This page is a sub-page of our page on Disambiguation.

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Adding apples or adding pears

Addition is the most fundamental operation of mathematics and it is the first one that normally confronts us when we start to learn about mathematical structures.

Unfortunately, there is a lot of confusion surrounding the meaning of addition. The underlying cause of this confusion is that the term ‘addition’ (and its representative symbol + ) has two different meanings that refer to two different types of operation.

A typical example of this confusion is the familiar saying “you cannot add apples and pears.” In fact you can, as I will explain below. However, the type of addition that is involved in adding apples and pears is different from the kind of addition that we encounter as kids. Let us recall how our first encounters with the concept of addition actually occurs.

Even before we start school, most of us have learned that three apples plus five apples is equal to eight apples, and that two pears plus four pears is equal to six pears. In elementary school we then learn that this type of addition has nothing to do with either apples or pears, and the take-away abstraction from this lesson is that three of anything plus five of the same thing is equal to eight of that same thing, which we represent as 3 + 5 = 8 , and two of anything plus four of the same thing is equal to six of that same thing, which we represent as 2 + 4 = 6 .

Discovering numbers by disregarding the apples and pears

This disregard of what is actually added represents a fundamentally important abstraction, a threshold concept, which in fact signals the birth of the concept of number. Instead of just adding apples or just adding pears, we have learned to add numbers.

Introducing abstract apples and abstract pears

Having thus forgotten about just adding apples or just adding pears by learning how to add numbers, a few years later we are again reminded of the apples and the pears when we are taught by analogy that 3x + 5x = 8x , and 2y + 4y = 6y . This represents a second major cognitive step towards abstract thinking.

Tacitly adding a mixture of abstract apples and abstract pears

However, not long after this threshold insight, the teacher suddenly starts adding abstract apples and abstract pears by writing things like 2x + 3y = 5 on the board. However, the teacher almost never explains that the familiar + sign here represents a different form of addition. Moreover, the combination of the + sign and the = sign is interpreted together to mean a linear equation in two ‘variables’ x and y , which, through the introduction of x and y coordinates, are given a geometric interpretation in the form of a straight line in the xy plane.

What is going on here?

Apart from tacitly introducing a different form of addition that allows the mixture of apples and pears, the teacher also introduces a different form of equality. When we write 3x + 5x = 8x , the = sign represents identical (or algebraic) equality, whereas when we write x + y = 5 , the = sign represents conditional (or equational) equality.

This reinterpretation of equality hides the fact that the + sign on the left hand side represents a different type of addition, a type of addition that allows adding apples and pears. This type of addition is called formal addition, and the result is called a formal (or direct) sum. Any time you carry apples and pears in the same bag, you have in fact created their formal sum. This sum keeps track of how many apples and how many pears that have gone into the bag, allowing you to make sense of expressions such as “I bought five apples plus three pears today. I’ve got them here in this bag”.

Structurally speaking, formal addition is actually performed on bags (= multisets) of apples and pears, and it is carried out by taking any finite number of such bags and putting them together in the same bag. Note that, even though we can distinguish between an apple and a pear in any such bag, the apples are indistinguishable from each other and so are the pears. We only know how many there are of each kind.

Here is a “mathematically styled” definition of the formal sum:

Let x_1 and x_2 be members of a set X . The formal sum x_1 + x_2 is the pair (x_1, x_2) taken in any order, i.e., (x_1, x_2) = (x_2, x_1) . This unordered pair is sometimes denoted by x_1 \oplus x_2 but often just by x_1 + x_2 which can be very confusing if you are new to this. As is often the case in mathematics, the meaning of an expression is deducible from the context, but only if you are familiar with the underlying concepts.

Note that if the pairs were not unordered we would not have x_1 + x_2 = x_2 + x_1 , which is an important property of addition: It obeys the commutative law.

Addition as a binary operation

In contrast to formal addition, the type of addition that we fist encounter as kids is a kind of binary operation. This is a fancy name given to an operation that takes two objects of the same type as input and turns (or maps) them into one object of that same type. In the first of the above examples, the two numbers 3 and 5 are taken as input, and the addition of them produces the output 8, which also is a number.

