The Fibonacci Numbers Fibonacci is perhaps best known for discovering a series of numbers that we now refer to as the Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,... This sequence was introduced in Fibonacci's first book, Liber abbaci, where it arose in the solution to a mathematical problem: "A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair from which the second month on becomes productive?" (Liber abbaci, chapter 12, p. 283-4) Solution: At month zero, there is one pair of rabbits. At the beginning of the first month (January 1st), there is one pair of rabbits that has mated, but not yet given birth. At the beginning of the second month, the original pair gives birth, giving rise to another pair of rabbits...and so on. After one full year, there are 377 pairs of rabbits.

The Fibonacci numbers form what we call a recurrent sequence. Beginning with 1, each term of the Fibonacci sequence is the sum of the two preceding numbers. 0+1=1 1+1=2 1+2=3 2+3=5 3+5=8 5+8=13 Here is a recurrence relation for the Fibonacci numbers: This equation identifies a situation, when n is greater than 2, where each term of a sequence is the sum of the two preceding terms, as is the case with the Fibonacci series. Spacer sequence | Article about spacer sequence by The Free Dictionary Spacer sequence | Article about spacer sequence by The Free Dictionary Also found in: Dictionary , Thesaurus , Medical , Legal , Wikipedia . sequence, in mathematics, ordered set of mathematical quantities called terms. A sequence is said to be known if a formula can be given for any particular term using the preceding terms or using its position in the sequence. For example, the sequence 1, 1, 2, 3, 5, 8, 13, … (the Fibonacci sequence) is formed by adding any two consecutive terms to obtain the next term.

The sequence − 1-2, 1, 7-2, 7, 23-2, 17, … is formed according to the formula (n2 − 2)/2 for the nth, or general, term. A sequence may be either finite, e.g., 1, 2, 3, … 50, a sequence of 50 terms, or infinite, e.g., 1, 2, 3, … , which has no final term and thus continues indefinitely. Special types of sequences are commonly called progressions progression, in mathematics, sequence of quantities, called terms, in which the relationship between consecutive terms is the same. An arithmetic progression is a sequence in which each term is derived from the preceding one by adding a given number, d, ..... Click the link for more information. . The terms of a sequence, when written as an indicated sum, form a series series, in mathematics, indicated sum of a sequence of terms. A series may be finite or infinite. A finite series contains a definite number of terms whose sum can be found by various methods. An infinite series is a sum of infinitely many terms, e.g. ..... Click the link for more information.

; e.g., the sum of the sequence 1, 2, 3, … 50 is the series 1 + 2 + 3 + … + 50. Sequence a fundamental concept of mathematics. A sequence is a set of elements of any nature that are ordered as are the natural numbers 1,2,…, n…. It can be written in the form x1, x2, …, xn, … or simply {xn}. The elements of which it is composed are called its terms. Different terms of a sequence may be identical. A sequence may be regarded as a function whose argument can take on only positive integral values—that is, a function defined on the set of natural numbers. To define a sequence, we can either specify its nth term or make use of a recurrence formula, by which each term is defined as a function of preceding terms. Fibonacci numbers, for example, are defined through a recurrence formula. The sequences most often encountered are those of numbers or functions.

For example, (1) 1, 2, …, n, … that is, xn = n If the terms of a sequence of numbers differ by an arbitrarily small amount from the number a for sufficiently large n, the sequence is said to be convergent, and a is called its limit. The limit of a sequence of functions is defined in a similar manner. For example, sequences (2) and (4) are convergent, and their limits are 0 and the function 1/(1 + x2), respectively. Sequences that are not convergent are said to be divergent. Sequences (1) and (3) are examples of divergent sequences. sequence Sequence (mathematics) | Article about Sequence (mathematics) by The Free Dictionary Sequence (mathematics) | Article about Sequence (mathematics) by The Free Dictionary Related to Sequence (mathematics): Finite sequence , Ordered sequence sequence, in mathematics, ordered set of mathematical quantities called terms. A sequence is said to be known if a formula can be given for any particular term using the preceding terms or using its position in the sequence.

For example, the sequence 1, 1, 2, 3, 5, 8, 13, … (the Fibonacci sequence) is formed by adding any two consecutive terms to obtain the next term. The sequence − 1-2, 1, 7-2, 7, 23-2, 17, … is formed according to the formula (n2 − 2)/2 for the nth, or general, term. A sequence may be either finite, e.g., 1, 2, 3, … 50, a sequence of 50 terms, or infinite, e.g., 1, 2, 3, … , which has no final term and thus continues indefinitely. Special types of sequences are commonly called progressions progression, in mathematics, sequence of quantities, called terms, in which the relationship between consecutive terms is the same. An arithmetic progression is a sequence in which each term is derived from the preceding one by adding a given number, d, ..... Click the link for more information. . The terms of a sequence, when written as an indicated sum, form a series series, in mathematics, indicated sum of a sequence of terms.

A series may be finite or infinite. A finite series contains a definite number of terms whose sum can be found by various methods. An infinite series is a sum of infinitely many terms, e.g. ..... Click the link for more information. ; e.g., the sum of the sequence 1, 2, 3, … 50 is the series 1 + 2 + 3 + … + 50. Sequence a fundamental concept of mathematics. A sequence is a set of elements of any nature that are ordered as are the natural numbers 1,2,…, n…. It can be written in the form x1, x2, …, xn, … or simply {xn}. The elements of which it is composed are called its terms. Different terms of a sequence may be identical. A sequence may be regarded as a function whose argument can take on only positive integral values—that is, a function defined on the set of natural numbers. To define a sequence, we can either specify its nth term or make use of a recurrence formula, by which each term is defined as a function of preceding terms. Fibonacci numbers, for example, are defined through a recurrence formula. The sequences most often encountered are those of numbers or functions.

