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What are Triangular Numbers?
These are the first 100 triangular numbers:
The sequence of the triangular numbers comes from
the natural numbers (and zero), if you always
add the next number:
1
1+2=3
(1+2)+3=6
(1+2+3)+4=10
(1+2+3+4)+5=15
...
You can illustrate the name triangular
number by the following drawing:
Formulas top
The general representation of a triangular number
is dn= 1 + 2 + 3 + 4 +...+ (n-2) + (n-1) + n,
where n is a natural number.
This sum is dn= n * (n + 1) / 2.
Proof:
dn= 1
+ 2 + 3 + ...+ (n-2) + (n-1) + n
dn= n
+ (n-1) + (n-2) +... + 3 + 2 + 1
------------------------------------------
Add both sides and combine the right terms in the pairs (n+1). There
are n terms.
2dn=n * (n+1)
dn= n * (n + 1) / 2, q.e.d.
There also is the recursion formula dn+1= dn+
n with d1=1.
Special Triangular
Numbers top
Even and odd triangular numbers
... ... |
You see:
The even triangular numbers in red and the odd
numbers in black form pairs in the usual sequence. |
The smallest square numbers
1=1²
d8=36=6²
d49=1225=35²
d288=41616=204²
d1681=1413721=1198²
d9800=480024900=6930²
d57121=1631432881=40391²
...
|
The smallest palindromic numbers
d10=55
d11=66
d18==171
d34=595
d36=666
d77=3003
d109, d132, d173 , d363,
...
|
Perfect numbers
A number which is equal to the sum of all its
divisors smaller than the number itself is called a perfect number.
The first perfect numbers are 6, 28 and 496.
They are triangular numbers like every perfect number.
The number of 666 top
The sum of seven Roman numerals is D+C+L+X+V+I=666. The letter M is
missing.
You also can write: DCLXVI=666.
666 is the largest triangular
number which you can form of the same digits (1, page 98).
666 is a Smith number. This means: The sum of digits
[6+6+6] is equal to the sum of the digits of the prime factors [2+3+3+(3+7)]
(1, page 200).
The number 666 appears in an unfavourable light,
because it is called the "number of the animal" in the bible.
Here is wisdom! Who has good brains, should think of the number
of the animal; because it is a human's number, and this is 666 (John's
revelation 13,18 in Luther's translation)
The number of the animal is a bad number in the interpretations of
the bible and is called the "number of the beast", "Satan's number",
or "Antichrist's number".
Consequently people looked in the names of the emperors Nero and Diokletian
for 666 and they found it, because they persecuted the Christians. In the
16th century, in the time of the religious wars, 666 was connected with
the name of Luther and on the other side with that of the pope.
The example of the pope uses the idea of the chronogram:
The pope is called VICARIUS FILII DEI (deputy of God). If you add the values
of the Roman numerals, you get 666 (VICARIVS
FILII DEI).
You are flooded with information on the internet by searching with 666,
if you like.
Counting Pairs top
You give eight squares with the numbers 0,1,2,3,4,5 and 6. They are
arranged to pairs.
There are 7+6+5++4+3+2+1=28 pieces. This is a triangular number.
... ...
|
There also are dominos with 36 or 45 pieces, if you add the squares
with 7 and 8 numbers. |
Everybody with each
other
... ...
|
If you join n points as often as possible, you get 1+2+3+...+(n-1)
lines.
An example with n=7 on the left. |
Shaking hands
Everybody shakes hands with each other. Result: You shake hands (1+2+3+...+(n-1))
times.
Prost
Everybody clinks glasses of champagne with each other.
Numbers
of recangles inside a nxn square top
... ...
|
There are 36 rectangles inside a 3x3 square, 14 of them are squared. |
Derivation for a nxn square:
Every rectangle is formed by pairs of vertical and horizontal lines.
There are n+1vertikal lines. You can arrange them to n(n+1)/2 pairs.
n+1 horizontal lines also have n(n+1)/2 pairs.
There are [n(n+1)/2]² combinations alltogether. If you
give n=3, you get 36.
You can easily generalize to the numbers of rectangular solids inside
a cube and even inside a rectangular solid.
