Imaginary and complex numbers

When Are We Ever Going to Use This? – Imaginary and Complex Numbers
The number √-9 may seem impossible, and it is when talking about real numbers. The reason is that when a number is squared, the product is never negative. However, in mathematics, and in daily life for that matter, numbers like these are used in abundance. Mathematicians need a way to incorporate numbers like √-9 into equations, so that these equations can be solvable. At first the going was tough, but as the topic gained more momentum, mathematicians found a way to solve what their predecessors deemed impossible with the use of a simple letter – i, and today it is used in a plethora of ways.

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History of Imaginary Numbers
During the early days of human mathematical history, when someone reached a point in a equation that contained the square root of a negative number, they froze. One of the first recorded instances of this was in 50 AD, when Heron of Alexandria was examining the volume of a truncated pyramid. Unfortunately for him, he came upon the expression which computes to . However, at his time, not even negative numbers were “discovered” or used, so he just ignored the negative symbol and continued on with his work. Thus, this first encounter with complex numbers was unsuccessful.
It is not until the sixteenth century when the dilemma of complex numbers returns, when mathematicians attempt to solve cubic and other equations of higher-order. The Italian algebraist Scipione dal Ferro soon encountered these imaginary numbers when solving higher degree polynomials, and he said that finding the solution to these numbers was “impossible”. However, Girolamo Cardano, also Italian, gave this subject some hope. During his mathematical career, he opened up the realm of negative numbers, and soon began analyzing their square roots. Although he admitted that imaginary numbers were pretty much useless, he shed some light on the subject. Fortunately, this little bit of light would soon turn into a full beam.
In 1560, the Bolognese mathematician Rafael Bombelli discovered a unique property of imaginary numbers. He found that, although the number √-1 is irrational and non-real, when multiplied by itself (squared), it produces both a rational and real number in -1. Using this idea, he also came up with the process of conjugation, which is where two similar complex numbers are multiplied together to get rid of the imaginary numbers and radicals. In the standard a+bi form, a+bi and a-bi are conjugates of each other. At this point, many other mathematicians were attempting to solve the elusive number of √-1, and although there were many more failed attempts, there was a little bit of success.
However, although I have been using the term imaginary throughout this paper, this term did not come to be until the 17th century. In 1637, Rene Descartes first used the word “imaginary” as an adjective for these numbers, meaning that they were insolvable. Then, in the next century, Leonhard Euler finalized this term in his own Euler’s identity where he uses the term ifor √-1. He then connects “imaginary” in a mathematical sense with the square root of a negative number when he wrote: “All such expressions as √-1, √-2 . . . are consequently impossible or imaginary numbers, for we may assert that they are neither nothing, not greater than nothing, nor less than nothing, which necessarily renders them imaginary or impossible.” Although Euler states that these numbers are impossible, he contributes with both the term “imaginary” and the symbol for √-1 as i. Although Euler does not solve an imaginary number, he creates a way to apply it to mathematics without much trouble. Throughout the years, there have been many skeptics of imaginary numbers; one is the Victorian mathematician Augustus De Morgan, who states that complex numbers are useless and absurd. There was a tug-of-war battle between those who believed in the existence of numbers such as i and those who did not.
Soon after Rene Descartes’ contributions, the mathematician John Wallis produced a method for graphing complex numbers on a number plane. For real numbers, a horizontal number line is used, with numbers increasing in value as you move to the left. John Wallis added a vertical line to represent the imaginary numbers. This is called the complex number plane where the x-axis is named the real axis and the y-axis is named the imaginary axis. In this way, it became possible to plot complex numbers. However, John Wallis was ignored at this time, it took over a century and a few more mathematicians for this idea to accepted. The first one to agree with Wallis was Jean Robert Argand in 1806. He wrote the procedure that John Wallis invented for graphing complex numbers on a number plane. The person who made this idea widespread was Carl Friedrich Gauss when he introduced it to many people. He also made popular the use of the term complex number to represent the a+bi form. These methods made complex numbers more understandable.
Throughout the 1800s, many mathematicians have contributed to the validity of complex numbers. Some names, to name a few, are Karl Weierstrass, Richard Dedekind, and Henri Poincare, and they all contributed by studying the overall theory of complex numbers. Today, complex numbers are accepted by most mathematicians, and are easily used in algebraic equations.
 

Construction Of Real Numbers

All mathematicians know (or think they know) all about the real numbers. However usually we just accept the real numbers as ‘being there’ rather than considering precisely what they are. In this project I will attempts to answer that question. We shall begin with positive integers and then successively construct the rational and finally the real numbers. Also showing how real numbers satisfy the axiom of the upper bound, whilst rational numbers do not. This shows that all real numbers converge towards the Cauchy’s sequence.
1 Introduction
What is real analysis; real analysis is a field in mathematics which is applied in many areas including number theory, probability theory. All mathematicians know (or think they know) all about the real numbers. However usually we just accept the real numbers as ‘being there’ rather than considering precisely what they are. The aim of this study is to analyse number theory to show the difference between real numbers and rational numbers.

