Addition and subtraction

1, 2, 3, 4, ... is what makes a lot, lot more

Addition is basically counting. When we add 3 to 5, for instance, we count up to 3, and then we count 5 ‘past’ 3, which takes us to 8. This is pretty intuitive, and it is what we teach children in the pre-school age. The addition symbol \(+\) in mathematics simply means, for numbers, continue counting up that much. The equality symbol \(=\) simply means, for numbers, that what's on one side of it has exactly the same value as what's on the other side. So, \(3+5=8\) means ‘count up until 3, then continue counting up 5 more, and that will have the same value as if you count up to 8.’ Because counting up is the natural way to count, we do not use a \(+\) sign before any number on its own, unless we want to emphasize that. If the sign is omitted, it is assumed to be a \(+\) sign, but the sign can only be omitted when there is no possible confusion. For example, we cannot write \(35\) to mean \(3+5\), because \(35\) also means the number ‘thirty-five.’ Clarity is mandatory in mathematics.

\[ \newcommand{\inv}[1]{\operatorname{inv}(#1)} \begin{array}{rcccccccc} 3 + 5 \rightarrow & \color{green}1 & \color{green}2 & \color{green}3 & \color{blue}1 & \color{blue}2 & \color{blue}3 & \color{blue}4 & \color{blue}5 \\ =8 \rightarrow & \color{red}1 & \color{red}2 & \color{red}3 & \color{red}4 & \color{red}5 & \color{red}6 & \color{red}7 & \color{red}8 \end{array} \]

Addition is essentially counting.

Natural numbers

Counting in essence requires that the things to be counted exist. We do not count non-existent objects. Things around us in nature do exist, and our desire to quantify them is what inspired counting, to start with. Therefore, whole numbers from 1 and greater are called in mathematics ‘natural numbers,’ and they are given the mathematical symbol \(\mathbb{N}\). When we say in mathematics that \(3 \in \mathbb{N}\) (pronounced ‘three is an element of the set of natural numbers’) we mean that 3 belongs to the natural numbers, in much the same way as we can say ‘My home \(\in\) the homes on my street,’ or ‘My thumb \(\in\) my fingers.’

Commutativity

Because of this, if we have a group of numbers, it doesn't matter which one we count first. They always will add up to the same final result. For instance, \(4+2=2+4=6\). Addition of more than 2 numbers will have the same property as well; e.g. \(2+3+4=3+2+4=4+2+3=9\)

‘Natural numbers’ are the whole numbers \(1,2,3,4,5,\dotsc\) going on forever.

\[ \begin{array}{rccccccccc} 2+3+4 \rightarrow & \color{green}1 & \color{green}2 & \color{blue}1 & \color{blue}2 & \color{blue}3 & \color{purple}1 & \color{purple}2 & \color{purple}3 & \color{purple}4 \\ =3+2+4 \rightarrow & \color{blue}1 & \color{blue}2 & \color{blue}3 & \color{green}1 & \color{green}2 & \color{purple}1 & \color{purple}2 & \color{purple}3 & \color{purple}4 \\ =4+2+3 \rightarrow & \color{purple}1 & \color{purple}2 & \color{purple}3 & \color{purple}4 & \color{green}1 & \color{green}2 & \color{blue}1 & \color{blue}2 & \color{blue}3 \\ =9 \rightarrow & \color{red}1 & \color{red}2 & \color{red}3 & \color{red}4 & \color{red}5 & \color{red}6 & \color{red}7 & \color{red}8 & \color{red}9 \end{array} \]

This property of addition, called ‘commutativity’ means that it doesn't matter which number we count first. It means that if we have an operation of addition of some number \(a\) and some other number \(b\), it is the same if we say \(a+b\) or \(b+a\). The numbers can ‘commute’ (i.e. move from one place to another) from one side of the plus sign to the other, and the result will be the same.

\[ \cssId{equation1.1}{ \begin{equation} \tag{1.1} \label{eq:additioncommutative} \color{blue}{a+b=b+a} \end{equation} } \]

A mathematical operation is said to be ‘commutative’ when the terms on both sides of the mathematical operator can interchange positions with no effect on the result.

