In this section, we will show some simple proofs of well-known and important statements. The reader will get acquainted to further proofs in the following chapters. In chapter 3.6 we will continue to practice this skill in proving several sum formulas.
Before we begin our first proof, let us refresh a few concepts, which will appear in the proven claim. We recall first the notions of rational and irrational real number.
A real number
Furthermore, let us recall the notions of coprime and non-coprime numbers.
If two integer numbers
Which of the following numbers are rational and which are irrational?
The numbers
Which of the following pairs of numbers are coprime or non-coprime?
1. non-coprime (both can be divided by
We can use the so-called contradiction. The idea at the basis of a proof by contradiction is
simple. One of the logic axioms says that every statement
The square root of
We assume the opposite, that is
It follows5 the equality
Since
Let us summarize the principle of the proof by contradiction. We want to be convinced of the truth of a certain claim
In the following, we illustrate the proof by mathematical induction. This type of proof is often used when we have infinitely many statements numbered by natural indices7, for example
proove the first claim
for any natural
A graphical representation of this procedure is shown in figure 2.1. The red arrows correspond to the induction step. The starting point, i.e. the proof of
Mathematical induction can be compared to demolish a domino spiral. Each domino piece represents a „statement“ and can be in two states. A piece can be standing, or falling (similarly a statement can be true, or false). If we want to find out if the assembled domino spiral falls, we have two options. We can check all the pieces one by one and see if they fall. The second option is to check:
if the first piece falls,
two adjacent pieces are located at such a distance that if the first one falls (the one closer to the first piece) then its neighbor also falls (analog of the induction step).
Then we automatically know that all the pieces would fall. Let us emphasize the substantial difference in these approaches. The second method (ie, mathematical induction) controls the state of only the first piece, while it does not checks whether the others are standing or not, unless they are adjacent.
Let us illustrate the proof by mathematical induction for the so-called binomial theorem. In the statement of the theorem, we use the abbreviated sum, i.e. the summation notation, which the reader can find described more in detail in 3.6. Factorials, bynomial coefficients and general combinatorics are treated in section 3.7.
For any real number
Let us verify that the equality being examined is true for the first
The equality
By direct calcuclation, we get8
In the equality marked by the exclamation point, we used the inductive assumption (the validity of the relation for
The claim of the binomial theorem contains well kown algebraic „formulas“
The above formulas represent special cases of the binomial theorem for a particular chosen
This calculation effectively proves the binomial theorem for
The importance ad utility of the binomial theorem can be further demonstrated on a concrete example
(someone would say „trick“). Let us imagine that we are going to count
What is the sum of the first
equal to? Proove your claim.
Another type of proof is the direct proof. So to speak, without any detours, straightforwardly, we derive the claim from the assumptions. Consider the following theorem.
Let us take
In other words, after multiplying the sum by the bracket
Alternatively, it would be possible to proove theorem 2.3 by mathematical induction (try it!).
There are well known special cases of theorem 2.3:
These formulas and the general formula 2.2 will be useful (not only) in calculating the limits in the future. What makes theorem 2.3 so useful?
It allows us to express the difference of same powers of two numbers by the difference of the numbers themselves.
In other words, if we have some information about the difference
The proven equality (2.2) also contained the formula for the sum of the first terms of a geometric sequence with ratio
and dividing by the non zero factor