When a number such 7 is said not to be div-3, we already know that this does not mean that 7 cannot be divided by 3. It only means that 7 divided by 3 does not give a whole number as a result (r=0).
But of course we can divided number 7 in 3 equal parts! The only problem is that, if we think in terms of apples, we will need a knife to cut some of them!
It may be better to change now apples by cakes, which can make pictures easier.
Imagine we have 7 cakes and 3 people. Each person wants the same amount of cake. What do we do?
7 cakes: +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+
We know how 7 = 2·3 + 1, which means that we can give two whole cakes to each person, and we will have a remaining cake.
+-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+
| | | | | | | | | | | | | |
| p1 | | p1 | | p2 | | p2 | | p3 | | p3 | | ? |
| | | | | | | | | | | | | |
+-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+
remainder
See how we have given the first two cakes to person 1 (p1), the next two cakes to p2 and the next two cakes to p3. But we said we want an exact division this time, so we still need to share the last cake among the three people. What do we do?
+-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-+-+-+
| | | | | | | | | | | | | | | |
| p1 | | p1 | | p2 | | p2 | | p3 | | p3 | | | | |
| | | | | | | | | | | | | | | |
+-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-+-+-+
1 2 3
Clearly, the last cake can be divided in three equal parts and each part can be given to each person.
Remember the concept of supplement (s)? Instead of writing D = q·d + r we divide the whole expression by d, obtaining D/d = q + r/d where r/d is s.
In this case, since 7 = 2·3 + 1, we also have 7/3 = 2 + 1/3. This means that the division, when exact, has two steps.
First, we share whole cakes as far as possible: we give 2 cakes to each person. This part deals with quotient q.
Second, we share *fractions* of the remaining cakes: we give to each person 1/3 of the cake.
Summary: to share D cakes among d people, we first find q and give a whole number q of cakes to each person. After this process, r cakes will be left, so that we give s to each person.
A number like 1/3 cannot be consider integer or whole. It is instead a *broken number* or a *fraction*.
We have already anticipated the symbol to write fractions, a bar that separates both "up" and "down" numbers. For example, in 1/3 we have used the forward slash / to separate 1 from 3.
Another choice is to use a horizontal bar to separate them. Here we arrive to one of the things that makes plain text writing in computer not the ideal choice for mathematics. In plain text, it is very easy to write 1/3, but how can we write a vertical construction in which 1 is at the top and 3 at the bottom, while being separated by a horizontal bar?
When drawing in a paper this is no issue at all. The same happens for the beautiful renderings of LaTeX. However, writing in plain text has a lot of benefits. See the "about" section of this site for a discussion of this.
Let's try to render a vertical fraction by using plain text.
1 1 1 1 1 - --- _ ___ 3 . 3 3 3 3
We don't need to choose one. Sometimes one choice will be better, and other times another choice will be more convenient.
I like to consider the last one, because it is so compact and because why using a horizontal line when you can always assume that a number on top of another are dividing? Aren't we assuming ab as a·b? So why cannot assume
a b
as a shorthand for a/b? A possible downside is that if you want to write an = beside it, the symbol = must go either at the top or at the bottom, never in a symmetrical way:
6 = 2 6 3 3 = 2 .
If you cannot stand this, you can write
6 6 = 2 --- = 2 . 3 3
If you know more advanced mathematics, you may recall expressions like binomial coefficients, in which two numbers are vertically positioned without a horizontal bar in the middle, but they can also be written in other ways (we will see this in due time).
My approach will be to use a single line of text whenever possible, like a/b, but many times we will use any of the other possibilities.
Let's write more about this symbol:
You may happen to be a real geek and have noticed that I have been using a regular slash / for division. However, unicode have an explicit "division slash" ∕ and a "fraction slash" ⁄. The unicode codes are, respectively, 002f, 2215 and 2044. Compare the results by yourself and decide which one you like most:
a / b 002f a ∕ b 2215 a ⁄ b 2044 .