Associativity

It follows directly from what's said previously that if we have more than 2 numbers added to each other, it doesn't matter which of the operations we do first. If we want to evaluate \(4+8+5\), for instance, it doesn't matter if we add 4 to 8 then add the result to 5, or if we add 8 to 5 then add the result to 4. The first case is expressed in math as \((4+8)+5\), which the second is expressed as \(4+(8+5)\). We use the parenthesis to say that we want to do what's inside the parenthesis first. In the expression \(a+b+c\), numbers on both sides of the plus sign can commute with no effect on the result; hence the order of evaluation does not matter.

\[ \overset{\curvearrowleft}{\color{blue}{a+b}}+c = b+\overset{\curvearrowleft}{a+c} = \color{blue}{b+c}+a \\ \therefore (a+b)+c = a+(b+c) \]

The symbol \(\therefore\) is pronounced ‘therefore,‘ and it means exactly that. It is usually used in mathematics to indicate that we reached a certain conclusion from what was discussed before. It is often coupled with another symbol \(\because\) that is pronounced ‘because,’ and also means exactly that. Whey they are combined, we start with the symbol \(\because\), list our reasons to reach the conclusion, and finish with the symbol \(\therefore\) right before the conclusion.

In mathematical terms, we say that if we have a chain of terms (numbers) on which we will do the same mathematical operation, we can ‘associate’ any two of them together and use the result with the others. It doesn't matter if we do the operation starting from right to left or from left to right, even though we write from left to right. This property of addition is called ‘associativity.’

\[ \cssId{equation1.2}{ \begin{equation} \tag{1.2} \label{eq:additionassociative} \color{blue}{a+b+c=(a+b)+c=a+(b+c)} \end{equation} } \]

A mathematical operation is said to be ‘associative’ if, given a chain of terms connected by the same operation, it doesn't matter if we evaluate them from right to left or from left to right.

Identity element

From early on, people realized the concept of ‘nothingness,’ and symbols for ‘nothing’ were found in many ancient languages. This nothing is now the number zero in mathematics. You cannot count ‘nothing’ in nature, because something has to first exist in order for you to count it. For this reason, the number zero is not considered a natural number. In mathematical language, we write this as \(0 \notin \mathbb{N}\) (which reads ‘zero is not an element of the set of natural numbers’), using the same symbol as the one shown above, except crossed out this time to indicate that it's the opposite. Crossing out symbols to indicate the opposite of their meaning (if this makes sense) is a widely used practice in mathematics.

Zero is not a natural number.

Now, if you add ‘nothing’ to any number, what would you end up with? Exactly! You'd end up with the same number, because you've added nothing at all. Because zero in mathematics is this ‘nothing,’ adding zero to any number does not change its value. If you add as many zeros as you want to any number, you'd still end up with the same number, because you've just added lots of nothing. Zero is, therefore, called in the mathematical operation of adding numbers ‘the addtivie identity,’ because if you add it to any number, the result will be identical to that number. Generally speaking, in mathematics, the value (e.g. zero) that does not change other values (e.g. other numbers) in a certain operation (e.g. addition) is called the ‘identity element’ for that particular operation. Not all mathematical operations have an identity element.

\[ \cssId{equation1.3}{ \begin{equation} \tag{1.3} \color{blue}{a + 0 = 0 + a = a} \end{equation} } \]

Zero is the ‘additive identity.’

More than one way to count

Subtraction is also counting, but in the opposite direction. Instead of counting up, we count down. So, when we want to subtract 3 from 7, we first count up to 7, then we count 3 from the ‘top,’ and what remains after is the result of subtraction. The subtraction sign \(-\) means that you should count that much in the opposite direction to ‘natural’ counting; i.e. count down instead of counting up. When we write \(7-3=4\), this means simply ‘count up to 7, then from there count 3 down, and what remains will have the same value as if you count o 4.’

\[ \begin{array}{rccccccc} 7 \rightarrow & \color{green}1 & \color{green}2 & \color{green}3 & \color{green}4 & \color{green}5 & \color{green}6 & \color{green}7 \\ -3 \leftarrow & ~ & ~ & ~ & ~ & \color{blue}3 & \color{blue}2 & \color{blue}1 \\ =4 \rightarrow & \color{red}1 & \color{red}2 & \color{red}3 & \color{red}4 \end{array} \]

It is easy to imagine that naturally you cannot take from something more than what you already have. Therefore, based on this understanding, there can be no number less than zero, because you cannot take anything from nothing. However, with the spread of the idea of trading, it became necessary for people to keep records of who owes what to whom. This introduced a new understanding of subtraction. I can indeed take something even when I don't have anything, because I will take from what you have, and I will ‘owe’ you that much. So, if I can take something that I don't have, how am I going to write this down in mathematics? Hence, the idea of ‘negative numbers’ came to be.

