Addition
Along with subtraction, multiplication, and division, addition is one of the four fundamental arithmetic operations
Of the four, addition is undoubtedly the most basic operation, considering that the other three are defined through a logic chain that starts from addition: subtraction is defined as the inverse of addition, multiplication as a recursive addition, division as the inverse of multiplication.
An operation based on the existence of a numeral system
Although it is a basic operation, performing an addition requires sophisticated mental skills. To understand why let’s imagine a concrete case. In a primitive society, thousands of years ago, a farmer collected the eggs laid by her hens and placed them in some containers of different sizes. One contains four eggs, another six, another ten, and so on. She wants to know how many eggs she has in all. She can count the eggs one after the other, moving from one container to another, but it is easy to get confused. It is more convenient to count the eggs in each container and then add up the partial sums.
However, to carry out this apparently trivial operation, first of all, she must know how to count. And counting is a very complex task. In fact, unless you stop at particularly low numbers (there are primitive societies that have names only for numbers up to 2 or 3), counting is not possible if you do not have a numeral system, i.e., a set of rules for combining elementary numbers to form larger ones. Thus, for example, we use a positional decimal system, where ten digits represent ones, tens, hundreds, and so on, depending on their position.
Given the level of abstraction necessary to make calculations based on these rules, the use of the abacus, a tool that can greatly facilitate the execution of arithmetic calculations, spread among many civilizations from the earliest times.
Returning to our hypothetical Neolithic hen farmer, the addition that she must perform to find out how many eggs in total her hens have laid presupposes that she knows numbers and master the rules of a numeral system within which it is possible to make additions.
Addition as an intuitive operation
The most intuitive form of addition is the juxtaposition of distinct sets of objects. It is visually evident that the more elements we add to a group, the larger, i.e., numerous, the group becomes. Probably, the idea of addition as an indeterminate increase of the elements in a set developed before the ability to determine the sum of the added objects accurately.
However, we can speak of addition in the modern, mathematical sense of the term only about the operation used to precisely determine the total number of items added. In its simplest form, this is equivalent to counting the elements of a set. In fact, the increments of 1 from any natural number to the next are additions and represent the basis on which to build additions in which we add numbers greater than 1.
A formal definition of addition
It is no coincidence that the formal definition of addition, formulated towards the end of the 19th century by the Italian mathematician Giuseppe Peano, is a recursive definition, which reduces any addition to a series of increments by 1, obtained with the repeated application of a function called “successor.”
Peano based its definition on two axioms:
- 𝑎+0=𝑎
- 𝑎+𝑆(𝑏)=𝑆(𝑎+𝑏)
The first appears intuitively true. Adding zero to any number returns the number unchanged: 7+0=7.
The second requires a bit of effort to understand what it states. Let’s clarify the terminology first. 𝑆(𝑏) is the successor of b, that is, the smallest integer that is greater than b. Therefore, the axiom states that adding the successor of b to a is equivalent to the successor of a plus b. For example, if 𝑎=7 and 𝑏=2, 𝑆(𝑏)=3 and 𝑆(𝑎+𝑏)=10.
But let’s see in practice how the iterative system proposed by Peano works. Let’s start with the simplest case. We want to add 1 to any natural number a. Since 1 is the successor of 0, by definition 𝑎+1=𝑎+𝑆(0). It follows that, according to the second axiom, 𝑎+𝑆(0)=𝑆(𝑎+0). But, according to the first axiom, 𝑆(𝑎+0)=𝑆(𝑎). Hence, 𝑎+1=𝑆(𝑎). E.g.: 7+1=𝑆(7)=8.
To obtain 𝑎+2 we proceed in the same way. By definition, 𝑎+2=𝑎+𝑆(1). It follows that, according to the second axiom, 𝑎+𝑆(1)=𝑆(𝑎+1). But, from the previous case, we know that 𝑎+1=𝑆(𝑎). Hence, 𝑎+2=𝑆(𝑆(𝑎)). E.g .: 7+2=𝑆(𝑆(7))=9.
The mechanism can go on indefinitely. By definition, 𝑎+3=𝑎+𝑆(2). It follows that, according to the second axiom, 𝑎+𝑆(2)=𝑆(𝑎+2). But, from the previous case, we know that 𝑎+2=𝑆(𝑆(𝑎)). Hence, 𝑎+3=𝑆(𝑆(𝑆(𝑎))). E.g .: 7+3=𝑆(𝑆(𝑆(7)))=10. In practice, the addition 7+3 is equivalent to finding the successor of the successor of the successor of 7.
Set definition of the sum of natural numbers
The addition between two natural numbers a and b can also be defined as an operation between finite sets, of which the natural numbers express the cardinality, that is, the number of elements in each set.
