When a number is divided into a given number of parts, it is done by the operation of *division*. For example, we have 12 apples and we want to share them among three people. We clearly need to divided 12 by 3, which gives 4. Each of the three people will get an equal number of apples (4).
From this it is clear that division is the inverse operation of multiplication, in a similar way as subtraction is the inverse of addition.
In this example, 12 is the *dividend*, or just D. The number of people, 3, is the *divisor* or just d. The result, which is 4, is called the *quotient*, or q.
In this case, since there are no apples left when we share equal amounts to each person, the *remainder*, or r, is 0. Think of the remainder as what is left over after the division.
Let's imagine that we need to divide the number 2 into 2 equal parts. Then, D = 2 and d = 2. Each part will be 1, so that D divided by d gives q = 1. Again, there is no remainder here, so r = 0.
When r = 0, notice that the multiplication of the quotient by the divisor must give our dividend back:
If r = 0 => q·d = D.
So, in fact, when we divide D/d to get q, what we are *really* doing is to *search* for values of q for which the multiplication q·d gives D. Whereas multiplication is a direct operation, division is an indirect process.
For example, we want to divide D=35 by d=7. We try q=3 and do q·d=3·7=21, which is not D. Then we test q=4 to get q·d=28, which is not D. We test q=5 to obtain q·d=5·7=35=D, so that q=5 is the true quotient.
Euler insists here in that (provided r=0), the dividend is a product. As another example, if D=63 and d=7, we can search for numbers (candidates for q) such that q·7=63. We know that our q is 9 in this case.
Imagine our D is D=ab, while our d is d=a. Using what we have learned, D = q·d, we have that ab = q·a, which makes clear that q = b. If we want to divided ab by b, then ab=q·b, which clearly indicates that q=a.
Expanding on the previous point, we may want to divide D=abc by d=a, so that, as D=q·d, we write abc=q·a, which indicates that q=bc. Or if we want to divided 12mn by 3m, we do D=q·d to obtain 12mn=q·3m, which gives q=4n.
Seeing division as a multiplication can seem quite trivial at first, but don't dismiss this point of view since it will lead to very powerful conclusions, especially when we will study Newton's method of fluxions and infinite series.
Of course, don't forget that these last calculations have all r=0. There are no leftovers after the division.
If a is to be divided by 1, the quotient will be a, since D can be written as D=a·1=q·d=q·1. Also, if a is to be divided by a, we can write D=1·a=q·a, which gives q=1. Again "trivial" calculations that allow us to train our minds to see division in a reverse angle.
Let's enter now into the territory of division with non-zero remainder. We are asked to divide D=24 by d=7.
It is clear that 7·3=21 and 7·4=28, so that there is no integer value of q for which we can divided 24 into seven equal parts.
Don't forget that we are dealing with integer numbers, so don't be tempted to say that the result is ~3.428etc.
Here we limit ourselves to conclude that the exact quotient will be some number greater than 3 and smaller than 4. The search for such exact number will lead us later to the concept of fractions. But not yet.
Imagine that we have 24 apples and 7 people. We must give to each person a whole number of apples, and every one has to receive the same amount.
We have no choice but to choose 7·3=21, so that we give 3 apples to each person. This implies that we have used only 21 out of the 24 apples that we have. Conclusion: 3 apples are left over, not given to anyone. This means that the remainder is r=3.
Let's perform this division in the way we do at elementary school. An important note here is that there are (that I know of) two main notations for such divisions. One notation is mainly used in Europe/Asia (I will call it Eurasian), while the other is mainly used in English speaking countries, so I will call it the Anglo version. I am sure there must be other notations in the world. I am also ignorant of the notation used in South America or in Africa, so if you are reading this and know about this more than me, please contact me and help me to improve this section
Let's go first for the Eurasian notation:
D |__d____ 24 |__7____
q 21 3
r -____
3
The method is: search for a number that multiplied by 7 gives the closest lower bound of 24, which is 3. Then multiply 3·7=21 and subtract this from the dividend, which gives the remainder, 3. Of course, the general method is a bit more complicated than that, but this simplified version is enough for this example.
Now let's write the same example in Anglo notation. The method does not change: only the way of writing it is different.
__q_____ __3_____
d| D 7 | 24
21
r -____
3
In both notations, we write our products under the dividend, so we always need the dividend to have free space below. The remainder will appear at the very bottom under the dividend. In this case r=3.
The divisor does not need free space to its right or left, so whatever choice is good.
The quotient needs free space on its right, since you know that when dealing with decimals, we may end up having a very long quotient. In both our notations there is free space to the right of the quotient, which is good.