Subtraction is counting backwards.

Negative numbers: I.O.U.

So, negative numbers are not natural, but rather invented by people to represent debt. The concept of subtraction can, then, be extended to any two whole numbers. You don't have to take a smaller number from a bigger number. You can take a bigger number from a smaller one, and you will end up with a ‘debt,’ which is equivalent to a negative number. All you have to do is to keep counting below the zero. Since we use the symbol \(-\) in subtraction to indicate counting ‘backwards,’ we can use the same symbol to indicate counting below the zero in this backward direction. We don't need a new symbol to indicate negative numbers.

\[ \begin{array}{rccccc|ccc} 3 \rightarrow & ~ & ~ & ~ & ~ & ~ & \color{green}1 & \color{green}2 & \color{green}3 \\ -8 \leftarrow & \color{blue}8 & \color{blue}7 & \color{blue}6 & \color{blue}5 & \color{blue}4 & \color{blue}3 & \color{blue}2 & \color{blue}1 \\ =-5 \leftarrow & \color{red}{-5} & \color{red}{-4} & \color{red}{-3} & \color{red}{-2} & \color{red}{-1} \end{array} \]

But does subtraction have the same properties as addition? Is it commutative and associative? In fact, it is not. The reason is very simple: taking 3 from 4 is not the same as taking 4 from 3! In the first case, you will end up with a surplus of 1, and in the second you will end up with a debt of 1. Therefore, we cannot say that for any two numbers \(a\) and \(b\) that \(a-b=b-a\). The two expressions are only equal when \(a=b\); i.e. when you take away all what you have and end up with nothing. In all other circumstances, they cannot be equal.

Addition is commutative; subtraction is not.

But because subtraction is counting down, in the opposite direction of addition, which is counting up, the value \(a-b\) and the value \(b-a\) are like ‘mirror images.’ Imagine that James wanted to do his art project, for which he needed buttons. He only happened to have 5 buttons. His friend, Mark, happened to have 7 buttons, but he needed none to finish his project. James found that he needed 10 buttons for his project. He asked Mark if he could borrow 5 buttons from his in order to have the 10 buttons he needed. Mark happily lent him the 5 buttons. Mark then had with him 2 buttons in excess of what he needed. This is essentially subtracting 5 from 7. Because 7 is larger than 5, we end up with something (in this case 2).

Imagine now that James did not have 5 buttons with him, but rather had 3 buttons; and that Mark did not have 7 buttons, but rather had 5 buttons. James then needed 7 buttons to complete his project. Mark did not have enough buttons to lend James, but because he is his best friend, he decided to see if he could borrow some buttons himself from other classmates. He finally found that Clara was willing to lend him the remaining 2 buttons. He got them from Clara and gave them, together with the 5 buttons he had, to his friend James. This is essentially subtracting 7 from 5. It would not have been possible for Mark to give James 7 buttons had he not borrowed from Clara. Therefore, Mark ended up with a ‘debt‘ of 2 buttons; i.e. \(-2\) buttons. Compare this with the previous example. Subtracting 7 from 5 resulted in 2 as a debt, whereas substracting 5 from 7 resulted in 2 as a surplus.

\[ \begin{array}{l|c|c|c} ~ & James & Mark & Clara \\ \hline \text{Case 1: } & 5 & 7 \\ \text{Borrow/Lend: } & 5 \leftarrow & \leftarrow 5 \\ ~ & 5+5=10 & \bbox[5px, lightgrey]{7-5=\color{blue}2} \\ \hline \text{Case 2: } & 3 & 5 \\ \text{Borrow/Lend: } & ~ & 2 \leftarrow & \leftarrow 2 \\ ~ & ~ & 5+2=7 \\ ~ & 7 \leftarrow & \leftarrow 7 \\ ~ & 3+7=10 & \bbox[5px, lightgrey]{5-7=\color{red}{-2}} \\ \end{array} \] \[ \begin{array}{rcc|ccccccc} 7 \rightarrow & ~ & ~ & \color{green}1 & \color{green}2 & \color{green}3 & \color{green}4 & \color{green}5 & \color{green}6 & \color{green}7 \\ -5 \leftarrow & ~ & ~ & ~ & ~ & \color{blue}5 & \color{blue}4 & \color{blue}3 & \color{blue}2 & \color{blue}1 \\ =2 \rightarrow & ~ & ~ & \color{red}1 & \color{red}2 \\ \hline 5 \rightarrow & ~ & ~ & \color{green}1 & \color{green}2 & \color{green}3 & \color{green}4 & \color{green}5 \\ -7 \leftarrow & \color{blue}7 & \color{blue}6 & \color{blue}5 & \color{blue}4 & \color{blue}3 & \color{blue}2 & \color{blue}1 \\ =-2 \leftarrow & \color{red}{-2} & \color{red}{-1} & \\ \end{array} \] \[ \cssId{equation1.4}{ \begin{equation} \tag{1.4} \color{blue}{a-b=-(b-a)} \end{equation} } \]