We call *numerator* to the top part of a fraction, and *denominator* to the bottom part.
If we want to express 2/3 of a cake, we may draw
+--+--+--+ |xx|xx| | |xx|xx| | |xx|xx| | +--+--+--+
where the part filled by x's represents the fraction 2/3.
The fraction 2/3 is called two thirds. 3/4 is three fourths. 7/3 is seven thirds. 3/8 is three eighths. 12/100 is twelve hundredths. And so on.
When the numerator is equal to the denominator, as in a/a, the result is clearly 1. No matter what we write at the top, we will get 1 if we write the same at the denominator: 1/1 = 2/2 = 3/3 = 4/4 = ... = 1.
This means that when the same factor appears at the top and at the bottom, we can cancel out them:
a --- = 1 . a
If the numerator is smaller than the denominator, the value of the fraction is less than 1. Example: 2/3 < 1.
If the numerator is greater than the denominator, the value of the fraction is greater than 1. Example 3/2 > 1.
Remember those inequalities we studied before? What about expanding our knowledge on them a little?
Imagine we have an expression like a = b and now we want to make the inverse of it, so that we write 1/a = 1/b.
This is possible because of this nice property (written in informal notation):
(1 / =) = (=) .
So when inverting an expression, never forget about the symbol in the middle! The same happened when we multiplied the whole expression by -1 or when we horizontally flipped the expression.
As you may guess, the inequality symbols are not so easy as the equal symbols:
(1 / >) = < (1 / <) = > .
This means that if we invert a > b we need to do (1/a) (1/>) (1/b) which gives 1/a < 1/b. For example, as 3 > 2, we see how 1/3 < 1/2. Or, in the case we had before, if 3/2 > 1, we have 2/3 < 1.
When the numerator is smaller than the denominator, like in 3/4, we are dealing with a single cake, which is divided in 4 equal parts and from which we take three of these parts.
However, when the numerator is greater than the denominator, like in 4/3, we are dealing with more than one cake. In this case, since 4/3 = 1 + 1/3 (remember: D/d = q + s), we need two cakes: one which we take in full and a second cake from which we only take 1/3.
In the latter case we may want to use a shorthand notation. Instead of writing q + s, or 1 + 1/3, you can write a regular 1 immediately followed by a smaller fraction 1/3. Again, plain text is not very well suited for this. Instead, we will enclose the supplement within brackets: 1[1/3]. This will tell us that the quotient is 1 and that the supplement is 1/3.
In our present example, 4/3 = 1[1/3]. But let's practise this more: 43/12 = 3[7/12]. It is always the best to work out this until you leave a fraction smaller than 1 within the brackets.
The geek in you may be tempted to write 2/5 as 0[2/5]. No problem! Go ahead! An equivalent expression would be just [2/5].
Anticipating things again, let's talk about this [supplement] notation when we work with decimals. Imagine you have 1.0012[1/3]. In this case, this is not equal to 1.0012 + 1/3. What this expression means is 1.0012 + 1/3·0.0001. In other words, the brackets after a given decimal indicate a supplement fraction of this last decimal position!
If you have 0.03[1/4] it means you have 0.03 and a quarter of 0.01. So 0.03[1/4] = 0.0325.
Maybe you think this is pure paranoia, but it is not. Such notation can give you powerful calculation skills in the future. As a little taste, try to multiply 0.0325 by 7. You can separate what is out of the brackets and what is inside them, so that 7·0.03 = 0.21 and 7·[1/4] = [7/4]. Then, 7·0.03[1/4] = 0.21[7/4]. See how the 7 infiltrates in both places! But now you don't want a number greater than 1 inside the brackets, and you know that [7/4]=0[7/4]=1[3/4], so notice how you need to add a 1 to the last decimal. In short: 7·0.03[1/4]=0.21[7/4]=0.22[3/4]. It is way more elegant than a raw 7·0.0325 = 0.2275. And it also helps keeping a fixed amount of decimals. There are higher powers you can achieve with this, as you can check in Newton's Method of Fluxions.
