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# Tag Archives: Numbers

## Proofs of Irrationality

“Irrational numbers are those real numbers which are not rational numbers!”

Def.1: Rational Number

A rational number is a real number which can be expressed in the form of $\frac{a}{b}$ where $a$ and $b$ are both integers relatively prime to each other and $b$ being non-zero.
Following two statements are equivalent to the definition 1.
1. $x=\frac{a}{b}$ is rational if and only if $a$ and $b$ are integers relatively prime to each other and $b$ does not equal to zero.
2. $x=\frac{a}{b} \in \mathbb{Q} \iff \mathrm{g.c.d.} (a,b) =1, \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus \{0\}$.

## A Trip to Mathematics: Part IV Numbers

If logic is the language of mathematics, Numbers are the alphabet. There are many kinds of number we use in mathematics, but at a broader aspect we may categorize them in two categories:
1. Countable Numbers
2. Uncountable Numbers
The names are enough to explain the properties of above numbers. The numbers which can be counted in nature are called Countable Numbers and the numbers which can not be counted are called Uncountable Numbers.

Well, this is not the correct way to classify the bunch of types of numbers. We have some formal names for special types of numbers, like Real numbers, Complex Numbers, Rational Numbers, Irrational Numbers etc.. We shall discuss these non-interesting numbers (let me say them non-interesting) at first and then some interesting numbers(those numbers are really interesting to learn). Although in this post I have concisely described the classification, I will rigorously discuss them later.
Let me start this discussion with the memorable quote by Leopold Kronecker:

“God created the natural numbers, and all the rest is the work of man.”

Actually, he meant to say that all numbers, like Real Numbers, Complex Numbers, Fractions, Integers, Non-integers etc. are made up of the numbers given by God to the human. These God Gifted numbers are actually called Natural Numbers. Natural Numbers are the numbers which are used to count things in nature.

Eight pens, Eighteen trees, Three Thousands people etc. are measure of natural things and thus ‘Eight’, ‘Eighteen’, ‘Three Thousands’ etc. are called natural numbers and we represent them numerically as ’8′, ’18′, ’3000′ respectively. So, if 8, 18, 3000 are used in counting natural things, are natural numbers. Similarly, 1, 2, 3, 4, and other numbers are also used in counting things —thus these are also Natural Numbers.

Let we try to form a set of Natural Numbers. What will we include in this set?

1?                    (yes!).
2?                    (yes).
3?                     (yes).
….
1785?                (yes)
…and          so on.

This way, after including all elements we get a set of natural numbers {1, 2, 3, 4, 5, …1785, …, 2011,….}. This set includes infinite number of elements. We represent this set by Borbouki’s capital letter N, which looks like $\mathbb{N}$ or bold capital letter N ($\mathbf{N}$ where N stands for NATURAL. We will define the set of all natural numbers as:

$\mathbb{N} := \{ 1, 2, 3, 4, \ldots, n \ldots \}$.

It is clear from above set-theoretic notation that $n$-th element of the set of natural numbers is $n$.
In general, if a number $n$ is a natural number, we right that $n \in \mathbb{N}$.
Please note that some mathematicians (and Wolfram Research) treat ’0′ as a natural number and state the set as $\mathbb{N} :=\{0, 1, 2, \ldots, n-1, \ldots \}$, where $n-1$ is the nth element of the set of natural numbers; but we will use first notion since it is broadly accepted.

Now we shall try to define Integers in form of natural numbers, as Kronecker’s quote demands. Integers (or Whole numbers) are the numbers which may be either positives or negatives of natural numbers including 0.
Few examples are 1, -1, 8, 0, -37, 5943 etc.
The set of integers is denoted by $\mathbb{Z}$ or $\mathbf{Z}$ (here Z stands for ‘Zahlen‘, the German alternative of integers). It is defined by
$\mathbb{Z} := \{ \pm n: n \in \mathbb{N} \} \cup \{0\}$
i.e., $\mathbb{Z} := \{\ldots -3, -2, -1, 0, 1, 2, 3 \ldots \}$.

