Number Systems, Sequences and Algorithms

1. Natural numbers (N) – number system that begins from 1 are natural numbers. Eg. 1, 2, 3, …,100 and so on. The first and the smallest natural number is 1 and the largest natural number cannot be obtained or defined. Consecutive natural numbers differ by 1.



2. Whole numbers (W) – number system that begins from a zero like 0, 1, 2, 3, 4, .…,100, …, 10,000 and so on. The first and the smallest whole number is zero and the largest whole number remains undefined or cannot be obtained. Other than zero (0) each of remaining whole numbers is also a natural number. Consecutive whole numbers differ by 1.



3. Integers (Z) – integers number system include numbers in the following manner ....., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ….. Every integer number is either a negative natural number (-2, -3 etc.) or a zero (0) or a natural number (4, 5, 6 etc.). The smallest and the largest integer number remains undefined or cannot be obtained. Consecutive integers differ by one.



4. Prime numbers – are natural numbers that is greater than 1 and is divisible by 1 and itself only. Eg. 2, 3, ….., 31, 33 and so on. The smallest prime number is 2 whereas the largest prime is undefined or cannot be obtained. Expect 2, every prime number is an odd natural number.



5. Composite numbers – is a number that is divisible by 1, by itself and atleast by one more number. Also, a natural number which is greater than 1 and is not a prime number is called a composite number. Eg. 4, 6, 8, ….20 etc.



6. Twin Primes – are those prime numbers that differ by two, such as 5 and 7, 11 and 13, 31 and 33 etc.



7. Prime triplet – it is the set of three consecutive prime numbers. There exists only such set i.e. 3, 5, 7.



8. Co-Primes – the pair of numbers which not divisible by any number other than 1 are called co-primes. Eg. 13 and 42, 33 and 57 etc.



9. Rational numbers (Q) – numbers that can be expressed as a fraction as $\dfrac{a}{b}$, where a and b are integers and b $\neq$ 0. Every fraction $\dfrac{3}{4}$, $\dfrac{4}{10}$ etc. is a rational number. Likely every integer, natural number, whole number and decimal number is also a rational number, eg. $\dfrac{-3}{1}$, $\dfrac{8}{1}$, 3.2 = $\dfrac{32}{10}$ etc. Zero (0) is also a rational number because it can be written in the form $\dfrac{0}{1}$, $\dfrac{0}{-8}$ etc.



10. Irrational numbers – numbers that cannot be expressed as $\dfrac{a}{b}$, where a $\in$ integer (I), b $\in$ integer (I) and b $\neq$ 0 are called irrational numbers. Irrational numbers are non-terminating and non-recurring in nature,
    
    eg. $\sqrt{2}$, $\sqrt{7}$, $\sqrt{2}$, $\sqrt{5}$ + $\sqrt{3}$ etc.
    
    Also, numbers which are not rational are irrational.



11. Real numbers – rational and irrational numbers taken together are called Real numbers
    
    R = Q $\cup$ $\bar{Q}$
    
    Every real numbers are either rational or irrational.



12. Fibonacci series – this series of numbers forms a sequence that starts with two values 0 and 1. Every subsequent number is the sum total of the preceeding two numbers.
    
    eg. $F_0 = 0, F_1 = 1, F_3 = 2, F_4 = 3, F_5 = 5, F_6 = 8$ and so on.
    
    The fabonacci series has been studied extensively and generalizeed in many ways, for eg., it has been tried and tested tostart with numbers other than 0 and 1, by adding more than two numbers to generate the next number or by adding objects instead of numbers.



13. Lucas numbers – Lucas numbers or the lucas series is an integer sequence named after the mathematician Edourd Anatole Lucas (1842-1891) who studied both the sequence and the closely related fabonacci number series. Lucas numbers and the Fabonacci series form complementary instances of Lucas sequences.
    Similar to Fabonacci numbers, each Lucas number is defined to be the sum of two immediate previous numbers thereby forming a fabonacci integer sequence. The first two Lucas numbers are $L_0 = 2$ and $L_1 = 1$ as opposed to the first two fabonacci numbers $F_0 = 0$ and $F_1 = 1$.
    
    eg. Lucas number series – 2, 1, 3, 4, 7, 11, 18 and so on.
    