Fractions like 2/3 in which the numerator is smaller than the denominator are called *proper* fractions. It makes sense, since it is only in this case that you are taking a part of the whole.
Fractions like 3/2 in which the numerator is greater than the denominator are called *improper* fractions, because they involve an integer number (in this case 1) plus a proper fraction (in this case 1/2). So the improper fraction 3/2 can be rewritten as 1[1/2], with a proper fraction within the brackets.
Remember that the bracket notation is my invention. If you are with pen and paper, the conventional notation is to substitute the brackets by writing the 1/2 with a smaller size. I, though, still enclose within brackets, sometimes between parentheses, the supplement. Otherwise I feel the temptation to multiply the big number by the fraction, which would be incorrect.
As it has been anticipated, you can think the fraction 5/7 as the concatenation of two processes. First, divide the cake into 7 parts, as to obtain the fraction 1/7. Later, take 5 of these equal parts, which means multiply 5 by 1/7.
5 1 --- = 5 · --- . 7 7
Again, time appears here in a lurking manner. First you divide according to the denominator. And later, you multiply according to the numerator.
Have you ever considered the meaning of the words numerator and denominator? Denominator comes from "giving a name", while numerator comes "counting".
So the denominator is the number "giving the name" thirds, which means we split the cake into three equal parts. Later, the numerator will "count" how many pieces do we take, in this case 2.
We usually say "numerator and denominator", as if the numerator, by being the one at the top, would be the "first". But nothing farther from the truth! It is the one at the bottom which comes first, so we should say "denominator and numerator". Why the one at the top must come first? Surely you can find a reason for that.
Since now we know that 5/7 is just a processed object, from which 1/7 is more fundamental entity, we can understand that the fractions 1/2, 1/3, 1/4, 1/5, 1/6, etc are the foundation of all others.
Observe as well how as the denominator increases, the value of the fraction diminishes.
As a particularly important case, consider the set of diminishing fractions 1/10, 1/100, 1/1000, 1/10000, etc. They will be the foundation of the decimal system.
Can we go up in such successions, in which the numerator is 1 and the denominator is an increasing number, such as we reach a moment for which the fraction is reduced to nothing? Euler answers "no". Because as long as the number of parts in which we divided the cake is finite, and we take one of these parts, this cannot be nothing. A fraction can never be equal to nothing.
From a practical perspective, though, if we take the 1/100000 of a meter, we may not appreciate its length. Our senses will not be able to resolve such a small magnitude. But with a good microscope, such magnitude will be visible.
Another thing would be to ask whether we have a microscope for an arbitrarily small magnitude. At least in theory. Then we enter into the realm of Quantum Mechanics, and the answer is very clear: an all powerful microscope could only resolve distances as short as the wavelength of the light you use to see. In order to see a hugely small distance, you would need hugely energetic photons of light. And certainly you cannot have photons with infinite energy.
The word itself, infinite, means without end. There is some danger in enclosing an infinite thing into a finite word. It is the equivalent paradox to give a name, which is something, to the concept of nothing!
When we have a finite symbol for what is not finite (∞) and when we have an existing symbol for what is not-existence (0) we can incur in many paradoxes.
We can summon the concept of infinity to be at our denominator. In that case, 1/∞ can be considered equal to 0, but keep in mind that ∞ is not a number, but a shorthand to remind you that you can never reach it.
Don't get the impression that the concept of infinity must be avoided in mathematics. Nothing farther from the truth. It is in fact one the most fruitful concepts of all! So give to it the greatest importance.
Only because we are in dangerous waters when we deal with it does not mean we must avoid it. On the contrary!
The facts that 1/∞ tends to 0 and that its inverse, 1/0 tends to ∞ are of the greatest relevance in mathematics.
Euler reminds us not to forget that "infinite" does not mean "the greatest", since otherwise we could not do 2·∞, and we certainly can do this. It is incontestable that the latter is twice the former. So never try to consider infinity as a given number, but as a tendency that never ends.