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Developments in calculus were mainly made in the seventeenth and eighteenth century. Examples from the literature can be given such as the proof that π cannot be rational by Lambert, 1971. During the development of calculus in the seventeenth century the entire set of real numbers were used without having them defined clearly. The first person to release a definition on real numbers was Georg Cantor in 1871. In 1874 Georg Cantor revealed that the set of all real numbers are uncountable infinite but the set of all algebraic numbers are countable infinite.
As you can see, real analysis is a somewhat theoretical field that is closely related to mathematical concepts used in most branches of economics such as calculus and probability theory. The concept that I have talked about in my project are the real number system.
2 Definitions
Natural numbers
Natural numbers are the fundamental numbers which we use to count. We can add and multiply two natural numbers and the result would be another natural number, these operations obey various rules.
(Stirling, p.2, 1997)
Rational numbers
Rational numbers consists of all numbers of the form a/b where a and b are integers and that b ≠ 0, rational numbers are usually called fractions. The use of rational numbers permits us to solve equations. For example; a + b = c, ad = e, for a where b, c, d, e are all rational numbers and a ≠ 0. Operations of subtraction and division (with non zero divisor) are possible with all rational numbers.
(Stirling, p.2, 1997)
Real numbers
Real numbers can also be called irrational numbers as they are not rational numbers like pi, square root of 2, e (the base of natural log). Real numbers can be given by an infinite number of decimals; real numbers are used to measure continuous quantities. There are two basic properties that are involved with real numbers ordered fields and least upper bounds. Ordered fields say that real numbers comprises a field with addition, multiplication and division by non zero number. For the least upper bound if a non empty set of real numbers has an upper bound then it is called least upper bound.
Sequences
A Sequence is a set of numbers arranged in a particular order so that we know which number is first, second, third etc… and that at any positive natural number at n; we know that the number will be in nth place. If a sequence has a function, a, then we can denote the nth term by an. A sequence is commonly denoted by a1, a2, a3, a4… this entire sequences can be written as or (an). You can use any letter to denote the sequence like x, y, z etc. so giving (xn), (yn), (zn) as sequences
We can also make subsequence from sequences, so if we say that (bn) is a subsequence of (an) if for each n∈ â„• we get;
bn = ax for some x ∈ â„• and bn+1 = by for some y ∈ â„• and x > y.
We can alternatively imagine a subsequence of a sequence being a sequence that has had terms missing from the original sequence for example we can say that a2, a4 is a subsequence if a1, a2, a3, a4.
A sequence is increasing if an+1 ≥ an ∀ n ∈ â„•. Correspondingly, a sequence is decreasing if
an+1 ≤ an ∀ n ∈ â„•. If the sequence is either increasing or decreasing it is called a monotone sequence.
There are several different types of sequences such as Cauchy sequence, convergent sequence, monotonic sequence, Fibonacci sequence, look and see sequence. I will be talking about only 2 of the sequences Cauchy and Convergent sequences.
Convergent sequences
A sequence (an) of real number is called a convergent sequences if an tends to a finite limit as n→∞. If we say that (an) has a limit a∈ F if given any ε > 0, ε ∈ F, k∈ â„• | an – a | If an has a limit a, then we can write it as liman = a or (an) → a.
Cauchy Sequence
A Cauchy sequence is a sequence in which numbers become closer to each other as the sequence progresses. If we say that (an) is a Cauchy sequence if given any ε > 0, ε ∈ F, k∈ â„• | an – am | Gary Sng Chee Hien, (2001).
Bounded sets, Upper Bounds, Least Upper Bounds
A set is called bounded if there is a certain sense of finite size. A set R of real numbers is called bounded of there is a real number Q such that Q ≥ r for all r in R. the number M is called the upper bound of R. A set is bounded if it has both upper and lower bounds. This is extendable to subsets of any partially ordered set. A subset Q of a partially ordered set R is called bounded above. If there is an element of Q ≥ r for all r in R, the element Q is called an upper bound of R
3 Real number system
Natural Numbers
Natural numbers (â„•) can be denoted by 1,2,3… we can define them by their properties in order of relation. So if we consider a set S, if the relation is less than or equal to on S
For every x, y ∈ S x ≤ y and/or y ≤ x
If x ≤ y and y ≤ x then x = y
If x ≤ y and y ≤ z then x ≤ z
If all 3 properties are met we can call S an ordered set.
(Giles, p.1, 1972)
Real numbers
Axioms for real numbers can be spilt in to 3 groups; algebraic, order and completeness.
Algebraic Axioms
For all x, y ∈ ℝ, x + y ∈ ℝ and xy ∈ ℝ.
For all x, y, z ∈ ℝ, (x + y) + z = x (y + z).
For all x, y ∈ ℝ, x + y = y + x.
There is a number 0 ∈ ℝ such that x + 0 = x = 0 + x for all x ∈ ℝ.
For each x ∈ ℝ, there exists a corresponding number (-x) ∈ ℝ such that x + (-x) = 0 = (-x) + x
For all x, y, z ∈ ℝ, (x y) z = x (y z).
For all x, y ∈ ℝ x y = y x.
There is number 1 ∈ ℝ such that x x 1 = x = 1 x x, for all x ∈ ℝ
For each x ∈ ℝ such that x ≠ 0, there is a corresponding number (x-1) ∈ ℝ such that x (x-1) = 1 = (x-1) x
A10. For all x, y, z ∈ ℝ, x (y + z) = x y + x z
(Hart, p.11, 2001)
Order Axioms
Any pair x, y of real numbers satisfies precisely one of the following relations: (a) x If x If x If x 0 then x z (Hart, p.12, 2001)
Completeness Axiom
If a non-empty set A has an upper bound, it has a least upper bound
The thing which distinguishes ℝ from is the Completeness Axiom.
An upper bound of a non-empty subset A of R is an element b ∈R with b a for all a ∈A.
An element M ∈ R is a least upper bound or supremum of A if
M is an upper bound of A and if b is an upper bound of A then b M.
That is, if M is a least upper bound of A then (b ∈ R)(x ∈ A)(b x) b M
A lower bound of a non-empty subset A of R is an element d ∈ R with d a for all a ∈A.
An element m ∈ R is a greatest lower bound or infimum of A if
m is a lower bound of A and if d is an upper bound of A then m d.
If all 3 axioms are satisfied it is called a complete ordered field.
John o’Connor (2002) axioms of real numbers