Now, if we again consider the statement of Kronecker, we might ask that how could we prepare the integer set $\mathbb{Z}$ by the set $\mathbb{N}$ of natural numbers? The construction of $\mathbb{Z}$ from $\mathbb{N}$ is motivated from the requirement that every integer can be expressed as difference of two positive integers (i.e., Natural Numbers). Let $a,b,c,d \in \mathbb{N}$ and a relation ρ is defined on $\mathbb{N} \times \mathbb{N}$ by $(a,b) \rho (c,d)$ if and only if $a+d = b+c$. The relation ρ is an equivalence relation and the equivalence classes under ρ are called integers and defined as $\mathbb{Z} := \mathbb{N} \times \mathbb{N} /\rho$. Now we can define set of integers by an easier way, as $\mathbb{Z}:= \{a-b; \ a,b \in \mathbb{N}\}$. Thus an integer is a number which can be produced by difference of two or more natural numbers. And similarly as converse defintion, positive integers are called Natural Numbers.
After Integers, we head to rational numbers. Say it again– ‘ratio-nal numbers‘ –numbers of ratio.

Image via Wikipedia

A rational number $\frac{p}{q}$ is defined as a ratio of an integer p and a non-zero integer q. (Well that is not a perfect definition, but as an introduction it is great for understanding.) The set of rational numbers is defined by $\mathbb{Q}$.
Once integers are formed, we can form Rational (and Irrational numbers: numbers which are not rational ) using integers.
We consider an ordered pair $(p,q):=\mathbb{Z} \times (\mathbb{Z} \setminus \{0 \})$ and another ordered pair $(r,s):=\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})$ and define a relation ρ by $(p,q) \rho (r,s) \iff ps=qr$ for $p,q,r,s \in \mathbb{Z}, \ q, r \ne 0$. Then ρ is an equivalence relation of rationality, class (p,q). The set $\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})/\rho$ is denoted by $\mathbb{Q}$ (and the elements of this set are called rational numbers).
In practical understandings, the ratio of integers is a phrase which will always help you to define the rational numbers. Examples are $\frac{6}{19}, \ \frac{-1}{2}=\frac{-7}{14}, \ 3\frac{2}{3}, \ 5=\frac{5}{1} \ldots$. Set of rational numbers includes Natural Numbers and Integers as subsets.
Consequently, irrational numbers are those numbers which can not be represented as the ratio of two integers. For example $\pi, \sqrt{3}, e, \sqrt{11}$ are irrationals.
The set of Real Numbers is a relatively larger set, including the sets of Rational and Irrational Numbers as subsets. Numbers which exist in real and thus can be represented on a number line are called real numbers. As we formed Integers from Natural Numbers; Rational Numbers from Integers, we’ll form the Real numbers by Rational numbers.
The construction of set $\mathbb{R}$ of real numbers from $\mathbb{Q}$ is motivated by the requirement that every real number is uniquely determined by the set of rational numbers less than it. A subset $L$ of $\mathbb{Q}$ is a real number if L is non-empty, bounded above, has no maximum element and has the property that for all $x, y \in \mathbb{Q}, x < y$ and $y \in L$ implies that $x \in L$. Real numbers are the base of Real Analysis and detail study about them is case of study of Real Anlaysis.
Examples of real numbers include both Rational (which also contains integers) and Irrational Numbers.

The square root of a negative number is undefined in one dimensional number line (which includes real numbers only) and is treated to be imaginary. The numbers containing or not containing an imaginary number are called complex numbers.
Some very familiar examples are $3+\sqrt{-1}, \sqrt{-1} =i, \ i^i$ etc. We should assume that every number (in lay approach) is an element of a complex number. The set of complex numbers is denoted by $\mathbb{C}$. In constructive approach, a complex number is defined as an ordered pair of real numbers, i.e., an element of $\mathbb{R} \times \mathbb{R}$ [i.e., $\mathbb{R}^2$] and the set as $\mathbb{C} :=\{a+ib; \ a,b \in \mathbb{R}$. Complex numbers will be discussed in Complex Analysis more debately.
We verified Kronecker’s quote and shew that every number is sub-product of postive integers (natural numbers) as we formed Complex Numbers from Real Numbers; Real Numbers from Rational Numbers; Rational Numbers from Integers and Integers from Natural Numbers. //
Now we reach to explore some interesting kind of numbers. There are millions in name but few are the follow:
Even Numbers: Even numbers are those integers which are integral multiple of 2. $0, \pm 2, \pm 4, \pm 6 \ldots \pm 2n \ldots$ are even numbers.

