Logic behind dividing negative numbers

Question

I've learnt in school that a positive number, when divided by a negative number, and vice-versa, will give us a negative number as a result.On the other hand, a negative number divided by a negative number will give us a positive number.

Given the following equations:

1. $\frac{-18}{2} = -9$
2. $\frac{18}{-2} = -9$
3. $\frac{-18}{-2} = 9$

This is how I would think the logic behind the equation would be:

1. If I have a debt of 18 dollars , owed to 2 people equally, I would owe each of them $9

   • $\frac{-18}2 = -9$

2. If I have 18 dollars, owed to 2 people equally, I would thus give them $9 each

   • $\frac{18}{-2} = -9$

However, I can't seem to come to terms with a negative number dividing by a negative number giving me a positive number as a result. What would be the logic behind it?

Also, I think that I have the logic/reasoning for the 2nd example wrong, as it is exactly the same as the reasoning for the first example?Could someone give me a better example of the logic behind the 2nd example?

I would really appreciate it if anyone could enlighten me.

Answer 1

If you have a debt of \$100, and that debt is paid in \$20 increments, you have:

$$\frac{-\$ 100}{-\$20} = 5\ \textrm{payments}.$$

Answer 2

Do not attempt to base your understanding of negative numbers on analogies with debts or with visualization. The sooner you grasp the concept of numbers as abstract objects, the better off you'll be.

The division $\frac a b$ means "what number, when multiplied by $b$, gives $a$?". Because of the way negative number multiplication is defined (and it is a made up, abstract definition, it doesn't "come" from anywhere), it is the case that multiplying a negative number $-b$ by the positive number $\frac a b$ gives you $-a$. Therefore, the number that needs to be multiplied by $-b$ to give $-a$ is the positive number $\frac a b$.

EDIT:

To respond to criticism, I'd like to clarify what my point of view with an example. Most people think of positive numbers as "abstract objects", in the sense I'm using here. "$5$" isn't "$5$ meters", it's just $5$. Of course, we're aware that they have concrete "representations", but the abstract point of view actually helps us there, because it gives us a bird's eye intuition on which things can be represented as numbers and which can't (anything that can be counted, basically). Now look at the ancient Greeks and their successors. They forced themselves to represent numbers as lengths. Multiplication of two numbers meant considering the area of a rectangle. Multiplciation of three meant considering the volume of a cuboid. This led to mathematicians saying patently ridiculous things like that multiplying four numbers together is illogical (specifically, that quartic equations are absurd, something that I think Cardano said).

Answer 3

Interpret multiplications as a way of scaling.

$2\times3$ would be the length $"2"$ scaled up thrice.

This provides a nice way to interpret negative multiplication as scaling and reversing.

$9$ meters forward times $-2$ would be $18$ meters backward.

Now define division as the inverse of multiplication. In other words, $-18/-2$ is defined to be the solution to the equation

$$(-2)x=-18$$

Now ask yourself, what length, should I double and flip to get $18$ backwards?

Clearly the answer is $9$ forwards or $+9$.

Note: The other answer here already has an interpretation with debts like you were asking for, I just posted it because the scaling thing is a nice way to look at it. (It also extends nicely to the complex numbers, showing you why they are not-so-imaginary)

Answer 4

You don't even have to think of negative as "debt".
How many $-2$'s do you need to add together to get $-10$? Five: $$\frac{-10}{-2} = 5$$

Answer 5

Division $a/b$ is really asking how many times $b$ goes into $a$. In terms of making sense out of division by comparing to practical problems where $b$ persons divide $a$ items it seems just as obscured to calculate $1.2/3.4$ since what is $3.4$ persons?

So regardless of possible practical interpretations that different answers may give, I would suggest to rather try to free yourself of the limiting framework of the practical world and think of it as asking how many times $b$ goes into $a$.

Answer 6

Think of it as $\dfrac{-18}{-2}=\dfrac{18\cdot(-1)}{(-2)\cdot1}=\dfrac{18}{-2}\cdot\dfrac{-1}{1}$

You say that you understand the logic behind $\dfrac{18}{-2}=-9$

Then you probably understand the logic behind $\dfrac{-1}{1}=-1$

Now simply take the results and multiply them: $(-9)\cdot(-1)=9$

Answer 7

To walk $18$ feet backwards in $2$ feet backward-steps requires $-18/{-}2 = 9\,$ steps, i.e. $-18 = 9(-2)$

Algebraically: scaling by $\,9\,$ the equation $\, {-}2+2 = 0\,$ $\Rightarrow$ $\,9(-2) + 9\cdot 2 = 0,\,$ so $\,9(-2) = -(9\cdot 2)$.