Assuming that we are adding non-negative numbers in the usual way, the operation of addition then is a binary operation, since the addition of two non-negative numbers always produces another non-negative number. In mathematics, the non-negative numbers are most often denoted by the symbol \mathbf N , and the binary operation of adding two non-negative numbers is denoted by

\mathbf N \times \mathbf N \, \stackrel {+} {\longrightarrow} \, \mathbf N.

Here the + sign represents a function that takes two non-negative numbers as input and produces their sum (which is another non-negative number) as output

An example of applying this function to two examples of non-negative numbers, such as 3 and 5, and producing the non-negative number 8 is denoted by

(3, 5) \, \stackrel {+} {\longmapsto} \, 8 .

This is another way to represent the familiar expression 3 + 5 = 8 . The point of this “functional notation” is to emphasize that this form of addition (binary operation) is a function that takes pairs of non-negative numbers and turns them into another non-negative number, namely their sum.

The two last expressions for the functional notation can be combined into the expression

{\mathbf N \times \mathbf N \ni (x, y) \, \mapsto \, x + y \in \mathbf N}

which uses the + sign in a more familiar place.

LEARNING TO ADD APPLES AND PEARS

A fruit basket as a function

Imagine that we have a set \mathcal F consisting of three kinds of fruit, apple, pear, and banana, i.e., \mathcal F = \{apple, pear, banana\} . A fruit basket f that consists of a mixture of instances of the types of fruits that are members of \mathcal F can be regarded as a function from the set of fruit types \mathcal F to the non-negative numbers \mathbf N :

\mathcal F \, \stackrel {f} {\longrightarrow} \, \mathbf N .

For example, if the fruit basket f consists of 5 apples, 3 pears, and 0 bananas, then we have

f(apple) = 5, f(pear) = 3, f(banana) = 0.

Adding fruit baskets

Now let us consider another fruit basket  \mathcal F \, \stackrel {g} {\longrightarrow} \, \mathbf N , which consists of 0 apples, 4 pears and 2 bananas, i.e. which has

g(apple) = 0, g(pear) = 4, g(banana) = 2.

Definition: The sum of the fruit baskets f and g , denoted f +g , is the fruit basket \mathcal F \, \stackrel {f+g} {\longrightarrow} \, \mathbf N defined by adding the number of fruits of corresponding types in the baskets f and g , i.e.,

(f+g)(x) \stackrel {\mathrm{def}}{=} f(x) + g(x), \forall x \in \mathcal F.

Hence, in our concrete example above, we get

(f+g)(apple) = f(apple) + g(apple) = 5+0 = 5,

(f+g)(pear) = f(pear) + g(pear) = 3+4 = 7,

(f+g)(banana) = f(banana) + g(banana) = 0+2 = 2.

Note: It is important to stress that there are two different  “plus functions” involved here. Although they are both of the same type (binary operation), they are in fact operating on different sets of objects. In order to express this clearly, we need a notation for the set of all functions from \mathcal F to \mathbf N . This set is denoted by {\mathbf N}^{\mathcal F} .  \mathcal F is called the domain and \mathbf N is called the codomain of the function f .

It is a good exercise to verify that this addition of functions fulfills the required properties of addition, namely:

associativity: (f + g) + h \equiv f + (g + h)

and

commutativity: f + g \equiv g + f.

Returning to the definition of the sum of two fruit baskets, the + sign on the right side of the definitional equality above denotes the familiar binary operation

{\mathbf N \times \mathbf N \ni (x, y) \, \mapsto \, x + y \in \mathbf N},

while the + sign on the left side denotes the binary composition that has just been defined on {\mathbf N}^{\mathcal F} , namely

{\mathbf N}^{\mathcal F} \times {\mathbf N}^{\mathcal F} \ni (f, g) \, \mapsto \, f + g \in {\mathbf N}^{\mathcal F}.

Observation: This is an example of a very important design principle in mathematics: Consider a set of functions with the same domain and co-domain. Whatever you can do to the values of these functions (in this case you can add them), you can also do to the functions themselves. We say that the structure of the co-domain induces the corresponding structure on the set of functions (from a common domain to that co-domain) by introducing a definition of the above type. In this case, the arithmetical structure of addition that is present in the common co-domain \mathbf N of the fruit basket functions induces the corresponding arithmetical structure of addition on the fruit basket functions themselves, since they all have the common domain \mathcal F .