Let 𝑁(𝑆) be the cardinality of a set S. Given two disjoint sets (two sets without elements in common) A and B, let 𝑁(𝐴)=𝑎 and 𝑁(𝐵)=𝑏. Then 𝑎+𝑏=𝑁(𝐴∪𝐵). The sum 𝑎 + 𝑏 corresponds to the cardinality of the set formed by the union of sets A and B. The following image visually illustrates the concept.
How addition works
In the positional decimal numeral system, the addition between two or more numbers, called addends, is performed by placing the numbers in columns with the ones corresponding to the ones, the tens to the tens, the hundreds to the hundreds, and so on. Any decimal digits are placed to the right of the ones, separated by a point.
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To the right of each addend, excluding the last, we put a + sign, the addition operator. On the right of the last addend, we place an equal sign (=). Finally, a horizontal rule separates the last addend from the result of the addition, called sum.
The operation is performed a column at a time. We start by adding up the figures in the rightmost column. If the total exceeds 9, we write the digit representing the ones at the bottom, under the horizontal rule, while the digit representing the tens forms the carry. We write it at the top, above the first addend, in the column immediately to the left of that which we have just added up. Then we add the carry to the figures in the column below.
We proceed in the same way from column to column until we have added up all the figures in the leftmost column. The number that appears at this point under the separation rule is the final sum of the addition. The following image illustrates the mechanism and provides the necessary terminology.
The addition of integers
So far, we have considered only the most basic form of addition, performed on natural numbers. But it is worth examining the rules to be applied in some other very common cases, for example, when the addends are integers, that is, numbers preceded by a sign, positive or negative.
There are essentially three cases to be considered:
- the addends are all positive;
- the addends are all negative;
- the addends are some positive, some negative.
The first case is the simplest. Addition works exactly the same way as it works when adding natural numbers.
If the addends are all negative, the sum is negative, too. It is obtained by adding the absolute values of the addends, as in the following example:
−37+(−54)+(−11)+(−32)
=−(|−37|+|−54|+|−11|+|−32|)
=−(37+54+11+32)=−134
The most complicated case is when the addends have different signs. If there are only two consecutive addends, we subtract the one with the lowest absolute value from the one with the highest absolute value. The sign of the sum is the sign of the number with the highest absolute value. Here are a couple of examples:
- −37+40=|40|−|−37|=40−37=3
- 54+(−55)=−(|−55|−|54|)=−(55−54)=−1
If there are more than two consecutive addends, we can proceed linearly, starting from the left and carrying out one addition at a time. But, since the addition is associative (changing the order in which we perform the additions does not alter the result), it is possible to proceed in another way as well. First, we add up the addends with the same sign. Then, we subtract the subtotal having the lesser absolute value from the other subtotal. The sign of the sum will be that of the subtotal having the greater absolute value, as in the following example:
−37+54+11+(−32)
=[−37+(−32)]+(54+11)
=−(69−65)=−4
It may seem strange, even counterintuitive, that we have to do a subtraction to obtain the sum of an addition. But the perplexity disappears if we reflect on what distinguishes the integers (i.e., the numbers that, together with zero, form the set ℤ). These numbers are the union of two elements: a sign (+ or −) and an absolute value.
One can imagine the addition between positive and negative integers as a sort of tug-of-war that takes place on the number line. Each number “pulls” with force corresponding to its absolute value in the direction indicated by its sign. The negative integers pull to the left, the positive ones to the right. Zero — which is neither positive nor negative — stands still in the center.
It follows that the addition between two opposite numbers — numbers that have the same absolute value but different signs — gives zero. They both “pull” with the same force but in opposite directions. This is why their sum is obtained from a subtraction!
If, on the other hand, the numbers are all negative, as in one of the examples proposed above, their sum will be the equivalent of a tug-of-war in which only one team participates, with no opponents. The endpoint will be on the side of negative numbers, away from zero by a distance equal to the sum of the absolute values of all the addends, as shown in the following image.
Finally, if the numbers are partly positive and partly negative, the result of the addition will be the difference between the total “strength” with which the positive numbers “pull” to the right and the total “strength” with which the negative ones pull to the left. In the example shown below, the two negative addends have a total “strength” of 10 units greater than the positive ones, so the sum of the addition is −10.
Properties of addition
Addition has three fundamental properties:
- commutative. The order of the addends does not change the sum. Given two numbers a and b, 𝑎+𝑏=𝑏+𝑎. Example: 4+9=9+4=13;
- associative. When the addends are more than two, the order in which the additions are performed does not change the sum. Given three numbers a, b and c, (𝑎+𝑏)+𝑐=𝑎+(𝑏+𝑐). Example: (2+7)+5=2+(7+5)→9+5=2+12=14;
- existence of the identity element. Adding 0 does not change the value of a number. For any number a, 𝑎+0=0+𝑎=𝑎. Example: 21+0=0+21=21.