Rational numbers
Axioms for Rational numbers
The axiom of rational numbers operate with +, x and the relation ≤, they can be defined on corresponding to what we know on N.
For on +(add) has the following properties.
For every x,y ∈ , there is a unique element x + y ∈
For every x,y ∈ , x + y = y + x
For every x,y,z ∈ , (x + y) + z = x + (y + z)
There exists a unique element 0 ∈ such that x + 0 = x for all x ∈
To every x ∈ there exists a unique element (-x) ∈ such that x + (-x) = 0
For on x(multiplication) has the following properties.
To every x,y ∈ , there is a unique element x x y ∈
For every x,y ∈ , x x y = y x x
For every x,y,z ∈ , (x x y) x z = x x (y x z)
There exists a unique element 1 ∈ such that x x 1 = x for all x ∈
To every x ∈ , x ≠ 0 there exists a unique element ∈ such that x x = 1
For both add and multiplication properties there is a closer, commutative, associative, identity and inverse on + and x, both properties can be related by.
For every x,y,z ∈ , x x (y + z) = (x x y) + (x x z)
For with an order relation of ≤, the relation property is For every x ∈ , either x For every x,y ∈ , where 0 For every x,y ∈ , x (Giles, pp.3-4, 1972)
From both the axioms of rational numbers and real numbers, we can see that they are about the same apart from a few bits like rational numbers do not contain square root of 2 whilst real numbers do. Both rational and real numbers have the properties of add, multiplication and there exists a relationship of 0 and 1.
4 Proofs
In this section I will be solving some basic proofs, most of my proofs have been assumed in the construction process and have been reduced.
Theorem:
Between any two real numbers is an rational number.
Proof
Let a ≠ b be a real number with a a. we can claim that b we would have > b – a.
John O’Connor (2002) axioms of real numbers
Theorem:
The limit of a sequence, if it exists, is unique.
Proof
Let x and x′ be 2 different limits. We may assume without loss of generality, that
x 0.
Since xn→ x, k1 s.t
| xn – x | Since xn→ x k2 s.t
| xn – x′| Take k = max{k1, k2}. Then n ≥ k,
| xn – x | | x′ – x | = | x′ – xn + xn – x |
≤ | x′ – xn | + | xn – x |
= x′ – x, a contradiction!
Hence, the limit must be unique. Also all rational number sequences have a limit in real numbers.
Gary Sng Chee Hien, (2001).
Theorem:
Any convergent sequence is bounded.
Proof
Suppose the sequence (an)®a. take = 1. Then choose N so that whatever n > N we have an within 1 of a. apart from the finite set {a1, a2, a3…aN} all the terms of the sequence will be bounded by a + 1 and a – 1. Showing that an upper bound for the sequence is max{a1, a2, a3…aN, a +1}. Using the same method you could alternatively find the lower bound
Theorem:
Every Cauchy Sequence is bounded.
Proof
Let (xn) be a Cauchy sequence. Then for
| xn – xm | Hence, for n ≥ k, we have
| xn | = | xn – xk + xk |
≤ | xn – xk | + | xk |
Let M = max{ | x1 |, | x2 |, …, | xk-1|, 1 + | xk | } and it is clear that | xn | ≤ M n, i.e. (xn) is bounded.
Gary Sng Chee Hien, (2001).
Theorem:
If (xnx, then any subsequence of (xn) also converges to x.
Proof
Let (yn) be any subsequence of (xn). Given any > 0, s.t
| xn – x | But yn = xi for some so we may claim
| yn – x | Hence, (
Gary Sng Chee Hien, (2001).
Theorem:
If (xn) is Cauchy, then any subsequence of (xn) is also Cauchy.
Proof
Let (yn) be any subsequence of (xn). Given any s.t
| xn – xm | .
But yn = xi for so we may claim
| yn – ym |
Hence (yn) x
Gary Sng Chee Hien, (2001).
Theorem
Any convergent sequence is a Cauchy sequence.
Proof
If (an) a then given > 0 choose N so that if n > N we have |an- a| N we have |am- an| = |(am- a) – (am- a)| |am- a| + |am- a| We use completeness Axiom to prove
Suppose X ∈ ℝ, X2 = 2. Let (an) be a sequence of rational numbers converging to an irrational
12 = 1
1.52 = 2.25
1.42 = 1.96
1.412 = 1.9881
1.41421356237302 = 1.999999999999731161391129
Since (an) is a convergent sequence in ℝ it is a Cauchy sequence in ℝ and hence also a Cauchy sequence in . But it has no limit in.
An irrational number like 2 has a decimal expansion which does not repeat:
2 =1.4142135623730
John O’Connor (2002) Cauchy Sequences.
Theorem
Prove that is irrational, prove that ≤ ℝ
Proof
We will get 2 as the least upper bound of the set A = {q Q | q2 Suppose x ∈ , x2 (x +)2
So if we pick then (x +)2 So x is not an upper bound of A. This shows least upper bound x cannot satisfy x2 Solving using the Newton’s method
xn+1 = (xn+ 2/xn)/2 and x1 = 1.
This gives ( 1 , 3/2 , 17/12 , 577/408 , 665857/470832 , … ) which is approximately ( 1, 1.5, 1.41667, 1.414215, 1,414213562, … )
John O’Connor (2002) Convergence in the real’s.
Theorem:
Let (xn) and (yn) be two Cauchy sequences. Then the following holds:
(i) (xn + yn) is Cauchy.
(ii) (xn yn) is Cauchy.
Proof
(i) Let any ε > 0 be given. Then k1, k2 s.t
| xn – xm | | yn – ym | Take k = max(k1, k2). Then
| xn – xm | But
| (xn + yn) – (xm + ym) | = | (xn – xm) + (yn – ym) |
≤ | xn – xm | + | yn – ym |
= ε n ≥ k.
Hence, (xn + yn) is also Cauchy.
(ii) Now, since (xn), (yn) is Cauchy, they are bounded by some X, Y ≠ 0. Let any
ε > 0 be given. Then k1, k2 s.t
| xn – xm | | yn – ym | Take k = max(k1, k2). Then
| xn – xm | | yn – ym | Hence,
| xn yn – xm ym | = | (xn yn – xm yn) + (xm yn – xm ym) |
≤ | xn yn – xm yn | + | xm yn – xm ym |
= | yn | | xn – xm | + | xm | | yn – ym |
≤ Y | xn – xm | + X | yn – ym |
=
Hence, (xn yn) is also Cauchy.
5 Conclusion
Real numbers are infinite number of decimals used to measure continuous quantities. On the other hand, rational numbers are defined to be fractions formed from real numbers. Axioms of each number system are examined to determine the difference between real numbers and rational numbers. Conclusion of the analysis of axioms resulted to be both real numbers and rational numbers contain the same properties. The properties being addition, multiplication and there exist a relationship of zero and one.
The four fundamental results are obtained from this study. First concept is that the property of real number system being unique and following the complete ordered field. Second is that if any real number satisfies the axioms then it is upper bound, whilst rational numbers are not upper bound. The third being that all Cauchy sequences are converges towards the real numbers. Finally found out that all real numbers are equivalence classes of the Cauchy sequence.
Appendices
List of symbols
â„• = Natural number
ℝ = Real number
= Rational number
∈ = is an element of
= There exists
= For all
s.t. = Such that
 

