Although operations with natural numbers are well-known, it is necessary to synthesize this knowledge using a bit more “mathematical manners.” For this reason, we will review the operations of addition, multiplication, and power of natural numbers and their properties.

The Origin of Basic Operations with Natural Numbers

Addition Operation

The foundation of the addition operation was reviewed in the class on Natural Numbers and Peano’s Axioms, because the successor of a natural number can also be presented as follows:

S(n) = n+1

As we said that 2=S(1), 3=S(2), 4=S(3), \cdots and so on, then we can interpret addition as the successive application of the succession operation.

n+1 =S(n),

n+2 =S(S(n)),

n+3 =S(S(S(n))),

\vdots

And in general:

n+m = \underbrace{S(S(\cdots S(}_{m\;times} n)\cdots))

Properties of Addition

If a,b,c\in\mathbb{N}, then from this we can derive the well-known properties of addition:

All these properties can be demonstrated by induction, but we will skip that work. However, I encourage you to try it as a way to practice the technique of induction.

Multiplication Operation

Similarly, the product of natural numbers is defined as a successive application of addition. Therefore, we have

n\cdot m = \underbrace{n+ n+ \cdots + n}_{m\;times}

Properties of Multiplication

And analogously its properties can be obtained

Furthermore, from the definition of multiplication, the “1” in natural numbers acquires the quality that transforms it into unit:

Combined Addition and Multiplication

When the operations of addition and multiplication are combined, we obtain the distributive property of addition with respect to multiplication

The Order Induced by Operations of Natural Numbers

From the operations of addition and multiplication we reviewed, an order relation is induced in natural numbers through the following definitions:

Properties of Order in Natural Numbers

Law of Trichotomy

From this, it follows that only one of the following three situations can occur:

1. a\lt b
2. a = b
3. a\gt b

If it happens that, for example, a is not less than b, then one of the two must occur: either a=b, or a\gt b, that is greater than or equal, and it would be written: a\geq b. And analogously, a\leq b. is written when it is less than or equal.

Transitive Property

If a,b and c are any natural numbers, then it is fulfilled that:

[(a\lt b) \wedge (b\lt c)] \rightarrow (a\lt c)

And analogously:

[(a\gt b) \wedge (b\gt c)] \rightarrow (a\gt c)

Monotonicity Property

There is a monotonicity property for both addition and multiplication, which is:

Inverse Operations: Subtraction and Division of Natural Numbers

Subtraction of Natural Numbers

If a,b,c\in\mathbb{N}, we say that the difference between a and b (in that order), written as a-b, is defined through the relation

a-b=c \leftrightarrow a= b+c

As we can see, such a relation is true only if a\gt b, because there is no c\in \mathbb{N} with which this relation can be satisfied if a\leq b.

Through the definition of subtraction, we have the well-known rule of “what is added on one side of the equation can be passed to the other side subtracting, and vice versa”.

Division of Natural Numbers

If a,b,c\in\mathbb{N}, we say that the division between a and b (in that order), written as a/b, is defined through the relation

a/b=c \leftrightarrow a= bc

From the definition of division, we have the rule of “what is multiplying on one side of the equation can be passed to the other side dividing, and vice versa”.

Just as for the subtraction a - b to exist it must be that a\gt b, for the division a/b to exist, it is necessary that a be “divisible” by b. This is represented by writing

a is divisible by b \; :=a|b \; := \; (\exists k \in \mathbb{N})(a = kb)

Powers of Natural Numbers

With natural numbers, powers can be defined. Raising a natural number b, called the base, to another natural number n, called the exponent, means multiplying b by itself n times. Thus

b^n = \underbrace{bb\cdots b}_{n\;times}

If a,b,n,m\in\mathbb{N}, by (double) induction, the following properties can be demonstrated:

1. \displaystyle b^nb^m=b^{n+m}
2. \displaystyle \frac{b^n}{b^m} = b^{n-m}, provided that n\lt m
3. \displaystyle (ab)^n=a^nb^n
4. \displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
5. \displaystyle (b^n)^m=b^{nm}

Proposed and Solved Problems

1. All the properties shown here can be demonstrated using mathematical induction (either simple or double), but I have not developed them because the resulting demonstration is unnecessarily long for these intuitive results. However, those following these classes can try to perform these demonstrations as an exercise. [Only proposed]
2. Is b^{n^m} (which is defined as b^{(n^m)}) the same as (b^n)^m? [Solution]
3. Using the properties seen, verify the equalities:
   a) (a+b)(c+d) = ac+ad+bc+bd [Solution]
   
   b) (a+b)(c-d) = ac-ad+bc-bd,; if c\gt d [Solution]
   
   c)(a-b)(c-d) = ac-ad-bc+bd,; if a\gt b, c\gt d [Solution] 
4. Demonstrate that
   
   a) (a+b)^2 = a^2 + 2ab + b^2 [Solution]
   
   b) (a-b)^2 = a^2 - 2ab + b^2; if c\gt d [Solution]
   
   c) (a+b)(a-b) = a^2-b^2; if c\gt d [Solution]
   
   d) (a+b)^3 = a^3 + 3a^2b+3ab^2+b^3 [Solution]
   
   e) (a-b)^3 = a^3 - 3a^2b+3ab^2-b^3; if c\gt d [Solution]

    

5. Prove by complete induction the following properties:
   
   a) 1+2+3+4+\cdots+n = \displaystyle \frac{n(n+1)}{2} [Solution]
   
   b) 1^2+2^2+3^2+4^2+\cdots+n^2 = \displaystyle \frac{n(n+1)(2n+1)}{6} [Solution]
   
   c) 1^3+2^3+3^3+4^3+\cdots+n^3 = \displaystyle \frac{n^2(n+1)^2}{4} [Solution]