When we want to add a number 3 to our previous value of, say, 5, we use the sign + in the middle of them and we say 5 + 3 or five plus three.
In 5 + 3 we can interpret a time sequence: before we had 5, and now we add 3, so later we will have 5 + 3 or just 8.
Of course, we can reverse the story and say 3 + 5 = 8, meaning that before we had 3, now we add 5 and later we end up having 8.
I think it is quite interesting to see this as a time sequence. On the one hand, the expression 5 + 3 = 8 can have a timeless quality, meaning time does not intervene here, and meaning that this relation is always true. However, the process of adding is exactly that, a process, and processes occur through time. I had something (5), then I change it by adding something. To add is to change, and to change involves time.
We notice how time lurks within these concepts without being explicitly written or considered.
We are not limited to add a single number to another. We can have a full chain of processes. Imagine we have 8 and then add 3. We get 13. Then we add 13, so we get 26. Then we add 1 and we get 37. Then we add 1 and we get 38. Then we add 3 and we get 41. Then we add 10 and we get 51. There is no need for such mess. We can produce the chain:
8 + 5 + 13 + 11 + 1 + 3 + 10 = 51
We could also use letters instead of numbers. This is a very important step in the understanding of algebra. A letter is considered a generalisation of a number, in the sense that it can contain any number inside.
In essence, a letter is a box in which we can place a number inside:
+---+ | 3 | +---+ a
Notice how we have placed the letter (a) of the box as a label to it, and then how we have placed a number (3) inside. This means that, for this case, a = 3.
But we could have left the box empty!
+---+ | | +---+ a
This means that a has no value yet. This is very useful, since, if we can solve a problem with empty letters, then we have the solution of all the particular cases in which the letters are filled with numbers. How convenient is to work with letters, then! As we will see, working with letters is the very essence of algebra.
Let's write a letter expression: a + b = c, or, with pictures:
+---+ +---+ +---+ | | + | | = | | +---+ +---+ +---+ a b c
In this case, if we fill two boxes with numbers, no matter which ones, the value inside the third will be determined.
We can also use letters in chains, as with numbers: a+b+c+d+e+...
We move now to the other process at hand here: subtraction. It is the inverse process of addition. Imagine that we (before) had 8 apples and we (now) give 3 to a friend. Immediately after this (later) we end up having 5 apples. This uses the sign - (minus) between the number from which we subtract and the number we subtract: 8 - 5 = 3.
Can we reverse the order of the terms to obtain the same result? Let's explore it. We begin by having (before) -5 apples, which could mean we owe 5 apples. Then we are given (addition) 8 apples. So the final result is to have 3 apples. The result is the same, but the point of view has changed significantly.
We can also subtract in chain, like 50 - 1 - 3 - 5 - 7 - 9. We can subtract term by term or we could also add all the subtracted terms, 1 + 3 + 5 + 7 + 9 = 25 and subtract this total from the original number, 50 - 25 = 25.
We can find the signs + and - mixed in a chain of operations. One approach to get the result is to collect all terms with + first and all terms with - later, and then subtract the result of the second group from the result of the first group. For example, in 12 - 3 - 5 + 2 - 1 we can collect 12 + 2 - 3 - 5 - 1 = 14 - 9 = 5.
Since the order of the terms in sums and subtractions does not matter, we have many choices for such an order. Sometimes a particular order will be particularly easier than other. For example, in 14 + 7 + 6 I feel that adding 14 + 6 = 20 first is easier, and then adding 7 is also easy. In 27 - 9 - 7 I would prefer to do 27 - 7 first. Each person will find different strategies better according to their taste.
Can we use letters (boxes) for chained additions and subtractions? Absolutely! For example
a - b - c + d - e
We can operate this in the order we want, and we can rewrite the expression in the order we wish as well.
A word of caution here: notice how these signs are typed *outside* the boxes. Keep this question in mind for a while.
Euler tells us to consider the sign before each number. In short, the number in "absolute" terms (meaning without the sign) and the sign the precedes it form a "team". If the positive sign is found before a number we call it a positive quantity. (See? We are using the word quantity for a number here.) We call negative to those numbers with a preceding minus. If we don't specify a sign, we assume it is a positive number. Then, -1 is negative, +1 is positive and 1 is positive.