    Though closely related Lucas and Fabonacci numbers exhibit distinct properties. In “Wythoff” array the Fabonacci series forms the first while the Lucas series forms the second row. Also, like the Fabonacci series, the ratio between two consequetive Lucas numbers converges (limits of a sequence) to the ‘Golden Ratio’.



14. Wythoff Array – is an infinite matrix of integers derived from the Fabonacci sequence and is named after Dutch mathematician William Abraham Wythoff. It was first defined by Morrison (1980) using Wythoff pairs, the cordinates of winning positions in Wythoff’s games, it can also be defined by using the Fabonacci numbers and Zeckendorf’s theorem or directly from the golden ratio and the recurrence relation defining the Fabonacci numbers. Every positive integer occurs exactly once in the array and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array. The famous ‘Wythoff’ array includes the following values :
    
    $\begin{array}{c c c c c c c} 1 & 2 & 3 & 5 & 8 & 13 & 21\\\\ 4 & 7 & 11 & 18 & 29 & 47 & 76\\\\ 6 & 10 & 16 & 26 & 42 & 68 & 110\\\\ 9 & 15 & 24 & 39 & 63 & 102 & 165\\\\ 12 & 20 & 32 & 52 & 84 & 136 & 220\\\\ 14 & 23 & 37 & 60 & 97 & 157 & 254\\\\ 17 & 28 & 45 & 73 & 118 & 191 & 309\\\\ \end{array}$
15. Golden Ratio – in mathematics, two quatities ar the golden ratio if their ratio is the same as the ratio of their sum to the larger of the quatities. The figure on the right. The following figure illustrates the geometric relation relationship expressed algebrically, for quatities ‘a’ and ‘b’ with a>b>0 :
    
    $\therefore \dfrac{a + b}{a} = \dfrac{a}{b} = \varphi$
    
    The Greek letter phi ($\varphi or \phi$) represents the golden ratio. Its value is :
    
    $\varphi = \dfrac{1 + \sqrt{5}}{2} = 1.6180339887…$



16. Recurring relation – In mathematics, a recurring relation is an equation that recursively (repeating in self-similar ways) defines a sequence or multidimensional array of values, once one or more initial terms are given. Each further term of the sequence is defined as a function of the preceeding terms.



17. Zeckendorf’s Theorem – is about representation of integers as sums of fibonacci numbers. It states that every positive integer can be represented uniquely as the sum of one or more distinct fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.
    
    eg. 100 = 89 + 8 + 3
    
    Other ways may include : 100 = 89 + 8 + 2 + 1 or 100 = 55 + 34 + 8 + 3.
    
    But these are not Zeckendorf representation because 1 and 2 are consecutive Fabonacci numbers as are 34 and 55 or any given positive integer, a representation that satisfies the conditions of Zeckendorf theorem can be found using a Greedy Algorithm, where the choice is made for the largest possible Fibonacci number at each stage.



18. Greedy Algorithm – this algorithm follows the problem solving heuristic of making the locally optimal chooice at each stage with a hope of finding a global optimum. Though it may not yield an optimal solution it yet may yield locally optimal solutions that approximate a global optimal solution in a reasonable time. For eg. A greedy strategy for a travelling salesman problem is following heuristic – “At stage visit an unvisited city nearest to the current city. This not only finds the best solution but also terminates a reasonable number of steps, otherwise finding an optimal solution typically requires unreasonably many steps. In mathematical ooptimization, greedy algoritms solve combinatorial problems having the properties of materials. Further, a greedy algorithm determines minimal number of coins to give while making a change :
    
    Eg. Rs.36 = 20 + 10 + 5 + 1.