Reintroducing Arithmetic: Addition, Its Derivative Operations, & Properties of Those Operations
You might think you understand arithmetic; you have probably been doing arithmetic for a long time. However, most people really do not understand the fundamental logical premises undergirding arithmetic. Why is $\frac{4}{0}$ undefined? Why is $\frac{0}{0}$ “not well defined”? Why is it that $x^0 = 1$ for any x? There are many important concepts in arithmetic, which you were probably told to know but which were never explained. Here, we will be reintroducing you to arithmetic as a theory of number, since arithmetic is, in some sense, the foundation of Number Theory. Here, we will not only give you the principles, but prove and explain them. Understanding this foundation of foundations to the edifice of mathematics will be instrumental to your ultimate success in mastering logical thinking and quantitative reasoning.
1. Addition & Its Derivative Operations: Subtraction, Multiplication, & Division
In many ways, arithmetic is not math as it is understood today. Rather, arithmetic is a tool used in mathematical thinking. The term got originated from the Greek word “arithmos” which simply means numbers. We will consider this our definition of arithmetic:
Arithmetic is the elementary branch of mathematics that specifically deals with the study of numbers and properties of traditional operations like addition, subtraction, multiplication, and division.
While we are familiar with four primary operations in arithmetic – addition, subtraction, multiplication, and divisions – it is important to realize and understand that all these operations are merely variants of addition. Let’s examine this claim by first intimately examining each of the arithmetic operations.
Besides the traditional operations of addition, subtraction, multiplication, and division arithmetic also include advanced computing of percentage, logarithm, exponentiation and square roots, etc. Arithmetic is a branch of mathematics concerned with numerals and their traditional operations.
Addition ( $+$ )
All of mathematics can be built from the simple notion of addition. This was shown by Italian mathematician Giuseppe Peano (1858-1932) in the 19th century. Thus, in some profound sense, if we understand addition well, we can understand all of mathematics!
Addition is an operation that combines two or more values or “numbers” into a single value. The process of adding some collection of n numbers together is called “summation” and the result of addition is called the “sum” of numbers.
Additive Identity:
The “quantity of nothing” we will call zero, and we will designate zero with 0. Zero or 0 is said to be the “identity element of addition” (aka the additive identity) because adding 0 to any value returns that value.
Ex: $0 + 11 = 11$
Additive Inverse:
There is also an “inverse elements of addition” (aka additive inverses). While there is only one additive identity (i.e. 0), for every number there is a unique additive inverse, which is the number’s corresponding negative value. The result of adding inverse elements will be an identity element that is 0.
Ex: $11 + -11 = 0$
Subtraction ( $-$ )
Subtraction can simply be thought of as the inverse of addition. Whatever process we do under addition, can be undone with subtraction and vice versa. We can also think of subtraction as addition of negative numbers, and so subtraction is no different than addition fundamentally.
How’s it Like Addition?
Since subtraction and addition are essentially the same thing, the principle of identity and inverse elements is the same for subtraction and addition.
Ex: Subtraction (as in $5 – 3 = 2$) is the same as negative addition (as in $5 + -3 = 2$)
Ex: Subtraction (as in $5 – 3 = 2$) is the same as negative addition (as in $5 + -3 = 2$)
Fun fact! The parts of a subtraction problem have technical names. These are not necessary to know, but if you like obscure words you might like to know that in a subtraction problem of two values, the first value (not necessarily the greatest value) is called the “minuend” and the second value (not necessarily the lesser value) is called the “subtrahend.”
Multiplication ( $\times$ )
Multiplication is a “multiplicity of additions,” namely when we multiply something we are adding it to itself multiple times. Ergo, the use of words like “times” and “multiple” when talking about multiplication. There are various ways to write multiplication with the most traditional way being with the use of an “$\times$” between two numbers. However, for the purposes of using “$\times$” as a variable in more advanced mathematics, we will not use “$\times$” to mean multiplication. Rather we will use parenthesize to designate multiplication of quantities.
The two values involved in the operation of multiplication are known as “multiplicand” and “multiplier.” Multiplication then combines two ideas that are multiplicand (the value to be multiplied) and multiplier (the number of times which something is multiplied or added it itself) to give a single product.
How’s it Like Addition?
Thus, multiplication is no different from addition; it is merely the addition of something to itself a certain number of times. The product of two values, let’s say “a” and “b,” written as $a(b)$, says that we have a number “a” which should be added to itself “b” number of times.
Ex: $8(4) = 8 + 8 + 8 + 8 = 32$
Multiplicative Identity:
Multiplication also has an identity element called the multiplicative identity. Since we are wished to return the same value under identity operations, the multiplication of something only once returns the value itself. So, the multiplicative identity is 1. Multiplying something by 1 says that we add nothing to it and merely retain the value itself.