Euler suggests (and we anticipated this before) to consider positive quantities as property and negative properties as debt. If I have +3 apples, it means I own three apples. If I "have" -3 apples, it means I owe three apples. This way of thinking is not strictly necessary, but it is undoubtedly useful.
What lies between property and debt? The concept of having *nothing* or owing nothing. We use the number 0 for that. In this sense, positive means more than nothing, and negative means less than nothing.
We can order the positive numbers in a geometric way, like
+1, +2, +3, +4, +5, ..., ---> infinity +∞ ,
where the symbol +∞ means positive infinity. But beware that ∞ is not exactly a number, since if it were, we could write the next number ∞ + 1, which would be absurd. Infinity is just a concept meaning that these numbers never end.
Or we can explicitly write the 0 in our previous list,
0, +1, +2, +3, +4, +5, ..., ---> infinity +∞
We define *natural numbers* as the set of all these positive numbers. Depending on the definition you use, the number 0 can be included in the concept of natural number or not. It does not really matter.
The set of all natural numbers is labelled by the letter ℕ.
We can also add negative numbers to such list, to draw
-∞ (infinity), ..., -3, -2, -1, 0, +1, +2, +3, ..., ---> infinity +∞
To this set, we call it *integer* numbers, and we use the label ℤ for them.
If you want to refer to our first list +1, +2, ..., without ambiguity, you can call them positive integers, so that no arbitrary definition of natural numbers will confuse you. The same for negative integers -1, -2, -3, ...
Integer numbers can also be called *whole* numbers, in the sense that they are not broken into smaller (decimal) parts.
From our last representation you can these numbers as a geometric landscape instead of thinking them in economy terms (property or debt). You can picture a line with equally spaced ticks and place whole numbers there, in order. Suddenly, numbers have some spatial meaning.
Adding a positive number means to travel to the right. Subtracting a positive number or equivalently, adding a negative number, means to travel to the left.
How tricky this language has become now! 12 - 3 means subtracting 3, so we are subtracting a positive quantity. But we can also see this as 12 + (-3) which means adding a negative quantity.
Remember that question about boxes we left suspended. Let's now expand the question and try to answer it:
We had signs placed outside the boxes, but are forced to place only positive numbers inside the boxes? Or are we allowed to place negative numbers inside as well? The answer is that we can have whatever signs anywhere, out and inside the boxes. Let's explore this.
Imagine we have a + b = c
+---+ +---+ +---+ | | + | | = | | +---+ +---+ +---+ a b c
Now we put only positive numbers inside, like 2 + 3 = 5
+---+ +---+ +---+ | 2 | + | 3 | = | 5 | +---+ +---+ +---+ a b c
No problems so far. Now consider substituting the 3 by a -3:
+---+ +---+ +---+ | 2 | + |-3 | = |-1 | +---+ +---+ +---+ a b c
Notice how box b has a positive sign outside but containing a negative number inside. We could have exactly the opposite, negative outside and positive inside:
+---+ +---+ +---+ | 2 | - | 3 | = |-1 | +---+ +---+ +---+ a b c
But beware of this! This expression is not included in a + b = c any more! Instead, it belongs to a - b = c. When we use letters, we can fill them with whatever value or sign, but what is outside the letters must be respected.
Here we are advancing a topic from next chapter: multiplication of signs. But it does not matter, since mathematics is not a linear thing! It is evident that the signs outside and inside the box can be swapped and the result will not change, since - multiplied by + equals + multiplied by -. For more about this, see next chapter.
Can we place boxes inside boxes? Absolutely! But at this point you may have seen how boxes play the same role as parentheses! The expression a + b = c is completely equivalent to () + () = () where each set of parentheses could have a label if you like. If we want to use the particular case 2 + 3 = 5, we can write (2) + (3) = (5), or if we want to be very redundant, we can write +(+2) + (+3) = +(+5), but this is not necessary at all. However, this baroque expression is instructive to learn to deal with negative numbers. We want to deal now with 2 - 3 = -1. We have many choices now. For example, +(+2) + (-3) = +(-1) or (+2) - (+3) = -(+1). There are many possible variants here.
The takeaway message is this: when using letters, consider them as containers in which we can put numbers of whatever sign inside, or even other boxes or other full expressions! When dealing with such substitutions, consider placing parentheses surrounding each letter.