Prime Numbers: History, Facts and Examples

Prime Numbers: An Introduction
Prime number is the number, which is greater than 1 and cannot be divided by any number excluding itself and one. A prime number is a positive integer that has just two positive integer factors, including 1 and itself. Such as, if the factors of 28 are listed, there are 6 factors that are 1, 2, 4, 7, 14, and 28. Similarly, if the factors of 29 are listed, there are only two factors that are 1 and 29. Therefore, it can be inferred that 29 is a prime number, but 28 is not.
Examples of prime numbers
The first few prime numbers are as follows:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.
Identifying the primes
The ancient Sieve of Eratosthenes is a simple way to work out all prime numbers up to a given limit by preparing a list of all integers and repetitively striking out multiples of already found primes. There is also a modern Sieve of Atkin, which is more complex when compared to that of Eratosthenes.
A method to determine whether a number is prime or not, is to divide it by all primes less than or equal to the square root of that number. If the results of any of the divisions are an integer, the original number is not a prime and if not, it is a prime. One need not actually calculate the square root; once one sees that the quotient is less than the divisor, one can stop. This is called as the trial division, which is the simplest primality test but it is impractical for testing large integers because the number of possible factors grows exponentially as the number of digits in the number to be tested increases.
Primality tests: A primality test algorithm is an algorithm that is used to test a number for primality, that is, whether the number is a prime number or not.

The AKS primality test is based upon the equivalence
(x – a)n = (xn – a) (mod n) for a coprime to n, which is true if and only if n is prime. This is a generalization of Fermat’s little theorem extended to polynomials and can easily be proven using the binomial theorem together with the fact that: for all 0 n = (xn – a) (mod n, x r – 1), which can be checked in polynomial time.

Fermat’s little theorem asserts that if p is prime and 1≤ a p -1≡ 1 (mod p)
In order to test whether p is a prime number or not, one can pick random a’s in the interval and check if there is an equality.

Solovay-Strassen primality test

For a prime number p and any integer a,
A (p -1)/2 ≡ (a/p) (mod p)
Where (a/p) is the Legendre symbol. The Jacobi symbol is a generalisation of the Legendre symbol to (a/n); where n can be any odd integer. The Jacobi symbol can be computed in time O((log n)²) using Jacobi’s generalization of law of quadratic reciprocity.
It can be observed whether or not the congruence
A (n -1)/2 ≡ (a/n) (mod n) holds for various values of a. This congruence is true for all a’s if n is a prime number. (Solovay, Robert M. and Volker Strassen, 1977)

This test is for a natural number n and in this test, it is also required that the prime factors of n − 1 should be already known.
If for every prime factor (q) of n − 1, there exists an integer a less than n and greater than 1 such as
a n -1 ≡1 (mod n)
and then
a n -1/q 1 (mod n)
then n is prime. If no such number can be found, n is composite number.