Ex: $8(1) = 8$ with no further additions to itself.
Multiplicative Inverse:
owever, while the additive inverse is the number that is the opposite of the number it is being added to, the multiplicative inverse is the “reciprocal” for some number, “a”, denoted as 1/a or $\frac{1}{a}$, which is a number that when multiplied by ‘a’ returns the multiplicative identity 1.
Ex: the multiplicative inverse of a fraction of two numbers ‘a’ and ‘b’ (that is a/b) is b/a.
$$\frac{a}{b}\bigg(\frac{b}{a}\bigg)=1$$
Ex: more concretely, the multiplicative inverse of a number, let’s say 5, is the reciprocal of 5, which is 1/5.
$$5\bigg(\frac{1}{5}\bigg)=\frac{5}{5}=1$$
Ex: similarly, the multiplicative inverse of a fraction, like 4/7, is 7/4
$$\bigg(\frac{4}{7}\bigg)\bigg(\frac{7}{4}\bigg)=\frac{28}{28}=1$$
Division ( $\div$ )
Division is the operation that computes the “quotient” of two numbers, and it is the inverse of multiplication. The two quantities involved in division are known as “dividend” and the “divisor” respectively. It is very important to realize that
a fraction is merely a division operation
$$8\div4=2\quad\text{and}\quad\frac{8}{4}=2$$
For many reasons which we will not talk about now, it is better that we represent all division problems henceforth as ratios or fractions rather than use the traditional division sign of arithmetic.
Thus: $7\div8$ is 7/8 or $\frac{7}{8}$ and similarly $4\div2$ is 4/2 or $\frac{4}{2}$ and so on.
How’s it Like Addition?
Division is also a variation of addition, since division is merely the inverse of multiplication, which is itself a variation of addition. So, we can think of division like a question that is asking about its associated multiplication.
Ex: 25/5 is asking “what number when multiplied by 5 is 25?” or “what number when added to itself 5 times is 25?” Of course, we can also read this problem in the more traditional way by saying “what number is the result of breaking 25 into 5 equal parts?” However, our goal here is to recognize the intrinsic underpinning of addition to all the arithmetic operations, and so I encourage you to think of division as not a question of breaking values into parts, but a reversal of multiplication, which is a summation of additions.
The fact that division is the inverse of multiplication is built into the idea of the “multiplicative inverse.” The multiplicative inverse of a number is the reciprocal of that number, as noted previously. To find reciprocals, we are required to think about inverse ratios or fractions, and since a fraction is a division problem, the multiplicative inverse has within itself division. Thus, there is NO different inverse for numbers under a division operation, since division is a kind of multiplication. Furthermore, since division is a kind of multiplication, division has the same identity.
Division and Fractions: One in the Same!
Since all division problems are expressible as fractions, one can think of the divided as the numerator and the divisor as the denominator from fraction anatomy. It will be best if we start to think of fractions in this way, rather than as something different from division.
Interesting Fact: The terms “numerator” and denominator” and be through of according to their derivative words “numeracy” and “denomination”. Numeracy means “countable” and “the concept of quantity, i.e how much of a thing there might be” and “denomination” means “a unit which defines the quantity or quality of something; that which gives something its name or quality/quantity.” Thus, the “numerator” is “how much of a thing you have” and the denominator is “what kind of things are we counting.” Ergo, 3/7 means that we have three pieces, each which are the denomination of a seventh (or 1/7).
2. Factorization, Divisibility, and Prime Numbers
3. Properties of Arithmetic:
The main properties of arithmetic you must know are:
- The Commutative Property
- The Associative Property
- The Distributive Property
- The Additive/Multiplicative Identity Property
- The Additive/Multiplicative Inverse Property
The identity and inverse properties you have already encountered in our discussion of additive and multiplicative inverses and identities. So, what we really need to focus on are the first three.
The Commutative Property:
The commutative property states that the numbers on which we operate can be moved or swapped from their positions without making any difference to the answer. The property holds for addition and multiplication, but not for subtraction and division.
So, the placement of adding numbers or multiplying number can be changed, and we will find the same results. Thus, for two numbers ‘a’ and ‘b’ we can say that
$$a + b = b + a \quad \text{and that} \quad a(b) = b(a)$$
Ex: $3+2 = 5$ and $2+3 = 5$, thus $2+3 = 3+2$
Ex: $3(4) = 3+3+3+3 = 12$ and $4(3) = 4+4+4 = 12$
The Associative Property:
The Associative Property is merely an expansion of the Commutative Property. Whereas in the Commutative Property we can move elements around (in addition and multiplication) and not change the result, the Associative Property say that in addition and multiplication problems with more than two elements, we can regroup the elements and solve without changing the answer. Simply put, the order in which the additions or multiplications are performed does not matter when the sequence of the numbers has not changed.