Finally, there is one reason for which I think the Anglo notation is superior: it allows free space to the right of the dividend. Although this is not the place to discuss divisions with decimal positions, we know from elementary school that in order to obtain more decimals, we need to elongate the dividend to the right. In this example, for the first decimal, we want to write 24 as 24.0. Then, for the second decimal position, we need 24.00, and so on. It is clear that the Eurasian notation has a conflict here. This is the notation I was taught in school, and I recall writing the dividend with a lot of blank space to its right in advance in order to avoid a mess. Of course, this is not a very big problem in practise, because as you need to develop to the right you also go down, as to form a bottom-right expanding motion that usually will not collide with neither the divisor not the quotient, but I see it as a design small failure. And that is my reason to prefer the Anglo notation.
As another example, let's try to share out 41 apples among 9 people:
____4____
9 | 41
36
-____
5
This means that we will give 4 apples to each person and we will have 5 apples not given.
Recall that when r = 0 we could write D = q·d. But this is not true any more for r ≠ 0. We need to add the leftovers in order to recover the dividend. So, in general, it is always true:
D = q·d + r .
Or, put into words:
Dividend = quotient·divisor + remainder.
It is easy to understate the relevance of this expression. It seems quite trivial, but it is the foundation of one of the most fascinating and difficult branches of mathematics: number theory, the branch dealing with whole numbers.
We can use it here as to give proof that a given division has been correctly performed. We need to multiply q·d and add r and see if it matches the value of D. In this case, 4·9 + 5 = 35 + 5 = 41 = D, which shows that the results were correct.
Also, instead of writing D = q·d + r we can divide the whole expression by the divisor d. We are anticipating the use of the symbol / for fractions, but I am sure you already know it anyway, so here it is:
D/d = q + r/d .
This is a very convenient way of understanding division. The division itself is D/d. What we obtain is a quotient q plus an extra term r/d. This term, in Newton's Method of Fluxions (which is a masterpiece that we will study in this site), is called the *supplement*, or just s if you like, so that D/d = q + s. So that you can understand division as a process that gives you a *tentative* result q plus a *correction* term s.
This separation between a main part of the result (q) and a supplemental part of the result (s) is at the base of all numerical calculations. It provides a way to numerically approximate to a result to any degree of accuracy. Although this is not the place to discuss this, keep in mind that the role of the supplement is key in mathematics, especially for numerical calculations and series.
For now, the main message is dividend = divisor·quotient + remainder or D/d = q + supplement. The two forms are very important.
In our previous example, not only 41 = 4·9 + 5, but 41/9 = 4 + 5/9, where s = 5/9. More on fractions in Chapter 7.
What about signs when dividing? Well, since division is the inverse process of multiplication, you can guess the rules from the rules you use to multiply.
Here forget about remainders and go back to D = q·d.
Since +·+ = +, we can consider that each sign corresponds to a letter in q·d = D, so that if D=+ and d=+, then q=+.
Since -·- = +, and since q·d = D, we can take D=+ and d=- and get q=-.
Since +·- = -, and since q·d = D, we can take D=- and now we have two choices. If we take d=+, then q=-, and if we take d=-, then q=+. But this last choice repeats the second case.
Summary:
D=+ and d=+ give q = +
D=- and d=- give q = +
D=+ and d=- give q = -
D=- and d=+ give q = -.
In analogy with multiplication, dividing equal signs gives a +, and dividing different signs gives a -.
Again, if you like difficult questions, you could say: in multiplication, although two equal numbers multiplied always give a positive result, as in a·a, we were told that there were some mysterious numbers i for which i·i = -1, so two equal entities multiplied can give a negative number. Can we have something similar in division? Is there a mysterious entity, let's call it j, such that j divided by j is equal to -1?
Since D=j, d=j and q=-1, we can apply (with r=0) D=q·d so that j=(-1)·j or j=-j. Do you know a number for which its negative version equal its positive version? I am sure you do. Of course, it is number 0.
Does it mean that we have shown that 0 divided by 0 is equal to -1? Absolutely not! These are dangerous waters. But why is not 0 divided by 0 equal to -1? Notice that we could as well take D=j, d=j and q=-2. We would arrive to the conclusion that, since D = q·d, j = (-2)·j or j=-j, which again would lead to j=0. So you would conclude that 0 divided by is now equal to -2 instead of -1. And it is clear that a well behaved operation cannot have two different results. Clearly, dividing by 0 is dangerous, and dividing 0 by 0 is also very tricky.
As a final example, Euler proposes to divide 18ab by -3a. This means D=18ab and d=-3a, so we apply D=q·d and write 18ab = q·(-3a). This can be rewritten as 18ab = (-3a)·(-6b), so that q = -6b.
This is enough for division of simple numbers. Let's move forward, still dealing with divisions.