Miller-Rabin primality test

If we can find an a such that
ad ≡ 1 (mod n), and
a2nd -1 (mod n) for all 0 ≤ r ≤ s – 1
then ‘a’ proves the compositeness of n. If not, ‘a’ is called a strong liar, and n is a strong probable prime to the base a. “Strong liar” refers to the case where n is composite but yet the equations hold as they would for a prime number.
There are several witnesses ‘a’ for every odd composite n. But, a simple way to generate such an ‘a’ is known. Making the test probabilistic is the solution: we choose randomly, and check whether it is a witness for the composite nature of n. If n is composite, majority of the ‘a’s are witnesses, therefore the test will discover n as a composite number with high probability. (Rabin, 1980)
A probable prime is an integer, which is considered to be probably prime by passing a certain test. Probable primes, which are actually composite (such as Carmichael numbers) are known as pseudoprimes.
Besides these methods, there are other methods also. There is a set of Diophantine equations in 9 variables and one parameter in which the parameter is a prime number only if the resultant system of equations has a solution over the natural numbers. A single formula with the property of all the positive values being prime can be obtained with this method. There is another formula that is based on Wilson’s theorem. The number ‘two’ is generated several times and all other primes are generated exactly once. Also, there are other similar formulas that can generate primes. Some primes are categorized as per the properties of their digits in decimal or other bases. An example is that the numbers whose digits develop a palindromic sequence are palindromic primes, and if by consecutively removing the first digit at the left or the right generates only new prime numbers, a prime number is known as a truncatable prime.
The first 5,000 prime numbers can be known very quickly by just looking at odd numbers and checking each new number (say 5) against every number above it (3); so if 5Mod3 = 0 then it’s not a prime number.
History of prime numbers
The most ancient and acknowledged proof for the statement that “There are infinitely many prime numbers”, is given by Euclid in his Elements (Book IX, Proposition 20). The Sieve of Eratosthenes is a simple, ancient algorithm to identify all prime numbers up to a particular integer. After this, came the modern Sieve of Atkin, which is faster but more complex. The Sieve of Eratosthenes was created in the 3rd century BC by Eratosthenes. Some clues can be found in the surviving records of the ancient Egyptians regarding their knowledge of prime numbers: for example, the Egyptian fraction expansions in the Rhind papyrus have fairly different forms for primes and for composites. But, the first surviving records of the clear study of prime numbers come from the Ancient Greeks. Euclid’s Elements (circa 300 BC) include key theorems about primes, counting the fundamental theorem of arithmetic and the infinitude of primes. Euclid also explained how a perfect number is constructed from a Mersenne prime.
After the Greeks, nothing special happened with the study of prime numbers till the 17th century. In 1640, Pierre de Fermat affirmed Fermat’s little theorem, which was later on proved by Leibniz and Euler. Chinese may have identified a special case of Fermat’s theorem much earlier. Fermat assumed that all numbers of the form 22n + 1 are prime and he proved this up to n = 4. But, the subsequent Fermat number 232+1 is composite; whose one prime factor is 641). This was later on discovered by Euler and now no further Fermat numbers are recognized as prime numbers. A French monk, Marin Mersenne looked at primes of the form 2p – 1, with p as a prime number. They are known as Mersenne primes after his name.
Euler showed that the infinite series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … is divergent. In 1747, Euler demonstrated that even the perfect numbers are in particular the integers of the form 2p-1(2p-1), where the second factor is a Mersenne prime. It is supposed that there are no odd perfect numbers, but it is not proved yet. In the beginning of the 19th century, Legendre and Gauss independently assumed that because x tends to infinity, the number of primes up to x is asymptotic to x/log(x), where log(x) is the natural logarithm of x.
Awards for finding primes
A prize of US$100,000 has been offered by the Electronic Frontier Foundation (EFF) to the first discoverers of a prime with a minimum 10 million digits. Also, $150,000 for 100 million digits, and $250,000 for 1 billion digits has been offered. In 2000, $50,000 for 1 million digits were paid. Apart from this, prizes up to US$200,000 for finding the prime factors of particular semi-primes of up to 2048 bits were offered by the RSA Factoring Challenge.
Facts about prime numbers

73939133 is an amazing prime number. If the last or the digit at the units place is removed, every time you will get a prime number. It is the largest known prime with this property. Because, all the numbers which we get after removing the end digit of the number are also prime numbers. They are as follows: 7393913, 739391, 73939, 7393, 739, 73 and 7. All these numbers are prime numbers. This is a distinct quality of the number 73939133, which any other number does not have. (Amazing number facts, 2008)

The only even prime number is 2. All other even numbers can be divided by 2. So, they are not prime numbers.

Zero and 1 are not considered to be prime numbers.

If the sum of the digits of a number is a multiple of 3, that number can be divided by 3.

With the exception of 0 and 1, a number is either a prime number or a composite number. A composite number is identified as any number that is greater than 1 and that is not prime.

The last digit of a prime number greater than 5 can never be 5. Any number greater than 5 whose last digit is 5 can be divided by 5. (Prime Numbers, 2008)

1/2
0.5
Terminates

1/3
0.33333…
Repeating block: 1 digit

1/5
0.2
Terminates

1/7
0.1428571428…
Repeating block: 6 digits

1/11
0.090909…
Repeating block: 2 digits

1/13
0.0769230769…
Repeating block: 6 digits

1/17
0.05882352941176470588…
Repeating block: 16 digits

1/19
0.0526315789473684210526…
Repeating block: 18 digits

1/23
0.04347826086956521739130434…
Repeating block: 22 digits

For some of the prime numbers, the size of the repeating block is 1 less than the prime.
These are known as Golden Primes.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
9 primes out of the 25 (less than 100) are golden primes; this forms 36% (9/25). (Amazing number facts, 2008)
Examples of mathematicians specialized in prime numbers
Arthur Wieferich, D. D. Wall, Zhi Hong Sun and Zhi Wei Sun, Joseph Wolstenholme, Joseph Wolstenholme, Euclid, Eratosthenes.
Applications of prime numbers
For a long time, the number theory and the study of prime numbers as well was seen as the canonical example of pure mathematics with no applications beyond the self-interest of studying the topic. But, in the 1970s, it was publicly announced that prime numbers could be used as a basis for creating the public key cryptography algorithms. They were also used for hash tables and pseudorandom number generators.
A number of rotor machines were designed with a different number of pins on each rotor. The number of pins on any one rotor was either prime, or co-prime to the number of pins on any other rotor. With this, a full cycle of possible rotor positions (before repeating any position) was generated.
Prime numbers in the arts and literature
Also, prime numbers have had a significant influence on several artists and writers. The French composer Olivier Messiaen created ametrical music through “natural phenomena” with the use of prime numbers. In his works, La Nativité du Seigneur (1935) and Quatre études de rythme (1949-50), he has used motifs with lengths given by different prime numbers to create unpredictable rhythms: 41, 43, 47 and 53 are the primes that appear in one of the études. A scientist of NASA, Carl Sagan recommended (in his science fiction ‘Contact’) that prime numbers could be used for communication with the aliens. The award-winning play ‘Arcadia’ by Tom Stoppard was a willful attempt made to discuss mathematical ideas on the stage. In the very first scene, the 13 year old heroine baffles over the Fermat’s last theorem (theorem that involves prime numbers). A popular fascination with the mysteries of prime numbers and cryptography has been seen in various films.
References
Amazing number facts, 2008. Retrieved April 28, 2008 from http://www.madras.fife.sch.uk/maths/amazingnofacts/fact018.html
Prime Numbers, 2008. Retrieved April 28, 2008 from http://www.factmonster.com/ipka/A0876084.html
Solovay, Robert M. & Strassen, V. (1977). “A fast Monte-Carlo test for primality”. SIAM Journal on Computing 6 (1): 84-85.
Rabin, M.O. (1980). Probabilistic algorithm for testing primality, Journal of Number Theory 12, no. 1, pp. 128-138.
 

Complex Numbers and their Applications

INTRODUCTION
A complex number is a number comprising area land imaginary part. It can be written in the form a+ib, where a and b are real numbers, and i is the standard imaginary unit with the property i2=-1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.