Euler tells us that we must keep in mind how all these expression lead to the same result = 0
+1 - 1, +2 - 2, +3 - 3, +4 -4 , etc.
He also reminds us that +2 - 5 = -3, so that if what you owe is "greater" (in absolute terms) than what you have, then your net balance is to owe something.
Concerning absolute values: the absolute value of a number is the number put with positive sign. We enclose the number between vertical bars to denote this. For example |-3| = |3| = 3. We can also call this the *magnitude*, so that the magnitude of -3 is 3. But since we have used the word magnitude before with another meaning, we don't want to mess up.
The previous section is way more interesting if we use letters. For example, a - a = b - b = c - c = 0. This is quite trivial. But what about a - b? Which sign does it have?
In order to deal with this, we need to introduce inequalities.
An equality like 2 + 3 = 5 uses the symbol =. This symbol separates the expression in two parts, the left hand (2 + 3) and the right hand (5). You can easily swap left and right and the expression remains true: 5 = 2 + 3. Such switch can be thought as a left-right mirror reflection (qualitatively speaking) and notice how the symbol = is invariant under such a horizontal flip.
Now consider the inequality 2 + 3 > 4. It must read from left to right and it means 2 + 3 is *greater than* 4. This new symbol, >, is not invariant under a horizontal flip. In fact, if we perform such flip, we obtain the symbol <, which means *smaller than*. Let's try to flip the whole expression: 4 < 2 + 3 which reads 4 is smaller than 2 + 3. It is still correct! So from now on, if you are too used to flip sides of an equation, beware if it is an inequality, since then the symbol must be flipped as well!
Anticipating again the multiplications from next chapter, what would happen if we would multiply *the whole expression* by (-1)?
In 2 + 3 = 5 we would get -2 - 3 = -5. The rule is just to multiply both sides. We have not touched the equal symbol, but in fact we have also multiplied it by -1. The thing here is that (-1) multiplied by = is equal to =. Sounds weird, doesn't it? To shed more light into this consider the next case:
In 2 + 3 > 4 we know multiply the left hand and get -2 - 3, we multiply the right hand and obtain -4, but if we don't multiply the symbol > by -1 we would get -2 - 3 > -4, which is not true!
Just a side note here: we have said that -5 is not greater than -4. Which is true. But it is common to fall here into the trap of thinking that -5 is "greater than" -4. If we think in absolute values, then yes, |-5| > |-4|, since this is just 5 > 4. But when there are signs, beware of them!
Imagine we have two holes, A and B:
overground
-----------+ +------------+ +------------------------
| | | | underground
| | | |
| | | |
| | | |
| | | |
| | +----+
| | B
| |
| |
| |
+----+
A
Here, hole A is deeper than hole B. That's clear. The confusion comes when we ask which hole is smaller? What do we mean by smaller? If by smaller we mean deeper, then hole A is smaller than B. If by smaller we mean "amount of digging that we have done", then hole B is smaller. The latter case is equivalent to consider the absolute value.
I think this is quite a natural confusion, since, if we picture negative numbers as holes, then it is weird to say that hole A is smaller than hole B. So, when you are asked about comparison between signed numbers, think better in terms of which is deeper, or even better, which one is more to the left or to the right. The geometric perspective in which negatives go to the left and positives to the right is incredibly useful. A number which is more to the right than another is just greater, and a number more to the left is smaller. As simple as that!
So it is clear that we need to multiply the sign > by (-1) which is equivalent to a horizontal flip. We obtain, then, that -2 - 3 < -4, which is the correct relation.
The final question is: what is the sign of a - b? The short answer is: it depends on a and b! There are three cases here:
What if they tell you that b < a? Then, you have two choices:
Just for fun, consider the following expressions and see if you understand them fully. They are not written in rigorous notation by any means. I have written them just to challenge you, but they also have meaning:
-= = =
-< = >
-> = <
(a = b) = (b = a)
(a > b) = (b < a)
(a < b) = (b > a)
-(a = b) = (-a [-=] -b) = (-a = -b)
-(a < b) = (-a (-<) -b) = (-a > -b)
-(a > b) = (-a (->) -b) = (-a < -b)