Above we used the Distributive Law (DL) $\ x(y+z) = xy + xz\,$ for $\,x = 9.\,$ This implies that such scalings are linear maps, e.g. for $\,x = 2\,$ it says that doubling the sum of $\,x\,$ and $\,y\,$ is the same as summing the doubles of $\,x\,$ and $\,y,\,$ i.e. $\,2(x+y) = 2x+ 2y,\,$ known intuitively at an early age.

When we extend positive numbers to include negative numbers, we strive to create a number system that preserves key properties (laws) of the positive numbers, since this will allow us to use the extended number system to correctly deduce theorems about the original number system. Therefore we preserve such linearity in the extended number system by requiring that it satisfy the Distributive Law. This then implies, as above, the standard properties of negative numbers, for example the Law of Signs.

"Number systems" may be abstracted by algebraic structures called rings (and related structures such as fields and (integral) domains). In a ring, the Distributive Law is the only axiom that relates its additive and multiplicative structure. After centuries of reflecting on the algebraic essence of "number systems", algebraists abstracted out the Distributive Law as a key property of "numbers". Every theorem of rings that effectively involves both addition and multiplication must employ the Distributive Law, since it is the only law relating the additive and multiplicative substructures. Thus, in a certain, sense, such linearity is a key property of our conception of "number", and a consequence of this linearity is the usual properties of negative numbers such as the Law of Signs.

Answer 8

When determining how to extend the definition of operations on a set to a superset, there are multiple factors that come into consideration. These include:

1. Logical consistency
2. Simplicity (I.e. Consistent semantics not actually forced by logic.)
3. Motivation from analogous mathematics
4. Motivation from real world problems

Having defined behavior, there is a related issue which is to see if there are real world situations that it models. One can then use that to better understand the mathematics or better understand the world.

Arithmetic being so basic, the third of these does not really apply in this case.

You can see that the other answers include both types 1,2 and 4.

For mathematics, logical consistency is of course paramount. That means that rules implied by type 1 are not really optional.

Now, (I am not aware of how to do this), it might be possible to carefully define operations on negative numbers in a manner other than the standard one. It would obvious that doing do would result in a load of special cases around negative numbers, essentially generating a different mathematical structure.

I'd suggest that the operations on negative numbers as shown in other answers of type 1 and 2 are much more significant than the type 4 cases (which includes 'debt' examples).

In sum: The 'debt' examples serve much more as real world models supporting understanding, but are not the reason for the definition.

Answer 9

Amongst other things, negative/negative = negative would violate the rule that anything (nonzero) number divided by itself is equal to 1.

• -2/-2 is 1.
• -4/-2 is 2.
• -6/-2 is 3...

And so on. One could also consider what is -2/-2 + 2/-2? This is equal to (-2+2)/-2, which is 0/-2, which is equal to 0. Therefore -2/-2 has the opposite sign to 2/-2. Since 2/-2 is negative, -2/-2 is positive.

Answer 10

$$\large{\textbf{Simple}}$$ How I think of negative numbers is with only the concept of $-1$ (This is just my view, I know there are others. I hope this makes sense to you). This idea is actually based off of basic algebraic theory (inverse).

So we can have $$\frac{-5}{2}=-1 * \frac{5}{2}$$ when we think of it this way it doesn't matter if the negative is on top or bottom. I.E. $$\frac{5}{-2}=\frac{5}{-1*2}=\frac{1}{-1}*\frac{5}{2}=-1*\frac{5}{2}$$ If you think about it this way the only tricky thing is $\frac{1}{-1}$, but it is much easier (imo) to reason why this is the same as $-1$ than it is for other fractions. You'll also have to trust me that $-1*-1=1$ (unless you read on)

$$\large{\textbf{Slightly more detailed}}$$ If you want to get into the algebraic theory I'll elaborate (a little). For a group we need a certain set of conditions. Closure, associativity, identity, and inverse operations must exist. We won't worry about closure.

An identity, $e$, results in $a * e = a$, where $a$ is an element in the group and * is the operator. Likewise an inverse works such that $a * a^{-1}=e$. With inverses we can understand why two inverses cancel one another.

So continuing with our example we can do this more abstractly. $$\frac{-a}{b}=-e*\frac{a}{b}$$ or similarly $$\frac{a}{-b}=-e*\frac{a}{b}$$ $\small\textrm{Where in our case the fraction operation is the same as inverse multiplication.}$

$$\large\textbf{Better explination}$$ I highly suggest reading $\textbf{A First Course in Abstract Algebra}$ by John B. Fraleigh. The first chapter will get you through this concept and is a short, but extremely useful, read.