For example, (1) 1, 2, …, n, … that is, xn = n If the terms of a sequence of numbers differ by an arbitrarily small amount from the number a for sufficiently large n, the sequence is said to be convergent, and a is called its limit. The limit of a sequence of functions is defined in a similar manner. For example, sequences (2) and (4) are convergent, and their limits are 0 and the function 1/(1 + x2), respectively. Sequences that are not convergent are said to be divergent. Sequences (1) and (3) are examples of divergent sequences. sequence 7.1 - Sequences and Summation Notation 7.1 - Sequences and Summation Notation A sequence is a function whose domain is the natural numbers. Instead of using the f(x) notation, however, a sequence is listed using the an notation. There are infinite sequences whose domain is the set of all positive integers, and there are finite sequences whose domain is the set of the first n positive integers. When you define a sequence, you must write the general term (nth term or an).

There are sometimes more than one sequence that is possible if just the first few terms are given. Defining a Sequence There are two common ways to define a sequence by specifying the general term. General Term, an The first is to use a form that only depends on the number of the term, n. To find the first five terms when you know the general term, simply substitute the values 1, 2, 3, 4, and 5 into the general form for n and simplify. Consider the sequence defined by the general term an = 3n-2 The first five terms are found by plugging in 1, 2, 3, 4, and 5 for n. 3(1) - 2 = 1 Therefore, the first five terms of the sequence are 1, 4, 7, 10, 13 Now consider the sequence defined by the general term an = 1 / n. The first five terms are 1/1, 1/2, 1/3, 1/4, and 1/5. Recursive Definition The second way is to recursively define a sequence. A recursive definition uses the current and/or previous terms to define the next term.

You can think of ak+1 being the next term, ak being the current term, and ak-1 being the previous term. Consider the sequence where a1 = 5 and ak+1 = 2 ak - 1.

You can read that last part as "the next term is one less than twice the current term" The first five terms are: 5 (by definition), 2(5) - 1 = 9 (twice the first term of 5 minus 1), 2(9) - 1 = 17 (twice the second term of 9 minus 1), 2(17) - 1 = 33 (twice the third term of 17 minus 1), 2(33) - 1 = 62 (twice the fourth term of 33 minus 1) Now consider the sequence defined by a1 = 2, a2 = 1, and ak+2 = 3ak - ak+1 You can read that last part as "the next term is 3 times the last term minus the current term" The first five terms are: 2 (by definition), 1 (by definition), 3(2) - 1 = 5 (3 times first term minus second term), Note that when k = 1, the sequence gets written as a1+2 = 3a1 - a1+1 which becomes a

3 = 3a1 - a2.

Since a1 = 2 and a2 = 1, this is where the 3(2) - 1 = 5 comes from. 3(1) - 5 = -2 (3 times second term minus third term), 3(5) - (-2) = 17 (3 times third term minus fourth term) Fibonacci Sequence One famous example of a recursively defined sequence is the Fibonacci Sequence. The first two terms of the Fibonacci Sequence are 1 by definition. Every term after that is the sum of the two preceding terms. The Fibonacci Sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... an+1 = an + an-1. Fibonacci sequences occur frequently in nature. For example, take a leaf on a stem of many plants (like cherry, elm, or pear trees). Count the number of leaves until you reach one directly in line with the one you selected. The total number of leaves (not including the first one) is usually a Fibonacci number.

If the left and right handed spirals on a pine cone, sunflower seed heads, or pineapples are counted, the two numbers are often consecutive Fibonacci numbers. Factorials - ! The symbol for factorial is ! (an explanation point). The factorial of a positive integer is the product of all positive integers less than or equal to that number. Zero factorial is a special case and 0! = 1 by defintion. 1! = 1 2! = 2*1! = 2*1 = 2 3! = 3*2! = 3*2*1 = 6 4! = 4*3! = 4*3*2*1 = 24 5! = 5*4! = 5*4*3*2*1 = 120 n! = n(n-1)! = n*(n-1)*(n-2)*...3*2*1 Notice that there is also a recursive definition in there. Any number factorial is that number times the factorial of one less than that number.

On the TI-82 and TI-83 calculators, the factorial key can be found under Math, Probability, menu choice 4. Simplifying ratios of factorials Consider 8!/5!. One way to work the problem would be to fully expand the 8! and fully expand the 5!. 8! 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 -- = ----------------------------- = 8 * 7 * 6 = 336 5! 5 * 4 * 3 * 2 * 1 Then you notice that there are common factors of 5, 4, 3, 2, and 1 in both the numerator and denominator that divide out. This leaves you with 8*7*6 in the numerator for an answer of 336. Another way to work the problem, however is to use the recursive nature of factorials. Since any number factorial is that number times the factorial of one less than that number, 8! = 8 * 7!, but 7! = 7 * 6!, and 6! = 6 * 5!. This means that 8!

= 8 * 7 * 6 * 5!. So, 8! / 5! Is 8*7*6*5!/5! = 8*7*6 = 336. The point is that there is no need to multiply the entire thing out when you're just going to be dividing a bunch of it out anyway. Here are some steps to simplifying the ratio of two factorials. Find the smaller factorial and write it down. It won't change, and it will be the part that is completely divided out. Take the larger factorial and start expanding it by subtracting one until the smaller number (that you've already written down) is reached. When you reach the smaller number, write it as a factorial and divide out the two equal factorials. Here's an example with a little bit more complicated ratio. Since 2n-1 is less than 2n+1, copy the 2n-1 factorial down on the bottom. Now, take 2n+1 times one less than 2n+1. One less then 2n+1 is 2n+1-1=2n.