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HISTORY OF COMPLEX NUMBERS:
Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them “fictitious”, during his attempts to find solutions to cubic equations. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.
The rules for addition, subtraction and multiplication of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton.
COMPLEX NUMBER INTERPRETATION:
A number in the form of x+iy where x and y are real numbers and i = -1 is called a complex number.
Let z = x+iy
X is called real part of z and is denoted by R (z)
Y is called imaginary part of z and is denoted by I (z)
CONJUGATE OF A COMPLEX NUMBER:
A pair of complex numbers x+iy and x-iy are said to be conjugate of each other.
PROPERTIES OF COMPLEX NUMBERS ARE:

If x1+ iy1 = x2 + iy2 then x1- iy1 = x2 – iy2
Two complex numbers x1+ iy1 and x2 + iy2 are said to be equal
               If R (x1 + iy1) = R (x2 + iy2)
               I (x1 + iy1) = I (x2 + iy2)

Sum of the two complex numbers is
               (x1 + iy1) + (x2 + iy2) = (x1+ x2) + i(y1+ y2)

Difference of two complex numbers is
               (x1 + iy1) – (x2 + iy2) = (x1-x2) + i(y1 – y2)

Product of two complex numbers is
               (x1+ iy1) ( x2 + iy2) = x1x2 – y1y2 + i(y1x2 + y2 x1)

Division of two complex numbers is
               (x1 + iy1) (x2 + iy2) = x1x2 + y1 y2)x22+y22 + iy1x2 ­ y2 x1x22+y22

Every complex number can be expressed in terms of r (cosθ + i sinθ)
               R (x+ iy) = r cosθ
               I (x+ iy) = r sinθ
               r = x2+y2 and θ = tan-1yx

REPRESENTATION OF COMPLEX NUMBERS IN A PLANE
The set of complex numbers is two-dimensional, and a coordinate plane is required to illustrate them graphically. This is in contrast to the real numbers, which are one-dimensional, and can be illustrated by a simple number line. The rectangular complex number plane is constructed by arranging the real numbers along the horizontal axis, and the imaginary numbers along the vertical axis. Each point in this plane can be assigned to a unique complex number, and each complex number can be assigned to a unique point in the plane.
Modulus and Argument of a complex number:
The number r = x2+y2 is called modulus of x+ iy and is written by mod (x+ iy) or x+iy
θ = tan-1yx is called amplitude or argument of x + iy and is written by amp (x + iy) or arg (x + iy)
Application of imaginary numbers:
For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as 2/3 and 1/8 are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Negative numbers such as -3 and -5 are meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits. Similarly, imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, cartography, vibration analysis, and many others.
APPLICATION OF COMPLEX NUMBER IN ENGINEERING:
Control Theory
Incontrol theory, systems are often transformed from thetime domainto thefrequency domainusing theLaplace transform. The system’spolesandzerosare then analyzed in the complex plane. Theroot locus,Nyquist plot, andNichols plottechniques all make use of the complex plane.
In the root locus method, it is especially important whether thepolesandzerosare in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are

in the right half plane, it will beunstable,
all in the left half plane, it will bestable,
on the imaginary axis, it will havemarginal stability.

If a system has zeros in the right half plane, it is anonminimum phasesystem.
Signal analysis
Complex numbers are used insignal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg (z) the phase.
If Fourier analysisis employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form
ω f (t) = z
where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.
Improper integrals
In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.
Residue theorem
The residue theorem in complex analysisis a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy and Cauchy’s integral formula.
The statement is as follows. Suppose U is a simply connected open subset of the complex plane C, a1,…, an are finitely many points of U and f is a function which is defined and holomorphic on U{a1,…,an}. If γ is a rectifiable curve in which doesn’t meet any of the points ak and whose start point equals its endpoint, then
Here, Res(f,ak) denotes the residue off at ak, and n(γ,ak) is the winding number of the curve γ about the point ak. This winding number is an integer which intuitively measures how often the curve γ winds around the point ak; it is positive if γ moves in a counter clockwise (“mathematically positive”) manner around ak and 0 if γ doesn’t move around ak at all.
In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero if it is large enough, leaving only the real-axis part of the integral, the one we were originally interested
Applications of Complex Numbers in Quantum mechanics
The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg’s matrix mechanics – make use of complex numbers.
The quantum theory provides a quantitative explanation for two types of phenomena that classical mechanics and classical electrodynamics cannot account for:

Some observable physical quantities, such as the total energy of a black body, take on discrete rather than continuous values. This phenomenon is called quantization, and the smallest possible intervals between the discrete values are called quanta (singular:quantum, from the Latin word for “quantity”, hence the name “quantum mechanics.”) The size of the quanta typically varies from system to system.
Under certain experimental conditions, microscopic objects like atoms or electrons exhibit wave-like behavior, such as interference. Under other conditions, the same species of objects exhibit particle-like behavior (“particle” meaning an object that can be localized to a particular region ofspace), such as scattering. This phenomenon is known as wave-particle duality.

Application of complex numbers in Computer Science.
Arithmetic and Logic in Computer Systems provides a useful guide to a fundamental subject of computer science and engineering. Algorithms for performing operations like addition, subtraction, multiplication, and division in digital computer systems are presented, with the goal of explaining the concepts behind the algorithms, rather than addressing any direct applications. Alternative methods are examined, and explanations are supplied of the fundamental materials and reasoning behind theories and examples.
This technological manual explores how software engineering principles can be used in tandem with software development tools to produce economical and reliable software that is faster and more accurate. Tools and techniques provided include the Unified Process for GIS application development, service-based approaches to business and information technology alignment, and an integrated model of application and software security. Current methods and future possibilities for software design are covered.
Application of complex numbers in Electrical Engineering:
The voltage produced by a battery is characterized by one real number (called potential), such as +12 volts or -12 volts. But the “AC” voltage in a home requires two parameters. One is a potential, such as 120 volts, and the other is an angle (called phase). The voltage is said to have two dimensions. A 2-dimensional quantity can be represented mathematically as either a vector or as a complex number (known in the engineering context as phasor). In the vector representation, the rectangular coordinates are typically referred to simply as X and Y. But in the complex number representation, the same components are referred to as real and imaginary. When the complex number is purely imaginary, such as a real part of 0 and an imaginary part of 120, it means the voltage has a potential of 120 volts and a phase of 90°, which is physically very real.
Application of complex numbers in electronic engineering
Information that expresses a single dimension, such as linear distance, is called a scalar quantity in mathematics. Scalar numbers are the kind of numbers students use most often. In relation to science, the voltage produced by a battery, the resistance of a piece of wire (ohms), and current through a wire (amps) are scalar quantities.
When electrical engineers analyzed alternating current circuits, they found that quantities of voltage, current and resistance (called impedance in AC) were not the familiar one-dimensional scalar quantities that are used when measuring DC circuits. These quantities which now alternate in direction and amplitude possess other dimensions (frequency and phase shift) that must be taken into account.
In order to analyze AC circuits, it became necessary to represent multi-dimensional quantities. In order to accomplish this task, scalar numbers were abandoned andcomplex numberswere used to express the two dimensions of frequency and phase shift at one time.
In mathematics, i is used to represent imaginary numbers. In the study of electricity and electronics, j is used to represent imaginary numbers so that there is no confusion with i, which in electronics represents current. It is also customary for scientists to write the complex number in the form a+jb.
In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.) This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, andvideosignals.
Introduce the formula E = I • Z where E is voltage, I is current, and Z is impedance.
Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. This seems odd at first, as the concept of using a mix of real and ‘imaginary’ numbers to explain things in the real world seem crazy!. To help you get a clear picture of how they’re used and what they mean we can look at a mechanical example…
We can now reverse the above argument when considering a.c. (sine wave) oscillations in electronic circuits. Here we can regard the oscillating voltages and currents as ‘side views’ of something which is actually ‘rotating’ at a steady rate. We can only see the ‘real’ part of this, of course, so we have to ‘imagine’ the changes in the other direction. This leads us to the idea that what the oscillation voltage or current that we see is just the ‘real’ portion’ of a ‘complex’ quantity that also has an ‘imaginary’ part. At any instant what we see is determined by aphase anglewhich varies smoothly with time.
We can now consider oscillating currents and voltages as being complex values that have a real part we can measure and an imaginary part which we can’t. At first it seems pointless to create something we can’t see or measure, but it turns out to be useful in a number of ways.
Applications in Fluid Dynamics
Influid dynamics, complex functions are used to describe potential flow in two dimensions. Fractals.
Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set
Fluid Dynamics and its sub disciplines aerodynamics, hydrodynamics, and hydraulics have a wide range of applications. For example, they are used in calculating forces and moments onaircraft, the mass flow of petroleum through pipelines, and prediction of weather patterns.
The concept of a fluid is surprisingly general. For example, some of the basic mathematical concepts in traffic engineering are derived from considering traffic as a continuous fluids.
Application of complex numbers in Relativity
In special and general relativity, some formulas for the metric on space time become simpler if one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but isused in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.
Application of complex numbers in​​​​​​​ Applied mathematics
In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = ert.
Application of complex numbers in​​​​​​​ Electromagnetism:
Instead of taking electrical and magnetic part as a two different real numbers, we can represent it as in one complex number
Application of complex numbers in​​​​​​​ Civil and Mechanical Engineering:
The concept of complex geometry and Argand plane is very much useful in constructing buildings and cars. This concept is used in 2-D designing of buildings and cars. It is also very useful in cutting of tools. Another possibility to use complex numbers in simple mechanics might be to use them to represent rotations.
BIBLIOGRAPHY
Websites:
​​​​​​​http://www.math.toronto.edu/mathnet/questionCorner/complexinlife.html
http://www.physicsforums.com/showthread.php?t=159099
http://www.ebookpdf.net/_engineering-application-of-complex-number-(pdf)_ebook_.html.
http:www.wikipedia.org.
http://mathworld.wolfram.com
http://euclideanspace.com
Books
Engineering Mathematics, 40th edition-B S Grewal.
Engineering Mathematics-Jain Iyenger.
Engineering Matematics-NP Bali
 