In such problems, we can use parenthesize to designate a hierarchy or order: whatever is inside the parenthesize, we will do first. Thus, we can change the location of parenthesize in an addition or multiplication problem without changing the ultimate result according to the Associative Property. Thus, for three numbers a, b, and c
$$(a + b) + c = a + (b + c) \quad \text{and} \quad (ab)(c) = a(bc)$$
$(4 + 5) + 6 = 4 + (5 + 6)$
$15=15$
$(4 \times 5) \times 6 = 4 \times (5 \times 6)$
$120=120$
The Distributive Property:
The distributive property deals with arithmetic problems that require both addition and multiplication concurrently. This property helps us to simplify the multiplication of a number by a sum (addition) or difference (subtraction). It “distributes” the multiplication expression as it simplifies the calculation. Thus, given three numbers a, b, and c we can say that
$$a (b + c) = a(b) + a(c) \quad \text{and} \quad a (b – c) = a(b) – a(c)$$
Ex: $4(5 + 6) = 4(5) + 4(6) = 20 + 24 = 44$
Ex: $4(5-6) = 4(5) – 4(6) = 20 -24 = -4$
The Identity Property:
The identity property is merely the formal name given to the arithmetic phenomenon of addition/subtraction and multiplication/division using the respective identity element, which was given previously. Recall that the “identity element” is an element that leaves other elements unchanged when combined with them. The identity element for the addition operation is 0 and for multiplication is 1.
Ex: For addition, $a + 0 = a$ and for multiplication $a(1) = a$
Inverse Property:
The inverse property is merely the formal name given to the arithmetic phenomenon of addition/.subtraction and multiplication/division using the respective inverse element, which was given previously. Recall that the “inverse element” is an element that “cancels” the opposing element under the arithmetic operation given. In addition/subtraction, the inverse is merely the negative of the given number (i.e. a number a and -a). In multiplication/division, the reciprocal for a number “a”, denoted by 1/a, is the inverse, since when it is multiplied by “a” we yield the multiplicative identity 1.
Ex: For addition/subtraction $a + -a = 0$ and for multiplication/division $a\frac{1}{a}=\frac{a}{a}=1$ or more generally
$$ \frac{a}{b}\frac{b}{a}=\frac{ab}{ba} \quad \text{and by the commutative property of multiplication} \quad \frac{ab}{ba}=\frac{ab}{ab}=1$$
Multiplication Property of Zero and Dividing by Zero:
One last thing to remember before we move onto more interesting properties of arithmetic is the Multiplication Property of Zero states that multiplying any number by 0 will return 0.
Ex: for a number ‘a’ , $a(0) = 0$, or more concretely $5(0) = 0$
What we have been doing so far actually builds up to this point. Perhaps you have heard that dividing by 0 is “undefined” or that one simply “cannot divide by zero.” Many math teachers will never explain why, and the reason for this is that most math teachers never understand that all these operations are merely forms of addition. However, if we understand all four arithmetic operations as addition, we can see why dividing by zero is not only not permissible but literally and matter-of-factly undefined.
First, do remember that we can divide zero by any number (excluding zero), as in 0/5 or 0/a. Dividing zero by any number (excluding zero itself) always returns zero. This is because dividing zero by something is actually the same as multiplying it by the reciprocal of the thing.
Ex: $\frac{0}{5}=0\frac{1}{5}=0$ since multiplying anything by zero returns zero by the Multiplication Property of Zero.
However, what if we ask the inverse questions: 5/0 ? What is this literally asking. Well, if we ask the classical division question of “breaking things” the undefined nature of this questions is not as apparent: “What is five broken into zero equal parts?” – you might say breaking 5 into 0 parts will return the whole 5. Right?…
Well let’s ask the same question in terms of the addition understanding of division: “What number must I add to itself zero times to get 5?” Hmm…..
Now think about this. If I asked, “What number must I add to itself one time to get 5?” you would answer 5, since five multiplied only once is 5 alone. But now I am asking what number when multiplied zero times returns five. There is no number when multiplied zero times that returns five:
Ex: $a(0) = 5$ is impossible, since any number times 0 returns zero, and the answer cannot be 0, since $0(0) = 0$, not 5.
Thus, there is no answer to the question of dividing something by zero. The operation is then undefined and is meaningless.