking about the number $-4$, and not the process of subtracting the number $4\ldots$

If you think in terms of money, the number "$4$" means you have $\$4$, and subtracting $4$ means you paid $\$4$; i.e., that $\$4$ are taken from you. In these terms, the number "$-4$" means that you owe $\$4$. Once this is clear, we can discuss your question:

Subtracting $-4$ means that a debt of $\$4$ is taken from you. That is the same as saying that you receive a positive credit of $\$4$.

Answer 11

Foobaz John wrote an answer I quite liked, however I'd like to give one that states less facts and maybe is friendlier for non-Mathematicians:

From an algebraic point of view, $\mathbb{Z}$ together with addition constitutes a so-called group structure, where we have a set (here: $\mathbb{Z}$) together with an operation on it, that takes two elements of the group and outputs another one (here: Addition or '$+$') and has certain properties.
Each group, by definition, has a neutral additive element (here: $0$), and for each element of the group, there is an additive inverse. This is defined as an element that, when we give our group operation an element together with its inverse, the output is the neutral additive element. Think $$ \begin{equation} l + k = 0 = k + l\;, \end{equation} $$ where $l \in \mathbb{Z}$ is arbitrary, and $k \in \mathbb{Z}$ is its additive inverse.

Now, by Mathematical convention, for any integer $l$, we write its additive inverse as $-l$. So this is pure notation to indicate an additive inverse in the algebraic sense. Also, by symmetry of the above equation, it is clear that the inverse of an inverse is again the original element, i.e. $-(-l)=l$, where the parentheses most likely stem from Mathematical conventions concerning formalism (and also serve readability, I would say).

Hence, we have $-(-l)=l$. Now, again by convention, we write $l+(-k)$ or $l - k$ if we want to add the inverse of $k$ on the right side (instead of, for instance, $l+-k$, which isn't commonly used in Mathematics for numerous reasons apart from it probably looking somewhat odd to most).

But then, with the above remarks, one gets $$ l - (-k) \overset{by\; def.}= l + (- (-k)) = l + k\;, $$ which in your example becomes $$ 2 - (-4) = 2 + (-(-4)) = 2 + 4\quad. $$

Answer 12

I don't know if this will make any sense in the end. If you imagine the same number line as mentioned in the other answers, where positives going to the right and negatives are going to the left. In this line, you'll start from 0 and your first instinct is to go right. Think this way: when you see a + or no sign at all, this means "keep direction" and if you see a -, that means that you have to change direction.

Going from the expression 2 + (-4). Since our starting point is 0, that will become 0 + 2 + (-4). Now, our first instinct is to go right and + means that we should keep our direction. So, 0 + 2 will land us at the point 2. Next we encounter a + (keep right) and then a - (change to the left). Now we have to go left, by four steps. That Will place us right at the -2.

Doing the same for 2 - (-4). Starting from 0 and going right, We land at 2 again. Now we encounter a - (change direction), which means we have to change direction, going left and, right next to it, we find another - (change direction), which makes us to go right again. Starting from 2, four steps to the right, we'll get to +6.

Honestly hope this helps.

Answer 13

All the other answers here are great, just another way to think about it:

$$4-(-3)$$

If we look at this algebraically, the minus sign really says $-1$. (Ones are often hidden) So, with parentheses added for clarity, we have:

$$4+(-1(-3))$$

Distribute the negative one and follow order of operations:

$$4+(3)$$ $$7$$

Therefore, $4-(-3)=7$.