‘Integers’ are the whole numbers \(\dotsc ,-4,-3,-2,-1,0,1,2,3,4, \dotsc\)

Mirror, mirror on the zero

So, for any two numbers \(a\) and \(b\), the ‘magnitude’ (i.e. the size or the quantity) of \(a-b\) is the same as that of \(b-a\) but in the other direction. If the negative sign indicates a change in direction, then \(a-b=-(b-a)\). The point at zero is where there's no debt or surplus, and therefore it is like a ‘mirror surface’ for the rest of the numbers. For each number on this side of the zero (the mirror) there is another corresponding number (image) on the opposite side of the mirror; i.e. for each natural number there is a corresponding negative whole number. Whole numbers, whether positive or negative or zero itself, are called ‘the set of integers’ in mathematics, and are denoted by the symbol \(\mathbb{Z}\). Since there are only two directions of counting (i.e. up and down), changing the direction two times will have no effect. If we are counting up, changing it two times will mean ‘down’ then ‘up.’ No effect! If we are counting down, changing it will mean ‘up’ then ‘down.’ No effect again! Therefore, we conclude that for any number \(a\):

\[ \cssId{equation1.5}{ \begin{equation} \tag{1.5} \color{blue}{a = -(-a)} \end{equation} } \]

Every ‘positive integer’ (i.e. natural number) has a corresponding ‘negative integer’ on the other side of the zero, like an object and its mirror image.

Absolute value

The ‘magnitude’ of those corresponding numbers is the same, but the ‘direction’ of counting from zero is different. This magnitude of any number \(x\) is called in mathematics the ‘absolute value’ of \(x\) and is written as \(|x|\). The absolute value of numbers is used frequently in mathematics, as we will see later. Because zero has no sign, it is meaningless to speak of flipping the sign of zero. Therefore, the absolute value of zero is simply zero. For any negative number \(x\), the value \(-x\) has the same magnitude, but in the opposite direction because of the \(-\) sign. So, for any negative number, \(-x\) has the effect of ‘removing’ the negative sign from \(x\).

\[ \cssId{equation1.6}{ \begin{equation} \tag{1.6} \color{blue}{ |x|= \begin{cases} x; & \quad x \gt 0 \\ -x; & \quad x \lt 0 \\ 0; & \quad x=0 \end{cases} } \end{equation} } \]

The ‘absolute value’ of an integer is its value with the negative sign, if any, removed.

Equation \(\href{#equation1.6}{1.6}\) is an example of what we call ‘multi-part definition,’ which is a definition composed of several different rules, one for each ‘range.’ In equation \(\href{#equation1.6}{1.6}\), we mean that \(|x|\) equals \(x\) when \(x\) is greater than 0, \(|x|\) equals \(-x\) when \(x\) is smaller than 0, and \(|x|\) equals 0 when \(x\) equals 0. In such definitions, there must be no ‘overlap’ in the ranges of the conditions. This means that we cannot have one range as \(x \ge 0\) and another as \(x \le 0\), because in this case, there will be an overlap where \(x=0\); hence there will be two possible rules of definitions when \(x=0\), which is not allowed in mathematics. There must be one and only one meaning for everything in mathematics.

The inverse

Now, what happens if you add one of the natural numbers with its mirror-image, negative number? Say that James had a debt (negative number) of 7 buttons, and he was given 7 buttons (positive number) by his teacher. He then used those 7 buttons to pay off his debt, so he ended up with no (zero) buttons. The same thing applies to all similar situations. Therefore, if you put together (add) any positive number and its corresponding negative number you'll end up with zero (the additive identity). In mathematics, we call this the ‘inverse’ of that number for that particular operation.