Odd Numbers: Odd numbers are those integers which are not integrally divisible by 2. $\pm 1, \pm 3, \pm 5 \ldots \pm (2n+1) \ldots$ are all odd numbers.

Prime Numbers: Any number $p$ greater than 1 is called a prime number if and only if its positive factors are 1 and the number $p$ itself.
In other words, numbers which are completely divisible by either 1 or themselves only are called prime numbers. $2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \ldots$ etc. are prime numbers or Primes. The numbers greater than 1, which are not prime are called Composite numbers.
Twin Primes: Consecutive prime numbers differing by 2 are called twin primes. For example 5,7; 11,13; 17,19; 29,31; … are twin primes.

Pseudoprimes: Chinese mathematicians claimed thousands years ago that a number $n$ is prime if and only if it divides $2^n -2$. In fact this conjecture is true for $n \le 340$ and false for upper numbers because first successor to 340, 341 is not a prime ($31 \times 11$) but it divides $2^{341}-2$. This kind of numbers are now called Pseudoprimes. Thus, if n is not a prime (composite) then it is pseudoprime $\iff n | 2^n-2$ (read as ‘n divides 2 powered n minus 2‘). There are infinitely many pseudoprimes including 341, 561, 645, 1105.

Carmichael Numbers or Absolute Pseudoprimes: There exists some pseudoprimes that are pseudoprime to every base $a$, i.e., $n | a^n -a$ for all integers $a$. The first Carmichael number is 561. Others are 1105, 2821, 15841, 16046641 etc.

e-Primes: An even positive integer is called an e-prime if it is not the product of two other even integers. Thus 2, 6, 10, 14 …etc. are e-primes.

Germain Primes: An odd prime p such that 2p+1 is also a prime is called a Germain Prime. For example, 3 is a Germain Prime since $2\times 3 +1=7$ is also a prime.
Relatively Prime: Two numbers are called relatively prime if and only their greatest common divisor is 1. In other words, if two numbers are such that no integer, except 1, is common between them when factorizing. For example: 7 and 9 are relatively primes and same are 15, 49.

Perfect Numbers: A positive integer n is said to be perfect if n equals to the sum of all its positive divisors, excluding n itself. For example 6 is a perfect number because its divisors are 1, 2, 3 and 6 and it is obvious that 1+2+3=6. Similarly 28 is a perfect number having 1, 2, 4, 7, 14 (and 28) as its divisors such that 1+2+4+7+14=28. Consecutive perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056 etc.

Mersenne Numbers and Mersenne Primes: Numbers of type $M_n=2^n-1; \ n \ge 1$ are called Mersenne Numbers and those Mersenne Numbers which happen to be Prime are called Mersenne Primes. Consecutive Mersenne numbers are 1, 3 (prime), 7(prime), 15, 31(prime), 63, 127.. etc.

Catalan Numbers: The Catalan mumbers, defined by $C_n = \dfrac{1}{n+1} \binom{2n}{n} = \dfrac{(2n)!}{n! (n+1)!} \ n =0, 1, 2, 3 \ldots$ form the sequence of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …

Triangular Number: A number of form $\dfrac{n(n+1)}{2} \ n \in \mathbb{N}$ represents a number which is the sum of n consecutive integers, beginning with 1. This kind of number is called a Triangular number. Examples of triangular numbers are 1 (1), 3 (1+2), 6 (1+2+3), 10(1+2+3+4), 15(1+2+3+4+5) …etc.

Square Number: A number of form $n^2 \ n \in \mathbb{N}$ is called a sqaure number.
For example 1 ($1^2$), 4 ($2^2$), 9($3^2$), 16 ($4^2$)..etc are Square Numbers.

Palindrome: A palindrome or palindromic number is a number that reads the same backwards as forwards. For example, 121 is read same when read from left to right or right to left. Thus 121 is a palindrome. Other examples of palindromes are 343, 521125, 999999 etc.

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## Fermat Numbers

Fermat Number, a class of numbers, is an integer of the form $F_n=2^{2^n} +1 \ \ n \ge 0$.

For example: Putting $n := 0,1,2 \ldots$ in $F_n=2^{2^n}$ we get $F_0=3$, $F_1=5$, $F_2=17$, $F_3=257$ etc.

Fermat observed that all the integers $F_0, F_1, F_2, F_3, \ldots$ were prime numbers and announced that $F_n$ is a prime for each natural value of $n$.