Finding Difference Between the Squares of any Two Natural Numbers

One of the basic arithmetic operations is finding squares and difference between squares of two natural numbers. Though there are various methods to find the difference between squares of two natural numbers, still there are scopes to find simplified and easy approaches. As the sequence formed using the difference between squares of two natural numbers follow a number patterns, using number patterns may facilitate more easy approach. Also, this sequence has some general properties which are already discussed by many mathematicians in different notations. Apart from these, the sequence has some special properties like sequence – difference property, difference – sum property, which helps to find the value easily. The sequence also has some relations that assist to form a number pattern.

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This paper tries to identify the general properties, special properties of finding difference between the squares of any two natural numbers using number patterns. A rhombus rule relationship between the sequences of numbers formed by considering the difference between squares of the two natural numbers has been defined. A new method to find a2 – b2 also has been introduced in some simple cases. This approach will help the secondary education lower grade students in identifying and recognizing number patterns and squares of natural numbers.
Mathematical Subject Classifications: (2010) 11A25, 11A51, 40C99, 03F50
DIFFERENCE BETWEEN SQUARES OF TWO NATURAL NUMBERS RELATIONS, PROPERTIES AND NEW APPROACH
Introduction
Mathematics, a subject of problem solving skills and applications, has wide usage in all the fields. Basic skills of mathematical applications in number systems used even in day – to – day life. Though calculators and computers have greater influences in calculations, still there is a need to find new easy methods of calculations to improve personal intellectual skills.
As there has been growing interest, in mathematics education, in teaching and learning, many mathematicians build simple and different methods, rules and relationships in various mathematical field. Though various investigations have made important contributions to mathematics development and education (2), there still room for new research to clarify the mutual relationship between the numbers and number patterns.
In natural numbers, various subsets have been recognized by ancient mathematicians. Some are odd numbers, prime numbers, oblong numbers, triangular numbers and squares. These numbers shall be identified by number patterns. Recognizing number patterns is also an important problem-solving skill. Working with number patterns leads directly to the concept of functions in mathematics: a formal description of the relationships among different quantities.
One of the basic arithmetic operations is finding squares and difference between squares of two natural numbers. Already many proofs and relationships were identified and proved in finding difference between squares of two natural numbers. We use different methods to find the difference between squares of two natural numbers. That is, to find a2 – b2. Though, this area of research may be discussed by early mathematicians and researchers in various aspects, still there are many interesting ways to discuss the same in teaching.
Teaching number patterns in secondary level education is most important issue as the students develop their analytical and cognitive skills in this stage. Different arithmetic operations and calculations need to be introduced in such way that they help the students in lifelong learning. Easy and simplified approaches will support the students in logical reasoning.
This paper tries to identify the general properties, special properties of finding difference between the squares of any two natural numbers using number patterns. Also, this paper tries to define the rhombus rule relationship between the sequences of numbers formed by the differences of squares of two natural numbers. A new method to find a2 – b2 also has been introduced in some simple cases.
These may be introduced in secondary school early grades, before introducing algebraic techniques of finding a2 – b2 to develop the knowledge and understanding of number patterns. This will help to recognize and apply number patterns in further level.
Literature Review
To find the difference between the squares of any two natural numbers, we use different methods. Also, we use various rules to find the square of a natural number. Some properties were also been identified by the researchers and mathematicians.
Methods used to find the difference between squares of two natural numbers
Direct Method
The difference between the squares of two natural numbers shall be found out by finding the squares of the numbers directly.
Example: 252 – 52 = 625 – 25 = 600
Using algebraic rule
The algebraic rule a2 – b2 = (a – b)(a + b) shall be applied to find the difference between the squares of two natural numbers.
Example: 252 – 52 = (25 – 5)(25 + 5) = 20 x 30 = 600
Method when a – b = 1(2)
“The difference between the squares of every two consecutive natural numbers is always an odd number, and that it is equal to the sum of these numbers.”
Example: 252 – 242 = 25 + 24 = 49
Methods used to find the square of a natural number
Using Algebraic Method
The algebraic rules shall be used to find the square of natural number other than the direct multiplication. In general, (a + b)2, (a – b)2 are used to find the squares of a natural number from nearest whole number.
Example: 992 = (100 – 1)2
= 1002 – 2(100)(1) + 12 = 10000 – 200 + 1
= 9801
Square of a number using previous number(8)
The following rule may be applied to find the square of a number using previous number.
(n + 1)2 = n2 + n + (n+1)
Example: 312 = 302 + 30 + 31 = 900 + 30 + 31 = 961
The Gilbreth Method of finding square(9)
The Gilbreth method uses binomial theorem to find the square of a natural number. The rule is
n2 = 100(n – 25) + (50 – n)2
Example: 992 = 100(99 – 25) + (50 – 99)2
= 7400 + 2401 = 9801
Other than the above mentioned methods various methods are used based on the knowledge and requirements.
Properties of differences between squares of the natural numbers
2.3.1. The difference between squares of any two consecutive natural numbers is always odd.
To prove this property, let us consider two consecutive natural numbers, say 25 and 26
Now let us find 262 – 252
262 – 252 = (26 + 25)(26 – 25) [Using algebraic rule]
= 51 x 1 = 51, an odd number
2.3.2. The difference between squares of any two alternative natural numbers is always even.
To prove this property, let us consider two alternative natural numbers, say 125 and 127
Now let us find 1272 – 1252
1272 – 1252 = (127 + 125)(127 – 125) [Using algebraic rule]
= 252 x 2 = 504, an even number
Some other properties were also identified and discussed by various mathematicians and researchers.
Number Patterns and Difference Between the Squares of Two Natural Numbers – Discussions and Findings
Some of the properties stated above shall be proved by using number pattern. Number patterns are interesting area of arithmetic that stimulates the logical reasoning. They shall be applied in various notations to identify the sequences and relations between the numbers.
3.1. Sample Table for the difference between squares of two natural numbers
To find the properties and relations that are satisfied by the sequences formed by the differences between the squares of two natural numbers, let us form a number pattern. For discussion purposes, let us consider first 10 natural numbers 1, 2, 3 … 10.
Now, let us find the difference between two consecutive natural numbers.
That is, 22 – 12 = 3; 32 – 22 = 5; and so on.
Then the sequence will be as follows: 3, 5, 7, 9, 11, 13, 15, 17 and 19.
The sequence is a set of odd numbers starting from 3.
i.e., Difference 1: {x| x is an odd number greater than or equal to 3, x Î N}
In the same way, let us form the sequence for the difference between squares of two alternative natural numbers.
That is, 32 – 12 = 8, 42 – 22 = 12, and so on.
Then the sequence will be: 8, 12, 16, 20, 24, 28, 32 and 36
Thus the sequence is a set of even numbers and multiples of 4 starting from 8.
i.e., Difference 2: {x| x is an multiple of 4 greater than or equal to 8, x Î N}
By proceeding this way, the sequences for other differences shall be formed.
Let us represent the sequences in a table for discussion purposes.
In Table 1, N is the natural number.
S is the square of the corresponding natural number.
D1 represents the difference between the squares of two consecutive natural numbers. That is, the difference between the numbers is 1.
D2 represents the difference between the squares of two alternate natural numbers. That is, the difference between the numbers is 2.
D3 represents the difference between the squares of 4th and 1st number. That is, the difference between the numbers is 3, and so on.
3.2. Relationship between the row elements of each column
Now, let us discuss the relationship between the elements of rows and columns of the table.
From the above table,
Column D1 shows that the difference between squares of two consecutive numbers is odd.
Column D2 shows that the difference between squares of two alternate numbers is even.
The other columns show that the difference between the squares of two numbers is either odd or even.
From the above findings, the following properties shall be defined for the difference between squares of any two natural numbers.
3.3. General Properties of the difference between squares of two natural numbers:
The difference between squares of any two consecutive natural numbers is always odd.
Proof: Column D1 proves this property.
This may also be tested randomly for big numbers.
Let us consider two digit consecutive natural numbers, say 96 and 97.
Now, 972 – 962 = 9409 – 9216
= 493, an odd number
Let us consider three digit consecutive natural numbers, say 757 and 758.
Thus, 7582 – 7572 = 574564 – 573049
= 1515, an odd number
This property may also be further tested for big numbers and proved. For example, let us consider five digit two consecutive natural numbers, say 15887 and 15888.
Then, 158882 – 158872 = 252428544 – 252396769
= 31775, an odd number
Apart from these, the property shall also be easily derived by the natural numbers properties. As the difference between two consecutive numbers is 1, the natural number property “The sum of odd and even natural numbers is always odd”, shall be applied to prove this property.
The difference between squares of any two alternative natural numbers is always even.
Proof: Column D2 proves this property.
This may also be verified for big numbers by considering different digit natural numbers as discussed above.
Apart from this, as the difference between two alternate natural numbers is 2, the natural numbers property “A natural number said to be even if it is a multiple of two” shall also be used for proving the stated property.
The difference between squares of any two natural numbers is either odd or even, depending upon the difference between the numbers.