For instance, the inverse of \(7\) in the previous case is \(-7\). Can you guess what the inverse of \(-7\) is? Correct! It is \(7\). So, simply speaking, if you take a number and flip its sign (the direction in which you count), you end up with its ‘additive inverse.’ And if you add any number to its additive inverse, you will always get zero. Generally speaking, for any particular mathematical operation (e.g. addition), doing this operation on an element (e.g. number) and its inverse (e.g. same number with changed sign) will result in the identity element (e.g. zero) for this operation. Not all mathematical operations have an inverse for all elements on which the can be done, even if they have an identity element.

For any integer, the ‘additive inverse’ is obtained by flipping its sign.

Let us indicate the inverse of the number \(b\) by \(\inv{b}\), so that instead of writing \(-b\) for the additive inverse of b, we will write \(\inv{b}\). We will look into the value of adding the number \(a\) to the additive inverse of the number \(b\).

\[ \inv{b} = -b \\ a + \inv{b} = a + (-b) \]

But what does \(a+(-b)\) even mean? Let us go back to the basics we explained above and see. It means ‘count up until \(a\), then count up... No wait..! Count down as much as \(b\).’ So, it is essentially telling us to subtract \(b\) from \(a\). This applies to any two numbers \(a\) and \(b\) regardless of their sign (i.e. whether they are positive or negative, because the \(-\) sign simply indicates to flip the direction of counting.) If \(a\) and \(b\) are to be negative numbers, then the expression \(a+(-b)\) will mean ‘count down until \(a\) (because it is negative; so, inherently down), then count up... No! down (because of the \(-\) sign)... No! up (because \(b\) is negative; so inherently down) as much as \(b\).’ This is also equivalent to subtracting \(b\) from \(a\), but the result will be in the opposite direction to the first case. This means that subtraction is indeed addition of the additive inverse.

Subtraction of \(b\) from \(a\) is essentially adding the additive inverse of \(b\) to \(a\).

Backwards or forwards in the opposite direction?

Therefore, all operations of subtraction can be considered operations of addition, provided that we treat them as ‘addition of the additive inverse.’ This simplifies things greatly, and tells us why subtraction (considered as its own operation) is not associative, while addition is. Consider this:

\[ \begin{aligned} (a-b)-c & = [a + \inv{b}] + \inv{c} \\ a-(b-c) & = a + \inv{b + \inv{c}} \end{aligned} \] \[ \cssId{equation1.7}{ \begin{equation} \tag{1.7} \color{blue}{ \text{For } a,b \in \mathbb{N}; \quad \inv{b} = -b \in \mathbb{Z} \\ a-b = a + \inv{b} } \end{equation} } \]

In the consensus on the meaning of \(a-b-c\), we mean to subtract \(b\) first from \(a\), and then subtract \(c\) from the result of the first subtraction. This means we evaluate subtraction from left to right. In other words, we strictly associate terms (numbers) in subtraction with those on their left. So, we can write:

\[ a-b-c-d-e = \color{red}\{\color{blue}[\color{green}(a-b\color{green})-c\color{blue}]-d\color{red}\}-e \\ \begin{aligned} 8-3-2-5-4 & = \color{red}{(\overset{\curvearrowleft}{\underset{\downarrow}{8}-3})}-2-5-4 \\ & = \phantom{(}5-2-5-4 \\ & = \color{red}{(\overset{\curvearrowleft}{\underset{\downarrow}{5}-2})}-5-4 \\ & = \phantom{(}3-5-4 \\ & = \color{red}{(\overset{\curvearrowleft}{\underset{\downarrow}{3}-5})}-4 \\ & = -2-4 \\ & = -6 \end{aligned} \]

Addition is associative. Subtraction is only left-associative.