Take that times one less than 2n, which is 2n-1. Since that 2n-1 is the same amount on the bottom, call the 2n-1 factorial and then divide it out, leaving the product of 2n and 2n+1 as the answer. (2n+1)! (2n+1)*(2n)*(2n-1)! ------- = ------------------- = 2n(2n+1) (2n-1)! (2n-1)! Sigma / Summation Notation Summation is something that is done quite often in mathematics, and there is a symbol that means summation. That symbol is the capital Greek letter sigma, and so the notation is sometimes called Sigma Notation instead of Summation Notation. The k is called the index of summation. k=1 is the lower limit of the summation and k=n (although the k is only written once) is the upper limit of the summation. What the summation notation means is to evaluate the argument of the summation for every value of the index between the lower limit and upper limit (inclusively) and then add the results together.

Examples: Substitute each value of k between 1 and 5 into the expression 3k-2 and then add the results together. [3(1)-2] + [3(2)-2] + [3(3)-2] + [3(4)-2] + [3(5)-2] = 1 + 4 + 7 + 10 + 13 = 35 [ 3(1) + 3(2) + 3(3) + 3(4) + 3(5) ] - 2 = [ 3 + 6 + 9 + 12 + 15 ] - 2 = 45 - 2 = 43 Although these look very similar, the answers are different. The second example is thrown in there to warn you about notation. Multiplication has a higher order of operations than addition or subtraction, so no group symbols are needed around the 3k. But since subtraction has the same precedence as addition, the subtraction of 2 does not go inside the summation.

In other words, be sure to include parentheses around a sum or difference if you want the summation to apply to more than just the first term. Properties of Summation The following properties of summation apply no matter what the lower and upper limits are for the index. For simplicity sake, I will not write the k=1 and n, but know that your index of summation is k in the following examples. You can factor a constant out of a sum. ∑cak = c∑ak Notice the ak has a subscript of k while the c doesn't. This means that the c is a constant and the a is function of k. The sum of a constant times a function is the constant times the sum of the function. The sum of a sum is the sum of the sums Ooh, that just sounds good. The summation symbol can be distributed over addition. ∑(ak + bk) = ∑ak + ∑bk The sum of a difference is the difference of the sums The summation symbol can be distributed over subtraction. ∑(ak - bk) = ∑ak - ∑bk Aren't those just beautiful sounding? Those should certainly be placed on a note card to help you remember them.

Numbers, Part II Types of Numbers, Part II Compiled by William Tappe FACTOR NUMBERS A number "a" that exactly divides another number "b" is called a factor (or divisor) of "b". Symbolically, this is often written as a I b, a divides b, where ac = b meaning that "a" is a factor/divisor of "b" or "b" is exactly divisible by "a" "c" times. A number may have more than one factor or multiples of the same factor. For instance, 36/2 = 18, 18/2 = 9, 9/3 = 3, and 3/3 = 1 indicating that both 2 and 3 divide 36 twice. Every composite number can be factored into a unique set of prime factors N = (p^a)(q^b)(r^c)....(z^i) where p, q, r,...z, are the various different prime factors and a, b, c,...i, the number of times p, q, r, or z occur in the prime factorization.

All of the factors of a number, less the number itself, are referred to as the aliquot parts, or proper divisors, of the number.. The total number of factors/divisors of a number N = p^a(q^b)r^c...z^i is given by d(N) = (a + 1)(b + 1)(c + 1)......(i + 1). Example: The number 60 = 2^2 x 3 x 5 resulting in the total number of divisors being ...............................d(N) = (2 + 1)(1 + 1)(1 + 1) = 12 namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 totaling 12 in all. The sum of the factors/divisors of the number N is given by .........................S(dN) = [p^(a+1) - 1] x [q^(b+1) - 1] x [r^(c+1) - 1]....

...........................................(p1 - 1)...........(p2 - 1)...........(pr - 1) Finding a number N having dN factors: Find the smallest number having 14 factors. We have 14 = 7 x 2, and subtracting unity from each we obtain 6 and 1 as exponents to apply to any primes we wish. To derive the smallest number, we must obviously apply our exponents to the smallest primes, namely 2 and 3 which results in N = 2^6 x 3^1 = 192. Recognize that any number of the form p^6(q^1), where p and q are primes greater than unity, also has exactly 14 divisors. Find the smallest number with 8 factors/divisors. We have 8 = 4x2 resulting from exponents of 3 and 1 which leads to 2^3x3^1 = 24 and 24 has factors of 1, 2, 3, 4, 6, 8, 12 and 24. We could also evaluate 8 = 8x1 making the exponents 7 and 0.

Applied to the prime of 2 yields N = 128 which also has 8 factors but is larger than 24. Product of the divisors of the number N: The product of all the divisors of a number N is given by which for our example yields H(d60) = 60/14 = 4 2/7. If the prime factors of the number N has all even exponents, N is the square of a natural number and sqrtN is rational. Only perfect squares have an odd number of factors/divisors. Sum of factors -There are many numbers, the factors of which, including 1 and the number itself, all add up to a perfect square. The smallest number with this characteristic is 3, since 1 + 3 = 4 = 2^2. Another example is 81, of which the factors 1, 3, 9, 27, and 81 add up to 121 = 11^2. FACTORIAL NUMBERS Factorial numbers are those derived from multiplying all the positive integers less than and equal to a given positive integer. The exclamation point is used to indicate factorial.