Scaling fruit baskets

Let us consider a general fruit basket  \mathcal F \, \stackrel {f} {\rightarrow} \, \mathbf N and let n \in \mathbf N .

Definition: The n-tuple fruit basket nf is the fruit basket \mathcal F \, \stackrel {nf} {\longrightarrow} \, \mathbf N defined by multiplying each number of fruits in the basket f with the number n , i.e.,

(nf)(x) \stackrel {\mathrm{def}}{=} n(f(x)), \forall x \in \mathcal F.

Hence, if we choose n = 2 , and consider our concrete fruit basket g defined above, then we can compute the “doubled fruit basket” 2g as

(2g)(apple) = 2(g(apple)) = 2\cdot 0 = 0 ,

(2g)(pear) = 2(g(pear)) = 2\cdot 4 = 8 ,

(2g)(banana) = 2(g(banana)) = 2 \cdot 2 = 4 .

Expressing a fruit basket as a linear combination of characteristic functions

We are almost ready to start adding apples and pears. But first we need to learn how to characterize each type of fruit with a corresponding characteristic function for this type.

Consider a fruit basket function {h : \mathcal F \, \rightarrow \, \mathbf N} with

h(apple) = 2, h(pear) = 4, h(banana) = 6 .

If we define the three “characteristic” fruit basket functions

\mathcal F \, \stackrel {\chi_{apple}} {\longrightarrow} \, \mathbf N \, , \, \mathcal F \, \stackrel {\chi_{pear}} {\longrightarrow} \, \mathbf N \, , \, \mathcal F \, \stackrel {\chi_{banana}} {\longrightarrow} \, \mathbf N

by

\chi_{apple}(apple) = 1, \chi_{apple}(pear) = 0, \chi_{apple}(banana) = 0 ,

\chi_{pear}(apple) = 0, \chi_{pear}(pear) = 1, \chi_{pear}(banana) = 0 ,

\chi_{banana}(apple) = 0, \chi_{banana}(pear) = 0, \chi_{banana}(banana) = 1 ,

we see that the function h can be written as the following “scale-and-add” combination (normally called linear combination) of these characteristic functions:

h = 2 \, \chi_{apple} + 4 \, \chi_{pear} + 6 \, \chi_{banana}.

In this expression the three characteristic functions to the right indicate or characterize the respective types (apple, pear, banana) in the fruit basket domain \mathcal F , while the coefficients (2, 4, 6) represent the number of fruits of each type that are present in the fruit basket h .

Identifying a fruit type with its characteristic function

The final step in making us able to add apples and pears is to identify each type of fruit with its characteristic function:

apple \longleftrightarrow \chi_{apple} \, , \, pear \longleftrightarrow \chi_{pear} \, , \, banana \longleftrightarrow \chi_{banana} \, .

Addition of apples and pears as a direct sum

This identification turns the (binary composition) sum of fruit basket functions

h = 2 \, \chi_{apple} + 4 \, \chi_{pear} + 6 \, \chi_{banana}

into a direct sum of fruits:

h = 2 \, apple \oplus 4 \, pear \oplus 6 \, banana

where the addition is now denoted by the direct sum symbol \oplus .

Note: It is important to stress that \oplus represents a new type of addition. It is not the usual binary operation type of addition, because the sum of fruit types is NOT a single fruit type. This is in contrast to the sum of (fruit basket) functions, which is a binary operation since the sum of two functions (and therefore the sum of any finite number of functions) is a single function of the same type as that of the participating terms.

In a direct sum the terms are just concatenated (in any order) through the use of the symbol \oplus . This is why a direct sum is also referred to as a formal sum.

Clarifying the addition of x and y in a linear equation of two variables

Let us go back to the interpretation of the linear equation 2x + 3y = 5 as a straight line in the xy plane. This interpretation is based on representing a point in the plane in terms of its (x, y)-coordinates, i.e., as a pair of real numbers (x, y) . But in fact, the symbols x and y are NOT real numbers. Instead, in this interpretation, x and y must be considered as coordinate functions that map a point in the plane to its x coordinate and its y coordinate (in some arbitrarily chosen coordinate system).