In mathematics, we call this ‘left assocativity.’ The term ‘associative’ without specifying direction means that you can associate any two terms in a chain of this particular operation, as stated above. It doesn't matter whether you do it from right to left or from left o right, or even if you choose terms randomly from within the chain. So, for addition, all the following expressions are equivalent:

\[ \begin{aligned} a+b+c+d+e & = \color{red}(a+b\color{red})+c+\color{blue}(d+e\color{blue}) \\ & = a+\color{red}[\color{blue}(b+c\color{blue})+d\color{red}]+e \\ & = a+\color{red}\{b+\color{blue}[c+\color{green}(d+e\color{green})\color{blue}]\color{red}\} \\ &= \color{red}(a+b\color{red})+\color{blue}[c+\color{green}(d+e\color{green})\color{blue}] \end{aligned} \]

So, while addition is associative, subtraction is only left-associative. But consider the case where we treat subtraction as addition of the additive inverse. In such case, we can treat subtraction as addition, and it becomes completely associative. Not only that, it becomes commutative, too!

\[ \begin{aligned} a-b-c-d & = a \overset{\curvearrowleft}{+} \inv{b} + \inv{c} + \inv{d} \\ & = \inv{b} + a + \inv{d} + \inv{c} = -b + a +(-d) + (-c) \\ & = \inv{b} + \color{red}(a + \inv{c}\color{red}) + \inv{d} = -b + \color{red}(a -c\color{red}) +(-d) \\ & = \inv{b} + a + \color{red}(\inv{c} + \inv{d}\color{red}) = -b + a + \color{red}(-c +(-d)\color{red}) \end{aligned} \]

Observing the previous expressions closely, we can see that the key to making subtraction associative and commutative is to stick the \(-\) sign to the number following it wherever it goes. We consider that there is a ‘hidden’ \(+\) sign before the \(-\) sign, that only appears if the \(-\) sign moves away. In the previous expressions, when we interchanged the \(a\) and the \(-b\), we kept the \(-\) sign stuck to the \(b\); we didn't just move the \(a\) and the \(b\) around and leave the \(-\) sign where it was. As a result, we ended up with \(-b+a\) rather than \(b-a\) which would have been a wrong step. Also, when we associated the \(-c\) with the \(-d\), we kept the \(-\) sign stuck to them inside the parentheses. As a result, we ended up with \((-c+(-d))\) instead of \(-(c+(-d))\) which would have been a wrong step.

Putting that in mind, what would be the equivalent of \((-c-d)\)? We know that it is not \(-(c-d)\) from the previous discussion. If we refer back to equation \(\href{\#equation1.5}{1.5}\) and we consider that the number \(s\) equals \(-c-d\), this means that \(s\) equals counting down below the zero as much as \(c\) then as much as \(d\), which is counting \(c+d\) below zero. Hence, there is a corresponding number \(-s\) that has the same absolute value as \(s\), but on the opposite direction of the zero. When we flip the sign of \(-s\), it will give us \(s\) again.

\[ s = -c-d \\ -s = c+d \\ s = -(-s) = -(c+d) \]

Applying this principle to any number of terms, we conclude that if we ‘take the \(-\) sign out’ of the parentheses, we have to flip the signs of all terms within the parentheses to keep the value unchanged. We've shown this for two negative numbers \(-b\) and \(-c\), so there remain three other cases, which we will examine:

\[ \begin{array}{c|c|c} \begin{aligned} s & = c+d \\ -s & = -c-d \\ s & =-(-s) \\ & = -(-c-d) \end{aligned} & \begin{aligned} s & = c-d \\ -s & = -c+d \\ s & =-(-s) \\ & = -(-c+d) \end{aligned} & \begin{aligned} s & = -c+d \\ -s & = c-d \\ s & =-(-s) \qquad \qquad \leftarrow \text{from } \href{#equation1.5}{1.5}\\ & = -(c-d) \end{aligned} \end{array} \]

In other words (although not as clear to understand), the additive inverse of the sum of numbers is the sum of the additive inverses of those numbers. Intuitively, if we count the numbers \(a,b,c, \dotsc\) following each other in a certain direction, that would be the expression \(a+b+c+\dotsb\), and if we want to obtain the additive inverse (mirror image on the other side of the zero) of that expression all we have to do is to count the same numbers following each other, but in the opposite (inverse) direction, so we count \(\inv{a},\inv{b},\inv{c}, \dotsc\), which is the sum \(\inv{a}+\inv{b}+\inv{c}+\dotsb\) The same concept applies if \(a,b,c,\dotsc\) were negative, because there are only two directions for counting.

\[ \cssId{equation1.8}{ \begin{equation} \tag{1.8} \color{blue}{ \begin{aligned} &\text{For } \inv{x}=-x; \quad x \in \mathbb{Z} \\ &\inv{a+b+c+ \dotsb} = \inv{a} + \inv{b} + \inv{c} + \inv{\dots} \end{aligned} } \end{equation} } \]