This is expressed by n! = n(n-1)(n-2)(n-3)..........(n-n+2)(n-n+1). To illustrate, 1! = 1, 2! = 2 x 1 = 2, 3! = 3 x 2 x 1 = 6, 4! = 4 x 3 x 2 x 1 = 24, 5! = 5 x 4 x 3 x 2 x 1 = 120. 0! is assumed to be 1. The first ten factorial numbers are 1, 2, 6, 24, 120, 720, 5040, 40,320, 362,880 and 3, 628,800. Factorials are used in many areas of mathematics, combinations, permutations, probability, and the binomial theorem to name a few. For instance, how many ways can you arrange the 3 books A, B, and C on a shelf?

Quite easily, ABC, ACB, BAC, BCA, CAB, and CBA or 6 ways which derives from 3! = 3x2x1 = 6. How many 9 digit numbers can be made from the digits 1 through 9? Taking the shortcut, 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800 different numbers. Each of the "arrangements" which can be made by taking some, or all, of a number of things from a group of things, is called a permutation. Each of the "groups", or selections, which can be made by taking some or all of a number of things from a group is called a "combination." To find the number of permutations of n dissimilar things taken n at a time, the formula is nPn = n! To find the number of permutations of n dissimilar things taken r at a time, the formula is nPr = n(n-1)(n-2)(n-3)..........(n-r+1).

To find the number of combinations of n dissimilar things taken r at a time, the formula is nCr = n!/[r!(n-r)!] which can be stated as n factorial divided by the product of r factorial times (n-r) factorial. A surprising application of factorials shows up in deriving the sum of 1/n! or 1/1!+1/2!+1/3+1/4!+.....1/n! which converges to a familiar number e = 2.71828.... Another place where factorials show up is in Wilson's theorem for determining whether a number "p" is prime. If the number "p" is prime it will evenly divide [(p-1)! + 1]. For instance, checking out a few numbers: [(p-1)! + 1]/p.................1.............1+............5..............20+.........103........630+.........329,891+ clearly showing that 3, 5, 7, and 11 are primes.

Unfortunately, checking the primeness of larger numbers becomes rather unwieldy rather quickly and is therefore not an effective tool for verifying whether a number is prime or not. FERMAT NUMBERS Pierre de Fermat once hypothesized that all numbers of the form F(n) = (2^(2^n)) + 1 were prime numbers. His conclusion was based on the results derived for n = 0, 1, 2, 3, and 4, which produced the primes of 3, 5, 17, 257 and 65,537. These numbers were called Fermat Numbers, F0, F1, F2, F3, and F4. However, Leonard Euler discovered in 1732 that F5 = (2^2)^5 + 1 = 4,294,967,297 = 6,700,416x641, a composite number. It was later shown that F8 through F20 were all composite. While never proved, it is widely accepted that all Fermat Numbers beyond F4 are composite.

FIBONACCI NUMBERS Leonardo Fibonacci, originally known as Leonardo of Pisa, was an Italian merchant and mathematician who contributed much to the field of algebra, Euclidian geometry, Diophantine equations, and number theory. Among his many writings was the Liber Abaci, published in 1202, which contained a famous problem about rabbits which led to what we refer to today as Fibonacci numbers or the Fibonacci sequence. The problem has been quoted many ways in historical literature but basically asks, "How many pairs of rabbits can be produced from a single pair in a year, each pair producing a new pair after the second month and every month thereafter? The accumulation of rabbits looks like the following. End of Month No..........1.....2.....3.....4.....5.....6......7.......8.......9.......10.....11.....12 Total............................1.....1.....2.....3.....5......8.....13.....21.....34.....55.....89.....144.....

As you can readily see, the sequence continues 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,........n, each succeeding term being the sum of the previous two terms expressed by Fn = F(n-1) + F(n-2). The initial terms are F1 = 1 and F2 = 1. Each successive pair of Fibonacci numbers are relatively prime, i.e., they have no common factors other than 1. Each Fibonacci number is defined in terms of the recursive relationship, Fn = [F(n-1) + F(n-2)]. To determine the 10th, 100th, or 1000th Fibonacci number, one would normally have to compute the previous 9, 99, or 999 numbers in order to compute the one desired. It is only natural therefore, to ask whether there is a simple, or complex, expression out there someplace that would allow us to calculate any Fibonacci number desired.

Search no more, and surprisingly, it involves the equally famous Golden Ratio or Golden Number, t = (1 + sqrt5)/2, and its reciprocal, 1/t. The expression is simply Fn = (t)^n - (-t)^-n = (t)^n - (-1/t)^n ............sqrt(5)................ sqrt(5) where t = the famous 1.618033988749894....... or simply 1.618, as we normally use it. The ratios of any one Fibonacci number to the previous number progressively close in on the Golden Ratio, 1.6180, = (sqrt5 + 1)/2. Surprisingly, individual terms of the Fibonacci sequence also derive from the Binet expression Fn = [((1 + sqrt5)/2)^n - ((1 -sqrt5)/2)^n] ................................sqrt(5) This amazingly simple expression involving square roots and powers of an irrational number does, in fact, produce the numbers in the Fibonacci sequence.

The sum of the squares of two adjacent Fibonacci numbers is equal to a higher Fibonacci number according to Fn^2 + F(n+1)^2 = F(2n+1). For instance, the 4thFn^2 + the 5thFn^2 = the F(2(4) + 1) = 9th Fn or 3^2 + 5^2 = 34, the 9th Fn. The product of two alternating Fibonacci numbers minus the square of the one in between is equal to +/- one as expressed by F(n-1)F(N+1) - Fn^2 = (-1)^n. For instance, 8x21 - 13^2 = 168 - 169 = (-1)^7 = -1. The sum of the cubes of two adjacent Fibonacci numbers minus the cube of the preceding one is equal to a higher Fibonacci number as expressed by Fn^3 + F(n+1)^3 = F3n.

