__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
....................P(fN) = N^[d(N)/2]
which for
our example of N = 60 becomes
P(fN) =
60^(12/2) = 46,656,000,000.
__Geometric
mean of the divisors of the number N:__
The
geometric mean is given by
....................G(dN) = sqrt[N}
__Arithmetic
Mean of the divisors of the number N:__
The
average, or arithmetic, mean of the divisors of the number N is
given by
..................M(dN) = S(dN)/d(N)
which for
our example yields M(f60) = 168/12 = 14.
__Harmonic
mean of the divisors of the number N:__
The
harmonic mean of the divisors of the number N is given by
....................H(dN) = N/M(dN)
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, 125, 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...................................3..............4..............5...............6..............7............8..................11
[(p-1)! +
1]....................3..............7.............25.............121.........721.......5041..........3,628,801
[(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
Pair
No..1.....................1.....1.....1.....1.....1.....1
Pair No.
2...................................1.....1.....1.....1
Pair No.
3..........................................1.....1.....1
Pair No.
4.................................................1.....1
Pair No.
5.................................................1.....1
Pair No.
6........................................................1
Pair No.
7........................................................1
Pair No.
8........................................................1
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.
__FRIENDLY
NUMBERS__
See Amicable
Numbers
__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
..............O...........O.O.........O.O.O.......O.O.O.O....O.O.O.O.O....O.O.O.O.O.O
...............................O................O.................O..................O......................O
.................................................O.................O...................O.....................O
....................................................................O...................O......................O
.........................................................................................O......................O
.................................................................................................................O
Odd.#....1..............3................5...................7....................9......................11
The successive
nesting of these right angled arrangements of dots graphically
create the sum of the odd numbers, 1+3+5+7=9+11+.....n = n^2, or
the perfect squares.
n................1..............2................3.................4...................5.......................6
.................O...........O.O.........O.O.O.......O.O.O.O....O.O.O.O.O....O.O.O.O.O.O
..............................O.O.........O.O.O.......O.O.O.O....O.O.O.O.O....O.O.O.O.O.O
.............................................O.O.O.......O.O.O.O....O.O.O.O.O....O.O.O.O.O.O
..............................................................O.O.O.O....O.O.O.O.O....O.O.O.O.O.O
...............................................................................O.O.O.O.O....O.O.O.O.O.O
....................................................................................................O.O.O.O.O.O
Squares1.................4................9..................16................25.....................36
__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.
N.......12.....13.....19.....23.....28.....31...32......44.......49.......68......70.....79.....82.....86.....91.....94.....97
^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
10----->1+0 =
1
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.
__HARDY-RAMANUJAN
NUMBERS __
See Taxicab
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 i^{2} = (-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.
Examples:
Letting Nn =
N(n-1) + N(n-2)
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__
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.
Continue This Article |
Back to Part I |