Answer 11

"18 divided by 2" means "How many 2s are there in 18?" Counting in 2s gives us:

• 1x2 = 2
• 2x2 = 2 + 2 = 4
• 3x2 = 2 + 2 + 2 = 6
• ..
• 9x2 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 2 = 18

The answer is 9

"-18 divided by -2" means "How many -2s are there in -18?" Counting in -2s gives us:

• 1x-2 = -2
• 2x-2 = -2 + -2 = -4
• 3x-3 = -2 + -2 + -2 = -6
• ..
• 9x-2 = -2 + -2 + -2 + -2 + -2 + -2 + -2 + -2 + -2 = -18

The answer is 9 again.

To figure out "-18 divided by 2" clearly counting forwards in 2s is only going to give us positive numbers, so we need to count backwards in 2s instead :

• -1x2 = -2
• -2x2 = -4
• ..
• -9x2 = -18, so the answer is -9

I think the hardest case to explain conceptually is in fact "18 divided by -2". We know that counting forwards in -2s will only give us negative numbers so we need to count backwards in -2s to get +18. i.e. we need -9 lots of -2 to get +18.

• -1x-2 = 2
• -2x-2 = 4
• -3x-2 = 6
• ..
• -9x-2 = 18

(Sorry about my lack of LaTeX)

Answer 12

Division can be thought of (in an algorithmic sense) as repeated subtraction. The question "what is 6/2" is exactly equivalent to the question "how many times must one subtract 2 from 6 to reach 0".

How many times must one subtract 2 from 18 to reach 0? Of course it's 9.

How many times must one subtract -2 from 18 to reach 0? Of course it's -9.

BTW this is a good way to help kids understand why division by zero is a singularity. How many times must one subtract 0 from X to reach 0? Any finite or inifinite number is not sufficient.

Answer 13

Before considering division, it may be helpful to first consider what the distributive law implies about negative integers and multiplication. In particular, if n is a positive integer and m is its additive inverse, then m+n equals zero, and mX+nX must equal (m+n)X which must then also equal zero. Consequently, mX must be the additive inverse of nX. If nX is positive, mX must be negative, and vice versa.

The quotient n/d is defined as being a value q, if one exists, such that qd=n. If e=-d, then qe=q(-d)=-(qd)=-n. Since qe=-n, q=(-n)/e. Since we also know that q=n/d, that implies that n/d=(-n)/(-d) in cases where a suitable value d exists.

Note that if one defines a form of division that requires an integer quotient but allows a non-zero remainder, that form of division can satisfy either -(n/d)=(-n)/d or (n+d)/d=(n/d)+1, but not both. The problem scenarios arise when the division has a remainder; if one only consider division to be defined in cases where there is no remainder, both equalities hold.

Answer 14

I caution against trying to model numeric relationships on real-world phenomena. To do so well, you must make sure that your mathematical model accurately captures the real world, which can be tricky and non-intuitive.

In this case, is a debt a negative asset? Is a negative debt an asset? Is negative owing an expectation of payment? So is $\frac{-18}{-2}$ a negative asset of $18 expected from two people? Clear as mud.

Instead, I recommend considering division as multiplication by a reciprocal (which I believe it always can be), so $\frac{-18}{-2}$ becomes $-18 \times \frac{1}{-2}$. If you factor out the $-1$ from $\frac{1}{-2}$, you get $-18 \times -1 \times \frac{1}{2}$. This becomes $18 \times \frac{1}{2}$, or $9$.

Answer 15

Although there are many answers here and much interesting discussion (even if I don't personally agree with the advice of abandoning practical analogies and intuition to progress in mathematics; I would say au contraire to that), I could not see a single answer or comment responsive to the OP's additional request for a clearer/more sensible concrete analogy for his second example, 18/-2 = -9, in the realm of money and debts etc.

I think that's a reasonable question but it's also a tough one, I think primarily because it's hard to interpret, for example, a negative number of people.

So here's the closest I could come: suppose you can pay for some product with a combination of coupons and cash. And suppose the effect of reducing the number of coupons you use by two is to increase the cash you need to pay by $18. What is the effect of one coupon?

Answer: it is $\frac{18}{-2} = -9$, i.e., each coupon reduces the amount of cash you have to pay by $9.

Does that help anyone perhaps a little? I for one would enjoy seeing other/better concrete examples where 18/-2 makes intuitive sense and gives an illuminating answer of -9.

Answer 16

You can understand why a negative divided by a negative is positive by thinking about division as repeated subtraction. In the problem -50/-2 for example, you start with -50 and think about how many groups of -2 you can take away. -50 divided into groups of -2 = 25 groups. So, -50/-2 = 25.