For instance, F4^3 + F5^3 - F3^3 = 3^3 + 5^3 - 2^3 = 27 + 125 - 8 = 144 = F12 = 144. The sum of the squares of a series of Fibonacci numbers starting with F1 and ending with Fn is equal to the product of Fn and F(n+1) as expressed by F1^2 + F2^2 + F3^2 + ......Fn^2 = FnF(n+1). For instance, 1^2 + 1^2 + 2^2 + 3^2 + 5^2 = 1 + 1 + 4 + 9 + 25 = 40 = 5x8. The sum of a series of Fibonacci numbers starting with F1 and ending with Fn is equal to a higher Fibonacci number as expressed by F1 + F2 + F3 + .......Fn = F(n+2) - 1. For instance, 1 + 1 + 2 + 3 + 5 + 8 = 20 = 21 - 1.

The sum of the first n even terms of Fibonacci numbers is given by F2 + F4 + F6 + F8 + ....F2n = F(2n+1) - 1. For instance, the sum of the first 6 even terms is therefore F(2(6) + 1) - 1 = F(13) - 1 = 233 - 1 = 232. The sum of the first n odd terms of Fibonacci numbers is given by F1 + F3 + F5 + ....F(2n-1) = F(2n) For instance, the sum of the first 6 odd terms is therefore F(2n) = F(12) = 144. Fn^2 + F(n+1)^2 = F(2n+1) F(n+1)F(n-1) - Fn^2 = (-1)^n Some other relationships of Fibonacci numbers are Fn^2 + F(n+1)^2 = F(2n+1) and F(n+1)F(n-1) - Fn^2 = (-1)^n.

The ratio of the 2nth Fibonacci number divided by the nth Fibonacci number is always an integer or F2n/Fn = K. For instance, F10/F5 = 55/5 = 11. The sum of any 4n consecutive Fibonacci numbers is evenly divisible by F2n. Example: The sum of 4(2) = 8 Fibonacci numbers is divisible by F2(2) = F4 = 3. 1+2+3+5+8+13+21+34 = 87/3 = 29. 2+3+5+8+13+21+34+55 = 141/3 = 47. 3+5+8+13+21+34+55+89 = 228/3 = 76 As any number may be represented by combinations of powers of 2, so may any number be represented by combinations of the Fibonacci numbers.

1 = 1, 2 = 2, or 1+1, 3 = 3 or 1+2, 4 = 3+1, 5 = 5 or 3+2, 6 = 5+1 or 3+2+1, 7 = 5+2, 8 = 5+3, 9=5+3+1, 10 = 5+3+2, 50 = 34+13+3, 100 = 89+8+3 or 55+34+8+3, and so on. The greatest common divisor of any two Fibonacci numbers is, itself, a Fibonacci number. Even more surprising is the fact that the g.c.d.(Fa, Fb) = c = F[g.c.d.(a,b)]. What this means is that the g.c.d. of the "a"th and "b"th Fibonacci numbers is the "c"th Fibonacci number where c = the g.c.d. of a and b. Every number can be represented by a unique sum of Fibonacci numbers.

The sum of any ten consecutive Fibonacci numbers can be determined by multiplying the 7th term of the ten terms by 11. Example: The sum of the first ten terms of the sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, is 11(13) = 143. ................The sum of the ten terms 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, is 11(55) = 605. Much more can be learned about the Fibonacci Sequence in another article in the Knowledge Database. FIGURATE NUMBERS Figurate numbers, often referred to as polygonal numbers, are numbers that can be represented by geometrically arranged groups of dots. They derive their names from the numerous geometric arrangements of dots, circles, etc., that they form. See polygonal numbers elsewhere in this collection for a more detailed description.

FRACTIONAL NUMBERS Fractions are the ratio of two integers, a/b, where the numerator, "a", may be any integer, and the denominator, "b" may be any integer greater than zero. If "a" is smaller than "b" it is a proper fraction. 3/5 is a proper fraction. If "a" is greater than "b" it is an improper fraction which can be broken up into an integer and a proper fraction. 8/5 is an improper fraction equaling. A mixed fraction is one where an integer is combined with a proper fraction. 1 3/5 is a mixed fraction. A common fraction is one wherein both the numerator and denominator are integers. A complex fraction is one wherein both numerator and denominator are fractions themselves. Partial fractions are fractions that sum to another specific fraction. For example, 1/3 + 1/4 = 7/12. A fraction is in its simplest form when the numerator and denominator are relatively prime, i.e., have no common factor other that 1.

A decimal fraction is a single fraction, or a series of fractions, in which the part immediately to the right of the decimal point is assumed to be the numerator of a fraction with some power of 10 as the denominator. For instance .75 = 75/100 or 7/10 + 5/100 and .625 = 6/10 + 2/100 + 5/1000. Rational decimal fractions may be converted to proper fractions: Given the decimal number N = 0.078078078... Multiply N by 1000 or 1000N = 78.078078078... Subtracting........................N = .078078078... ....................................999N = 78 making N = 78/999 = 26/333.