Let P \in\mathcal P denote a point P in the plane \mathcal P . Then we have the coordinate functions \mathcal P \, \stackrel {x} {\rightarrow} \, \mathbf R and \mathcal P \, \stackrel {y} {\rightarrow} \, \mathbf R acting on the point P and producing its coordinates as a pair of real numbers (x(P), y(P)) \in \mathbf R \times \mathbf R in accordance with

\mathcal P \ni P \, \mapsto \, x (P) \in \mathbf R,

\mathcal P \ni P \, \mapsto \, y(P) \in \mathbf R.

Therefore, x + y must be interpreted as a (binary operation type) sum of these two coordinate functions, i.e., \mathcal P \, \stackrel {x+y} {\rightarrow} \, \mathbf R is the function defined by

\mathcal P \ni P \, \mapsto \, (x+y) (P) \stackrel {\mathrm{def}}{=} x(P)+y(P) \in \mathbf R .

The polynomial x+y as a direct sum of the symbols x and y

But what if we have no coordinate system, but only the sum x+y of two symbols x and y ? Then there are no functions involved and we must interpret the sum as a direct sum of the symbols x and y , i.e., we must add x and y in the same way as we have just learned to add apples and pears. In algebra, such a direct sum is called a polynomial sum, and the expression x+y , now interpreted as x\oplus y , is referred to as a polynomial (of degree one).

MULTIPLYING FRUIT BASKETS

The distributive law

The product of m sums of fruits is equal to the sum of all possible products of m fruits, where each of these factors is chosen from a different sum and the factors are taken in the order that they appear in the product of sums.

For example, the product of the two sums apple \oplus banana and cherry \oplus date is given by

(apple \oplus banana) \times (cherry \oplus date)

= (apple \times cherry )\oplus (apple \times date) \oplus (banana \times cherry) \oplus (banana \times date).

The product of m scaled sums (= linear combinations) of fruits is equal to the scaled sum (= linear combination) of all possible products of m fruits, where each of these “factor fruits” is chosen from a different sum, the factors are taken in the order that they appear in the product of scaled sums, and each m -product is scaled with the product of the corresponding coefficients, i.e., the product of the coefficients of the appearing factors.

For example, the product of the two scaled sums n_{a} \, apple\oplus n_{b} \, banana and n_{c} \, cherry \oplus n_{d} \, date is given by

(n_{a} \, apple\oplus n_{b} \, banana) \times (n_{c} \, cherry \oplus n_{d} \, date)

=n_{a}n_{c} (apple \times cherry )\oplus n_{a}n_{d}(apple \times date) \oplus n_{b}n_{c}(banana \times cherry) \oplus n_{b}n_{d}(banana \times date).

Here is a concrete example:

(2 \, apple \oplus 4 \, pear) \times (3 \, apple \oplus 1 \, pear)

= 2 \cdot 3 \, (apple \times apple) \oplus 2 \cdot 1 \, (apple \times pear) \oplus 4 \cdot 3 \, (pear \times apple) \oplus 4 \cdot 1 \, (pear \times pear)

= 6 \, (apple \times apple) \oplus 2 \, (apple \times pear) \oplus 12 \, (pear \times apple) \oplus 4 \, (pear \times pear).

Introducing ‘unit’ as a special type of fruit

(2 \, unit \oplus 5 \, apple) \times (4 \, apple \oplus 3 \, pear) =

= 2 \cdot 4 \, (unit \times apple) \oplus 2 \cdot 3 \, (unit \times pear) \oplus 5 \cdot 4 \, (apple \times apple) \oplus 5 \cdot 3 \, (apple \times pear) =

= 8 \, (unit \times apple) \oplus 6 \, (unit \times pear) \oplus 20 \, (apple \times apple) \oplus 15 \, (apple \times pear).

DESIGNING FRUIT ALGEBRAS

General rules to make an algebra work:

unit \times unit \equiv unit ,

unit \times apple \equiv apple,

unit \times pear \equiv pear.

Definition: An addition and multiplication structure on apples and pears that obeys these rules is called a (non-commutative) algebra on apples and pears.

A commutative algebra on apples and pears

Commutative rule: pear \times apple \equiv apple \times pear.

Definition: An algebra on apples and pears that obeys the commutative rule is called a commutative algebra on apples and pears.

An anti-commutative algebra on apples and pears

Anti-commutative rule: pear \times apple \equiv - \, apple \times pear.

Definition: An algebra on apples and pears that obeys the anti-commutative rule is called an anti-commutative algebra on apples and pears.