In writing to Prof. Mersenne, Fermat confidently announced:

I have found that numbers of the form $2^{2^n}+1$ are always prime numbers and have long since signified to analysts the truth of this theorem.

However, he also accepted that he was unable to prove it theoretically. Euler in 1732 negated Fermat’s fact and told that $F_1 -F_4$ are primes but $F_5=2^{2^5} =4294967297$ is not a prime since it is divisible by 641.
Euler also stated that all Fermat numbers are not necessarily primes and the Fermat number which is a prime, might be called a Fermat Prime. Euler used division to prove the fact that $F_5$ is not a prime. The elementary proof of Euler’s negation is due to G. Bennett.

# Theorem:

The Fermat number $F_5$ is divisible by $641$ i.e., $641|F_5$.

# Proof:

As defined $F_5 :=2^{2^5}+1=2^{32}+1 \ \ldots (1)$

Factorising $641$ in such a way that $641=640+1 =5 \times 128+1 \\ =5 \times 2^7 +1$
Assuming $a=5 \bigwedge b=2^7$ we have $ab+1=641$.

Subtracting $a^4=5^4=625$ from 641, we get $ab+1-a^4=641-625=16=2^4 \ \ldots (2)$.

Now again, equation (1) could be written as
$F_5=2^{32}+1 \\ \ =2^4 \times {(2^7)}^4+1 \\ \ =2^4 b^4 +1 \\ \ =(1+ab-a^4)b^4 +1 \\ \ =(1+ab)[a^4+(1-ab)(1+a^2b^2)] \\ \ =641 \times \mathrm{an \, Integer}$
Which gives that $641|F_n$.

Mathematics is on its progression and well developed now but it is yet not confirmed that whether there are infinitely many Fermat primes or, for that matter, whether there is at least one Fermat prime beyond $F_4$. The best guess is that all Fermat numbers $F_n>F_4$ are composite (non-prime).
A useful property of Fermat numbers is that they are relatively prime to each other; i.e., for Fermat numbers $F_n, F_m \ m > n \ge 0$, $\mathrm{gcd}(F_m, F_n) =1$.

Following two theorems are very useful in determining the primality of Fermat numbers:

# Pepin Test:

For $n \ge 1$, the Fermat number $F_n$ is prime $\iff 3^{(F_n-1)/2} \equiv -1 \pmod {F_n}$

# Euler- Lucas Theorem

Any prime divisor $p$ of $F_n$, where $n \ge 2$, is of form $p=k \cdot 2^{n+2}+1$.

Fermat numbers ($F_n$) with $n=0, 1, 2, 3, 4$ are prime; with $n=5,6,7,8,9,10,11$ have completely been factored; with $n=12, 13, 15, 16, 18, 19, 25, 27, 30$ have two or more prime factors known; with $n=17, 21, 23, 26, 28, 29, 31, 32$ have only one prime factor known; with $n=14,20,22,24$ have no factors known but proved composites. $F_{33}$ has not yet been proved either prime or composite.

## Problem1: Smallest Autobiographical Number:

A number with ten digits or less is called autobiographical if its first digit (from the left) indicates the number of zeros it contains,the second digit the number of ones, third digit number of twos and so on.

For example: 42101000 is autobiographical.

Find, with explanation, the smallest autobiographical number.

Solution of Problem 1

## Problem 2: Fit Rectangle:

A rectangle has dimensions $39.375$ cm $\times 136.5$ cm.

• Find the least number of squares that will fill the rectangle.
• Find the least number of squares that will fill the rectangle, if every square must be the same size and Find the largest square that can be tiled to completely fill the rectangle.

Solution of Problem 2

## Solutions of Problem 1:

The restrictions which define an autobiographical number make it straightforward to find the lowest one. It cannot be 0, since by
definition the first digit must indicate the number of zeros in the number. Presumably then, the smallest possible autobiographical number will contain only one 0.If this is the case, then the first digit must be 1. 10 is not a candidate because the second digit must indicate the number of 1s in the number–in this case, 1. So If the
number contains only one zero, it must contain more than one 1.
(If it contained one 1 and one 0, then the first two digits
would be 11, which would be contradictory since it actually contains two 1s).
Again, presumably the lowest possible such number will contain the lowest
possible number of 1s, so we try a number with one 0 and two 1s. It will be of the form: 12-0–..
Now, there is one 2 in this number, so the first three digits must be 121. To meet all the conditions discussed above, we can simply take a 0 onto the end of this to obtain 1210, which is
the smallest auto-biographical number.