GENERATING NUMBERS Generating number(s) is the term often applied to the values assigned to "m" and/or "n" and used to generate unique sets of numbers such as Pythagorean Triples, the integer sides of primitive, or non-primitive, Pythagorean triangles, and Heronian Triples, the integer sides of non-right angled Heronian triangles. All primitive Pythagorean Triples of the form x^2 + y^2 = z^2 derive from the expressions x = m^2 - n^2, y = 2mn, and z = m^2 + n^2 where "m" and "n" are arbitrary positive integers of opposite parity (one odd one even), "m" and "n" have no common factor, and "m" is greater than "n". (For x, y, & z to be a primitive solution, "m" and "n" can have no common factors and cannot both be even or odd.

Violation of any of these limitations will produce non-primitive Pythagorean Triples.) Pythagorean Triples can also be derived from x = 2n + 1, y = 2n^2 + 2n, and z = 2n^2 + 2n + 1 where n is any positive integer. These expressions create only triangles where the hypotenuse exceeds the larger leg by one. Another set of expressions that produce Pythagorean Triples is x = n^2, y = (n^2 - 1)^2/2, and z = (n^2 + 1)^2/2. For any positive integer m, 2m, m^2 - 1, and m^2 + 1 are Pythagorean Triples. All Heronian Triple triangles can also be derived from three simple relationships. Denoting the sides by x, y, and z, the altitude to z by h, z = z1 + z2 with z1 adjacent to x and z2 adjacent to y, we can write that h^2 = x^2 - z1^2 = y^2 - z2^2.

Introducing the variables m, n, and k, every rational integral triangle can be derived from x = n(m^2 + k^2), y = m(n^2 + k^2), and z = (m + n)(mn - k^2) where m, n, and k are positive integers and mn is greater than k^2. The altitude to z is given by h = 2mnk making the area A = mnk(m + n)(mn - k^2). See Heronian Numbers elsewhere in the collection. GNOMON NUMBERS Gnomon numbers, from the family of figurate numbers, are the right angled arrangements of a series of dots representing the odd numbers. n.............1..............2................3...............4.................5....................6 GOLDEN NUMBER The Golden Number, or Golden Ratio, is considered the proportion of the most pleasing rectangle. Consider a rectangle one unit wide with length of x, x being greater than one. The sides of the rectangle are said to be in the Golden Ratio if you cut away a square with sides of one unit and the remaining rectangle has the same proportions as the original rectangle.

Expressed mathematically, the width to length ratio becomes 1/x = (x - 1)/1 which produces x^2 - x - 1 = 0, the root of which is x = [1 + sqrt(5)]/2 = 1.618. To create such a rectangle, draw unit square ABCD, A lower left, B lower right, C upper right, and D upper left. Bisect AB at point E. With radius EC and E as center, swing an arc from C to point F on AB extended. Then AF/AB = AB/BF = 1.618. Surprisingly, the successive ratios of any Fibonacci number to the previous Fibonacci number progressively close in on the Golden Ratio, 1.6180, = (sqrt5 + 1)/2. Even more surprising is the fact that the Fibonacci numbers themselves derive from Fn = (t)^n - (-t)^-n = (t)^n - (-1/t)^n ............sqrt(5)................ sqrt(5) where t = the famous (1 + sqrt5)/2 = 1.618033988749894.......

or simply 1.618, as we normally use it. If the Golden Ratio fascinates you, consider the following: If the Golden Ratio µ = 1.618, µ^2 = 2.618, 1/µ = .618, µ = 1 + 1/µ, 2 = µ + 1/µ^2, µ = 1/(µ-1), µ^2 = µ + 1, µ^3 = µ^2 + 1, or more generally, µ^n = µ^(n-1) + µ^(n-2). GYRATING NUMBERS Gyrating numbers are numbers whose digits increase and decrease in a continuous repetitive cycle. 12,321, 2,357,532, 234,323,432, 12,345,432,123,454,321, are examples of gyrating numbers. HAPPY NUMBERS Happy numbers are those numbers that, after successively adding the squares of the digits of the number, the end result is the number 1.

The number 23 is a happy number. Square each of its digits and add, i.e., 2^2 + 3^2 = 13; then 1^2 +3^2 = 10; then 1^2 +0^2 = 1. Since the final number is a 1, the number 23 is a happy number. There are only 17 2 digit happy numbers, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, and 97.

By definition, 100, 130, 103, 310, 301, 190, 109, 910, 901, 230, 203, 320, 302, 280, 208, 820, 802, 440, 404, 490, 409, 940, 904, 680, 608, 860, 806, 700, 790, 709, 970, and 907 are the only three digit happy numbers. Adding more zeros enables the creation or 4, 5, 6, etc. digit numbers using the same two digits. Note the first set of square/sum results for the 17 two digit happy numbers.

^2.....1+4...1+9..1+81.4+9..4+64..9+1.9+4.16+16.16+81.36+64.49+0 etc. Sum...5......10.....82....13.....68.....10....13.....32........97.....100.....49....130....68....100....82.....97....130 Combining the first square/sum results of two of the two digit numbers will result in 4 digit happy numbers.

For example, from 28------>4+64 = 68-->36+64 = 100-->1 44---->16+16 = 32---->9+4 = 13--->1+9 = 10-->1 68-->36+64 = 100-->1+0 = 1 we see that 68 + 32 = 100-->1 making the numbers 2844, 2484, 2448, 4248, 4428, 4482, 8244, 8424 and 8442 all happy numbers. How many other 4 digit happy numbers can you find? Combining the results of three of the two digit numbers will produce 5 and 6 digit numbers.

For example 23--->2+9 = 13-->1+9 = 10-->1+0 = 1 28-->4+64 = 68-->36+64 = 100-->1+0 = 1 19-->1+81 = 82-->64+4 = 68-->36+64 = 100-->1+0 = 1 Note that 1 + 13 + 68 = 82 making the numbers 12,328 and 102,328, and all the permutations of their digits, happy numbers.