Gauss Sum top
There is a story about the famous mathematician Karl Friedrich Gauß
(1777-1855), when he was a child. He should add the numbers 1 to 100. The
teacher thought, that he would be busy with it for a long time. But Karl
Friedrich found the sum 5050 after some minutes. Instead of adding the
numbers one after the other, he made pairs of numbers and could multiply:
1+2+3+4+...+50+51+...+97+98+99+100
= (1+100) + (2+99) + ... + (50+51)
= 50*101
= 5050.
[(3), page 22f.]
Position in Pascal's
Triangle top
... ...
|
Pascal's triangle makes a contribution to many fields of the number
theory.
The red numbers are triangle numbers.
You even can find the sum of the triangular numbers easily.
Example: 1+3+6+10+15=35 |
You can express the triangular numbers as binomial coefficients
Figurate Numbers top
You can generalize the triangular numbers and
go further to quadrilateral, pentagons, ...
triangular numbers
square numbers
pentagonal numbers
hexagonal numbers
heptagonal numbers
octogonal numbers
...
|
n*(n+1)/2
n²
n*(3n-1)/2
n*(4n-2)/2
n*(5n-3)/2
n*(3n-2)
...
|
1 3 6 10 15 21 28...
1 4 9 16 25 36 49...
1 5 12 22 35 51 70...
1 6 15 28 45 66 91...
1 7 18 34 55 81 112...
1 8 21 40 65 96 133...
...
|
It is fun to find out which triangular numbers also appear in the new sequences.
You can generalize from 2d- (triangle numbers)
to higher dimensions:
triangular numbers
tetrahedral numbers
hypertetrahedral numbers
...
|
n*(n+1)/2
n*(n+1)*(n+2)/6
n*(n+1)*(n+2)*(n+3)/24
...
|
1 3 6 10 15 21...
1 4 10 20 35 56...
1 5 15 35 70 126...
...
|
Here you also have the question, which triangular numbers repeat in the
new sequences.
There is the famous theorem:
The sum of two successive numbers is a square number.
Proof: Add dn and dn+1. The result is (n+1)².
See also the drawings with the triangles above.
Triangular Numbers
on the Internet top
German:
blogger.de
153
Jutta Gut
Figurierte
Zahlen - die Arithmetik der Spielsteinchen
Werner Winzen (Universität Dortmund)
Geometrische
Zahlen (.pdf-Datei)
Wikipedia
Dreieckszahl,
Sechshundertsechsundsechzig,
Gaußsche
Summenformel
English:
Alexander Bogomolny (cut-the-knot)
There
exist triangular numbers that are also square
American Scientist - the magazine of Sigma Xi, The Scientific Research
Society (Reprint)
Brian
Hayes: Gauss’s Day of Reckoning
Eric W. Weisstein (world of mathematics)
Triangular
Number (Figurate
Number, Heptagonal
Triangular Number, Octagonal
Triangular Number, Pentagonal
Triangular Number, Pronic
Number, Square
Triangular Number)
Mark Warner
Triangular
Numbers
Mathpages.com
Square Triangular
Numbers
Michael Gilleland (Merriam Park Software)
The Number of the
Beast: A Numerological Primer
Mike Keith
The Number of the
Beast
Patrick De Geest (World of Numbers)
Palindromic Triangulars
Peter Macinnis
Enquiring
into triangular numbers
Shyam Sunder Gupta
Fascinating
Triangular Numbers
Wikipedia
Triangular
number
153 (number)
"The number of fish caught by Simon Peter, "yet the net was not broken"
(Holy Bible, John 21:11)"
References top
(1) Martin Gardner: Die magischen Zahlen des Dr. Matrix, Frankfurt
am Main 1987 [ISBN 3-8105-0713-X]
(2) Jan Gullberg: Mathematics - From the Birth of Numbers, New York
/ London (1997) [ISBN 0-393-04002-X]
(3) Walter Lietzmann: Lustiges und Merkwürdiges von Zahlen und
Formen, Göttingen 1969
Thank you Gail from Oregon Coast for supporting me in my translation
Feedback: Email address on my main page
This
page is also available in German.
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
2004 Jürgen Köller
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