## Solution of Problem 2:

We solve the second and third parts of the question
first:

We convert each number to a fraction and get a common denominator, then find the gcd (greatest common divisor) of the numerators.

That is, with side lengths $39.375$ cm and $136.5$ cm , we convert those numbers to fractions (with a common
denominator):
$39.375 = \dfrac{315}{8}$.

$136.5 = \dfrac{273}{2} = \dfrac{1092}{8}$.

Now we need to find the largest common factor of 315 and 1092.
Which is 21. So $\dfrac{21}{8}=2.625$ is the largest number that divides evenly into the two numbers $39.375$ and $136.5$.
There will be $\dfrac{1092}{21} \times \dfrac{315}{21} = 52 \times 15 = 780$ squares, each one a $2.625$ cm $\times 2.625$ cm square needed to fill the rectangle (52 in each row,with 15 rows).

Now we shall solve the first part.
Number of squares lengthwise is 52 and breadthwise is 15. Now we will combine these squares in order to find least number of squares to fill the rectangle. First three squares would be of
dimension 15 by 15. In this way length of 45 units is utilized. Now the rectangle which is left with us excluding three squares is 7 by 15. Again in the same way we can make two squares of dimension 7 by 7. In this way breadth of 14 units is utilized.
Now we are left with the rectangle of dimension 7 by 1.
These can further be subdivided into seven squares each of
dimension 1 by 1. In this way the least number of squares to fill the
rectangle is 3 + 2+ 7 = 12. The required answer is 12.
Note that the three numbers 3, 2, and 7 are involved in the Euclidean Algorithm for finding the g.c.d.!

Source: Internet

## The Collatz Conjecture : Unsolved but Useless

The Collatz Conjecture is one of the Unsolved problems in mathematics, specially in Number Theory. The Collatz Conjecture is also termed as 3n+1 conjecture, Ulam Conjecture, Kakutani’s Problem, Thwaites Conjecture, Hasse’s Algorithm, Syracuse Problem.

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## Six Puzzles

Assume that the English letters are digits (from 0 to 9 ) and they satisfy the given relations, then you have to solve each equation for these letters.

For Example:
$ABCDE \times ABCDE = FDBABCDE$ can have a solution:
$09376 \times 09376 = 87909376$

Similarly, Try these:

1. $ABCDEEABCD$ $\times$ $FEC$ = $AAAAAAAAAAAA$

2. $(A+B+C+D+E)$ $\times$ $(A+B+C+D+E)$ $\times$ $(A+B+C+D+E)$ = $ABCDE$

3. $6 \times ABCDEF =DEFABC$

4. $ABCDEABCDEABCDE$ = $C \times CCCCCGGGGH \times ABCDE$

5. $ABCDEABCDE$ $= FF \times GHGF \times 86485$

# Solution:

1. $8547008547 \times 104$ $=888888888888$
2. $(1+9+6+8+3) \times (1+9+6+8+3) \times (1+9+6+8+3)$ $=19683$
3. $6 \times 14857$ $857142$
4. $283512835128351$ $=3 \times 3333366667 \times 28351$
5. $8648586485$ $=11 \times 9091 \times 86485$

Puzzle Idea: Mr. Sawinder Singh, Gurdaspur, Punjab (INDIA)
Note:
There may be many other solutions for these puzzles too.

## What is a Prime Number?

An integer, say $p$, [ $\ne {0}$ & $\ne { \pm{1}}$ ] is said to be a prime integer iff its only factors (divisors) are $\pm{1}$ & $\pm{p}$.

## As?

Few easy examples are:
$\pm{2}, \pm{3}, \pm{5}, \pm{7}, \pm{11}, \pm{13}$ …….etc…this list increases upto infinity & mathematicians are trying to find the larger one than the largest, because primes numbers/integers are not symmetrical (as any one cannot guess the next prime after one.) As of now the biggest prime number found is (probably) $M-47$, called as Mersenne 47. This has an enormous value of $2^{43112609} -1$. It is very hard to write it on paper because it consists of $12978189$ digits.
»M47 was Invented in 2008. (more…)