12,328-->1 + 4 + 9 + 4 + 64 = 82-->64 + 4 = 68-->36 + 64 = 100-->1 + 0 = 1 102,328-->1 + 0 + 4 + 9 + 4 + 64 = 82-->64 + 4 = 68-->36 + 64 = 100--> 1 + 0 = 1 Infinitely many other happy numbers can be created by combining other same level iterative results as shown in these examples. For example (10,101,013), (10,101,923), (101,010,104,444), and all their permutations are happy numbers. HERONIAN NUMBERS Heronian numbers are the groups of 3 unique integers that define a Heronian triangle. They are sometimes referred to as Heronian Triples. A Heronian, or arithmetical, triangle is one whose sides, at least one altitude, and area, are all integers.

The area of a Heronian triangle is defined by A = sqrt[s(s-a)(s-b)(s-c)] where a, b, c, are the three sides and s = ½(a+b+c). Another surprising characteristic of Heronian triangles is that the area can also be derived from A = rs, where r is the radius of the circle inscribed inside the triangle r = sqrt[(s-a)(s-b)(s-c)/s], also an integer. The radii of both the circumscribed and inscribed circles are also integers. See Taxicab numbers. Heronian Triples can be derived from three simple relationships, much the same as Pythagorean Triples derive from three simple expressions. Denoting the sides by x, y, and z, the altitude to z by h, z = z1 + z2 with z1 adjacent to x and z2 adjacent to y, we can write that h^2 = x^2 - z1^2 = y^2 - z2^2.

Introducing the variables m, n, and k, every rational integral triangle can be derived from x = n(m^2 + k^2), y = m(n^2 + k^2), and z = (m + n)(mn - k^2) where m, n, and k are positive integers and mn is greater than k^2. The altitude to z is given by h = 2mnk making the area A = mnk(m + n)(mn - k^2). A couple of examples will illustrate their application. Lets take m = 3, n = 2, and k = 1 for example. Using the three expressions, x = 20, y = 15, z = 25, and h = 12. From Heron's area formula, the area equals 150 making the altitude h to side z = 12 and z is divided up into 9 and 16 by the altitude. How elegant. Lets try m = 4, n = 3, and k = 1. Here, we end up with x = 51, y = 40, and z = 77.

The area works out to be 924, the altitude to z being 24 and z divided up into lengths of 32 and 45. Lets try m = 5, n = 3, and k = 2. This yields x = 87, y = 65, and z = 88 for an area of 2640, an altitude of 60 and z broken into lengths of 25 and 63. Since the sides of a Heronian triangle are integral, the integer altitude from a vertex to the opposite side forms two integer right triangles whose sum, or difference, equal the given triangle, or a smaller Heronian triangle, respectively. Knowing this, it is possible to form any desired Heronian triangle, which is not a right triangle, by combining two integral right triangles of different shapes. A check of these, and other, results will show that each Heronian triangle is made up from two Pythagorean Triple triangles with the common side as the altitude. A more extensive article regarding Heronian Triangles may be found in the Knowledge Database under the topic "Math - Heronian Triangles".

IMAGINARY NUMBERS An imaginary number is one of the form bi where b is a real number and i is the imaginary element derived from i2 = (-1). Imaginary numbers were defined to enable the solutions of equations requiring the square roots of negative numbers. INFINITE NUMBERS Generally speaking, infinite means without limit. Therefore, an infinite number is one that is greater than all other numbers, the largest of all numbers, a limitless quantity, etc. There are an infinite number of numbers. A so called infinite number is called infinity yet infinity is not a number. It is symbolized by the figure 8 lying on its side (which cannot be produced here). Some very large numbers have been given names such as 10^100 = one googol and 10^googol = one googolplex. There are many others and, given any definition of a specific large number and its contrived name, anyone can derive a larger one, none of which, in the end, is infinite.. INTEGER NUMBERS The integers are any of the combined set of positive and negative whole numbers, zero, +/-1, +/-2, +/-3, ...etc.

The positive numbers, 1, 2, 3, ..., are called the natural or counting numbers. An integer is a real whole number containing no fractional parts. The natural numbers are traditionally referred to as the positive integers while their negatives are called the negative integers. IRRATIONAL NUMBERS An irrational number is one that cannot be expressed by an integer or as the ratio of two integers. Irrational numbers are infinite non-repeating decimals, numbers whose decimal representation goes on forever without repeating any pattern such as sqrt(2), sqrt(3), sqrt(6), etc. Transcendental numbers such as "Pi" and "e" are irrational numbers. KEITH NUMBERS You are probably familiar with the famous Fibonacci number sequence where every number, after the second, is the sum of the two preceding terms. Letting Fn denote the nth term, the sequence can be defined by the recursive expressions F1 = 1, F2 = 1, F(n) = F(n-1) + F(n-2) for n = or > 3.

The result is the familiar 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. Fibonacci sequence. Consider how one might create a similar type sequence where each term is the sum of the "n" preceding terms. If the first "n" single digit terms of the sequence ultimately appear as an "n" digit number of the sequence utilizing the same "n" digits of the first "n" terms in the same order, this "n" digit number is referred to as a Keith number. 2 - 8 - 10 - 18 - 28 - 46 - 74 - 120 making 28 a Keith number 1 - 9 - 7 - 17 - 33 - 57 - 107 - 197 - 304 - 501 making 197 a Keith number. I believe there are only 71 known Keith numbers below 10^19.