Proof: The other columns of Table 1 prove this property.
In Table 1, as D3 represents the sequence formed by the difference between two natural numbers whose difference is 3, an odd number, the sequence is also odd. Thus, the property may be proved by testing the other Columns D4, D5, …
Also, the addition, subtraction and multiplication properties of natural numbers prove this property.
Example:
112 – 62
Here the difference (11 – 6 = 5) is odd.
So, the result will be odd.
i.e. 112 – 62 = 121 – 36 = 85, an odd number
122 – 82
Here the difference (12 – 8 = 4) is even.
So, the result will be even.
i.e. 122 – 82 = 144 – 64 = 80, an even number
3.4. Special Properties of the difference between squares of the two natural numbers
Table 1 also facilitates to find some special properties stated below.
Sequence Difference Property
Table 1 shows that the sequences formed are following a number pattern with a common property between them. Let us consider the number sequences of each column.
Let us consider the first column D1 elements. D1: 3, 5, 7, 9, 11 … …
As D1 represents the difference between the squares of two consecutive natural numbers, let us say, a and b with a > b, the difference between them will be 1.
That is a – b = 1
Let us consider the difference between the elements in the sequence.
The difference between the numbers in the sequence is 2.
Thus the difference between the elements of the sequence shall be expressed as, 2 x 1. Thus, Difference = 2(a – b)
Now, let us consider the second column D2 elements. D2: 8, 12, 16, 26, … … …
As D2 represents the difference between the squares of two alternative natural numbers, the difference between the natural numbers, say a and b is always 2. That is a – b = 2
If we consider the difference between the elements in the sequence, the difference is 4.
Thus, the difference between the elements in the sequence shall be expressed as 2 x 2.
That is, difference = 2 (a – b)
In the same way, D3: 15, 21, 27, 33, … … …
D3 represents the difference between squares of the 4th and 1st numbers, difference is 3. That is a – b = 3
The difference between the numbers in the sequence is 6.
Thus, difference = 2 x 3 = 2(a – b)
All other columns also show that the difference between the numbers in the corresponding sequence is 2 (a – b)
Thus, this may be generalized as following property:
“The difference between elements of the number sequence, formed by the difference between any two natural numbers, is equal to two times of the difference between those corresponding natural numbers.”
Difference – Sum Property:
From Table 1, we shall also identify another relationship between the elements of the sequence formed.
Let us consider the columns from table 1 other than D1.
Consider D2: 8, 12, 16, 20
This sequence shall be formed by adding two numbers of Column D1.
i.e. 8 = 3 + 5
12 = 5 + 7
16 = 7 + 9
20 = 9 + 11
And so on.
Thus, if the difference between the natural numbers taken is 2, then the number sequence of the difference between the two natural numbers shall be formed by adding 2 natural numbers.
Consider D3: 15, 21, 27
This sequence shall be formed by adding three numbers from Column D1.
i.e. 15 = 3 + 5 + 7
21 = 5 + 7 + 9
27 = 7 + 9 + 11
And so on.
Thus, if the difference between the natural numbers taken is 3, then the number sequence of the difference between the two natural numbers shall be formed by adding 3 natural numbers.
This may also be verified with respect to the other columns.
Table 2 shows the above relationship between the differences of the squares of the natural numbers.
Now the above relation shall be generalized as
“If a – b = k > 1, then a2 – b2 shall be written as the sum of ‘k’ natural numbers”
As Column D1 elements are odd natural numbers, this property may be defined as
“If a – b = k > 1, then a2 – b2 shall be written as the sum of ‘k’ odd natural numbers”
As these odd numbers are consecutive, the property may be further precisely defined as:
“If a – b = k > 1, then a2 – b2 shall be written as the sum of ‘k’ consecutive odd natural numbers”
3.5. New Method to find the difference between squares of two natural numbers
Using the above difference – sum property, the difference between squares of two natural numbers shall be found as follows.
The property shows that, a2 – b2 is equal to sum of ‘k’ consecutive odd numbers. Now, the principal idea is to find those ‘k’ consecutive odd numbers.
Let us consider two natural numbers, say 7 and 10.
The difference between them 10 – 7 = 3
Thus, 102 – 72 = sum of three consecutive odd numbers.
102 – 72 = 100 – 49 = 51
Now, 51 = Sum of 3 consecutive odd numbers
i.e., 51 = 15 + 17 + 19
Let we try to find these 3 numbers with respect to either the first number, let us say, ‘a’ or the second number, say, ‘b’.
Assume, for ‘b’
As general form for odd numbers is either (2n + 1) or (2n – 1), as b 15 = 2(7) + 1 = 2b + 1
17 = 2(7) + 3 = 2b + 3
19 = 2(7) + 5 = 2b + 5
Thus, 102 – 72 shall be written as the sum of 3 consecutive odd numbers starting from 15.
i.e. starting from 2b + 1
This idea may also be applied for higher digit numbers. Let us consider two 3 digit numbers, 101 and 105. Let us find 1052 – 1012
Here the difference is 4. Thus 1052 – 1012 shall be written as the sum of 4 consecutive odd numbers.
The numbers shall be found as follows:
Here b = 101
The first odd number = 2b + 1 = 2(101) + 1 = 203
Thus, the 4 consecutive odd numbers are: 203, 205, 207, 209
So,
1052 – 1012 = 203 + 205 + 207 + 209 = 824
This shall be verified for any number of digits. Let us consider two 6 digit numbers 100519, 100521. Let us find 1005212 – 1005192
Here the difference is 2. Thus 1005212 – 1005192 shall be written as the sum of two odd numbers.
Applying the same idea,
The first odd number = 2(100519) + 1 = 201039
Thus the 2 consecutive odd numbers are: 201039, 201041
1005212 – 1005192 = 201039 + 201041 = 402080
The above result shall be verified by using other methods.
For example: 1052 – 1012
1052 – 1012 = 11025 – 10201 = 824 (Using Direct Method)
1052 – 1012 = (105 + 101) (105 – 101) = 206 x 4 = 824 (Using Algebraic Rule)
Thus, this idea shall be generalized as follows:
“a2 – b2shall be found by adding the (a – b) consecutive odd numbers starting from 2b + 1”
This shall also be found using the first term ‘a’. As a > b, let us consider (2n – 1) form of odd numbers.
From Table 1, 102 – 62 = 13 + 15 + 17 + 19 = 64
Here, 2a – 1 = 2(10) – 1 = 19
2a – 3 = 2(10) – 3 = 17
2a – 5 = 2(10) – 5 = 15
2a – 7 = 2(10) – 7 = 13
Thus, as the difference between the numbers is 4, 102 – 62 shall be written as the sum of four consecutive odd numbers in reverse order starting from 2a – 1.
Thus proceeding, this may be generalized as,
“a2 – b2shall be found by adding the (a – b) consecutive odd numbers starting from 2a – 1 in reverse order”
Finding the first number of each column
Let us check the number pattern followed by the first numbers of each column. From Table 1, the first numbers of each column are: 3, 8, 15, 24 …
Let us find the difference between elements of this sequence.
The difference between two consecutive terms of this sequence is 5, 7, 9 …
i.e. D2 – D1 = 8 – 3 = 5; D3 – D2 = 7; D4 – D3 = 9 and so on.
As D2 represents the difference between two alternate natural numbers, (say a and b) which implies that the difference between a and b is 2.
Now, 5 = 2 (2) +1
i.e. 2 times of the difference between the numbers + 1
In the same idea, D3 – D2 = 15 – 8 = 7
As D3 represents the difference between squares of the 4th and 1st natural numbers, (say a and b) which implies that the difference between a and b is 3.
Thus, 7 = 2(3) + 1
This also shows that the difference shall be found by
= 2 times of the difference between the numbers + 1
Thus,
“The first term of the each column shall be found by adding the previous column first term with 2 times of the difference between the numbers + 1”
Finding the elements row – wise
The elements of the table shall also be formed in row wise.
If we check the elements of each row, we can find that they follow a number pattern sequence with some property.
Let us consider the elements of row when N = 5: 20, 40, 60, 80
20 = 2 x 5 x 2
Here, 5 represent the row natural number.
2 represent the difference between the elements using which the column is formed.
Thus Row element = 2 x N x difference
In the same way, 40 = 2 x 5 x 4
= 2 x N x difference
Thus, the elements shall be formed by the rule:
“Row Element = 2 x N x difference”
This shall be applied for middle rows also.
For example, let us consider the row between 5 & 6:
The elements in this intermediate row are: 11, 33, 55, 77, 99
Here N is the mid value of 5 & 6. i.e. N = 5.5
Let us consider the elements and apply the above stated rule.
11 = 2 x N x difference
= 2 x 5.5 x 1
In the same way other elements shall also be formed.
Thus the elements of the table shall be formed in row wise using the stated rule.
Rhombus Rule Relation
Let us consider the elements in D2, D3 and D4.
Consider the elements in the rhombus drawn, 24, 33, 39 and 48
24 + 48 = 72
33 + 39 = 72
Thus the sums of the elements in the opposite corners are equal.
The other column elements also prove the same.
Thus, Rhombus Rule Relation:
“Sum of the elements the same row of the sequence of alternative columns is equal to the sum of the two elements in the intermediate column”
Application of the Properties in Finding the Square of a number
The square of a natural number shall be found by various methods. Here is one of the suggested methods.
This method uses nearest 10’s and 100’s to find the square of a number.
This method is also based on the algebraic formula a2 – b2 = (a – b)(a + b)
If a > b, b2 = a2 – (a2 – b2)
If b > a, b2 = a2 + (b2 – a2)
Example: Square of 32
As we need to find 322, let us assume b = 32.
The nearest multiple of 10 is 30. Let a = 30
Here b > a. b2 = a2 + (b2 – a2)
322 = 302 + (322 – 302)
Using the Difference – Sum Property,
322 = 900 + 61 + 63 = 1024
Example 2: Square of 9972
Let b = 997
Nearest multiple 10 is 1000. Let a = 1000
Here a > b, so b2 = a2 – (a2 – b2)
9972 = 10002 – (10002 – 9972)
Using Difference – Sum Property,
9972 = 1000000 – (1995 + 1997 + 1999)
= 994009
Conclusion
Though this method shall be applied to find the difference between squares of any two natural numbers, if the difference is big, it will be cumbersome. Thus, this method shall be used for finding the difference between squares of any two natural numbers where the difference is manageable. The properties shall be used for easy calculation.
This properties and approach shall be introduced in secondary school lower grade levels, to make the students to identify the number patterns. This approach will surely help the students to understand the properties of squares, difference and natural numbers. The new approach will surely help the students in developing their reasoning skills.
Limitations
As number systems, number patterns and arithmetic operations have wide applications in various fields, the above properties, rules and relations shall be further studied intensively based on the requirements. Thus, new properties and relations shall be identified and discussed with respect to other nations.