Subscript notation

Sometimes in mathematics, we use a small number below and to the right of a name to indicate that it is one of a family. This small number is called ‘subscript.’ So, if we have a family of natural numbers, for instance, composed of 5 numbers, we can call them \(a_1, a_2, a_3, a_4, a_5\). The subscript indicates they are different from each other (they are not all the same number), and the common letter indicates they are from the same family. To give a real-life example, let us consider a family of five people: the father, Peter; the mother, Tanya; the eldest daughter, Catherine; the son, Jacob; and the youngest daughter, Miranda. We can say that each member of this family will be represented by the letter \(F\), and to distinguish them from each other, each will have a different subscript. So, we can say:

\[ \begin{aligned} & F_n \in \text{Family}; \quad n = 5 \\ & F_1 = \text{Peter} \\ & F_2 = \text{Tanya} \\ & F_3 = \text{Catherine} \\ & F_4 = \text{Jacob} \\ & F_5 = \text{Miranda} \end{aligned} \]

Under certain circumstances, this way of expressing mathematical entities makes mathematical expressions easier to understand. This way is called ‘subscript notation.’ Using this subscript notation to express equation \(\href{#equation1.7}{1.7}\), we can write:

\[ \cssId{equation1.9}{ \begin{equation} \tag{1.9} \color{blue}{ \text{For } x, a_n \in \mathbb{Z}; \quad n \in \mathbb{N}; \quad \inv{x}=-x \\ \inv{a_1 + a_2 + \dotsb + a_n} = \inv{a_1} + \inv{a_2} + \dotsb + \inv{a_n} } \end{equation} } \]

The inverse of the sum of integers is the sum of their inverses.

Equation \(\href{#equation1.9}{1.9}\) means that for any number \(x\) and any group of numbers \(a\) expressed in subscript notation, where their subscript is a natural number (i.e.whole number with value of 1 or more) and both \(x\) and the \(a\) numbers are integers (whole numbers of any value, positive or negative), and considering that \(\inv{x}\) is the additive inverse of \(x\), the inverse of the sum of all numbers \(a\) is equal to the sum of the individual inverses of each of those numbers. The dots (\(\dots\)) in the expression indicate that we should continue with the sequence \(a_1, a_2, a_3, a_4, \dotsc\) until we finally reach \(a_n\). We don't know beforehand how many integers we have, since it depends on the value of \(n\), and we want to say that it applies to any value of \(n\). Therefore, we use dots in such conditions. Mathematical language can be intimidating at first, but it's like any foreign language you try to learn. With practice, you are going to speak ‘mathematiquese’ perfectly!

For all of you; not each of you!

An important point to consider now is that the absolute value of the sum of integers is not equal to the sum of the absolute values of those integers. Does it sound confusing?! Let's elaborate more. If we have a group of integers, some of them are positive and some are negative, and we wish to add them up and get the absolute value of this sum, will we get the same result if we get the absolute value of each of them individually, then sum up all those absolute values? The answer is: No! Remember that getting the absolute value involves getting rid of the sign completely. If we get rid of the sign, then we will not be adding those integers to each other, but rather adding the positive integers to the additive inverses of the negative integers. We will have altered the numbers, and the sum will not be the same. Instead of counting the positive ones in the up direction and the negative ones in the down direction, we will be counting them all in the up direction.

The absolute value of the sum of integers is not the sum of their absolute values.

\[ \cssId{equation1.10}{ \begin{equation} \tag{1.10} \color{blue}{ \text{For } n \in \mathbb{N}; \quad a_n \in \mathbb{Z} \\ |a_1+a_2+ \dotsb +a_n| \color{red}{\neq} |a_1| + |a_2| + \dotsb + |a_n| } \end{equation} } \]

Although we may find situations in which the previous equation may hold, the equation does not always hold. For instance, we find that \(|4+8|=|12|=12=|4|+|8|\). We also find that \(|-5-7|=|-12|=12=|-5|+|-7|\). But in other situations, the equation does not hold; e.g. \(|-6+4|=|-2|=2 \color{red}\neq |-6|+|4|=10\). In mathematics, an equation has to hold under all circumstances specified. Therefore, the rule is what is mentioned above in \(\href{#equation1.10}{1.10}\).

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‘Math from the ground up’ by Rafeek Mikhael licensed under Creative Commons Attribution-NonCommercial-ShareAlike International 4.0 license.

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