LEAST DEFICIENT NUMBERS A least deficient number N has been defined as one where the sum of all of its factors/divisors is equal to one less than twice the number or s(N) = 2N - 1. All the powers of 2 are least deficient numbers. (See perfect, semi-perfect, multiply-perfect, quasi-perfect, deficient, abundant, super abundant) LUCAS NUMBERS The Lucas numbers, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199,.......n, each succeeding term being the sum of the previous two terms as expressed by Ln = L(n-1) + L(n-2). The initial terms are L1 = 1 and L2 = 3. The individual terms of the Lucas sequence also derive from the Binet expression Ln = [(1 + sqrt5)/2]^n + [(1 - sqrt5)/2]^n. MERSENNE NUMBERS Mersenne numbers are numbers of the form Mn = (2^n - 1) where n = a natural number.

When n is a prime number, p, (2^p - 1) often produces a prime number referred to as a Mersenne Prime. Mersenne primes follow no logical pattern of evolution. All perfect numbers derive from P = [2^(p-1)](2^p - 1) when (2^p - 1) is a prime. The 35th Mersenne Prime, was discovered in late 1996, being [2^1,398,269 - 1] and having 420,921 digits. MONODIGIT NUMBERS Numbers consisting of a single repeating digit are often referred to as monodigits, e.g., 22, 333,333. 7,777,777,777, 999,999,999,999,999,999,999, etc. Multipowered numbers are the powers of several different numbers.

Examples: 256 = 16^2 = 4^4 = 2^8 4096 = 16^3 = 8^4 = 2^12 16,777,216 = 4096^2 = 256^3 = 64^4 = 16^6 = 8^8 = 4^12 = 2^24 1,152,921,504,606,846,976 = 1,073,741,824^2 = 1,048,576^3 = 32,768^4 = 4096^5 = 1024^6 = 64^10 = 32^12 = 16^15 = 8^20 = 4^30 = 2^60 = 2^(3x4x5) MULTIPLY- PERFECT NUMBERS Multiply-perfect numbers are extensions of the family of numbers that are either deficient, perfect, or

 abundant.

They are numbers where the sum, sa(N), of its aliquot parts (proper divisors) is a multiple of the number itself or sa(N) = kN. (In the language of the Greek mathematicians, the divisors of a number N were defined as any whole numbers smaller than N that, when divided into N, produced whole numbers. The factors/divisors of a number N, less the number itself, are referred to as the aliquot parts, aliquot parts, or proper divisors, of the number.) Equivalently, N is also multiply-perfect if the sum, s(N), of all its divisors is equal to (k+1)N or s(N) = (k+1)N. Multiply-perfect numbers are sometimes referred to as k-tuptly perfect numbers.

For example, the prime factorization of 120 is 2^3(3^1)5^1 which leads to 120 having 16 factors/divisors, namely, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120, the sum of which is 360 or 3 times 120.

Another is 672, the prime factorization of which is 2^5(3^1)7^1 which leads to 672 having 24 factors/divisors, namely 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 168, 224, 336 and 672, the sum of which is 2016 or 3 times 672. Others are 523,776 and 1,476,304,896. (See perfect, semi-perfect, quasi-perfect, deficient, least deficient, abundant, super abundant) NARCISSISTIC NUMBERS Narcissistic numbers are those "n" digit numbers that can be derived from the sum of "nth" powers of the digits of the number.

For example, if N = 100A + 10B + C = A^3 + B^3 + C^3, the number N is narcissistic. The most basic example of this is the 153 = 1^3 + 5^3 + 3^3. Three other well known examples are 370, 371, and 407. Can you find any others? NATURAL NUMBERS The natural numbers are traditionally referred to as the positive integers, the integers being any of the combined set of 99positive and negative whole numbers, zero, +/-1, +/-2, +/-3, ...etc. The positive numbers, 1, 2, 3, ..., are called the natural or counting numbers. An integer is a real whole number containing no fractional parts. The negatives are called the negative integers. OBLONG NUMBERS Oblong numbers are numbers that represent the total number of dots contained within a rectangular arrangement of dots with the number of columns being one more than the number of rows. Coincidentally, the oblong numbers are the sums of the first "n" even integers as defined by the expression On = n(n + 1).

They are also twice the triangular numbers and sometimes referred to as rectangular or pronic numbers. OCTAHEDRAL NUMBERS Octahedral numbers derive from stacking two successive square pyramids. Essentially, it evolves by creating mirror images of square pyramids. The square pyramidal numbers are 1, 5, 14, 30, 55, 91, etc. Adding the first square pyramid to the second creates the octahedral number of 6. Adding the second square pyramid to the third creates the octahedral number of 19. Subsequent octahedral numbers are 44, 84, 146, etc. ODD NUMBERS Odd numbers are numbers not evenly divisible by 2 or that leave a remainder of 1 when divided by 2.. The nth odd number is given by No = 1n - 1. The sum of the set of "n" consecutive odd numbers beginning with 1 is given by So = n^2.

The sum of the se7t of m consecutive odd numbers starting with n1 and ending with n2 is given by So(n1-n2) = (n2)^2 - (n2 - 1)^2. The sum of the squares of the odd numbers, i.e., 1^2 + 3^2 + 5^2 + .....+ n^2 = (4n^3 - n)/3. An odd number multiplied by another odd number, or raised to any power, results in another odd number. An odd number multiplied by an even number results in an even number. All prime numbers are odd with the exception of the number 2, the only even prime, making it the oddest prime.. ORDINAL NUMBERS Ordinal numbers denote the order of elements in a set by describing the position of the element in the set. Your finishing position in a race, second, third, etc., is an ordinal number. Your house number is an ordinal number.

Question: What name is given to a number sequence in which each term is the sum of the two preceding terms?
Answer: Fibonacci