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Types of Numbers
Compiled by William Tappe http://www.mathgoodies.com/articles/numbers.html
Some basic definitions of terms normally encountered in the classroom are given first.
Numbers - The Basics
Even
Probably the easiest number to define, an even number is any number that is evenly
divisible by 2.
Odd
Odd numbers are numbers not evenly divisible by 2 or that leave a remainder of 1 when
divided by 2..
Composite
A whole number that can be divided evenly by numbers other than 1 or itself. Example: 9 can be
divided evenly by 3 (as well as 1 and 9), so 9 is a composite number. But 7 cannot be divided evenly
(except by 1 and 7), so it is NOT a composite number (it is a prime number).
Prime
A prime number is
1--a positive number "p" that has but two positive divisors/factors, 1 and "p" or
2--any natural number larger than 1 that cannot be written as the product of two smaller
numbers.
(The strict interpretation of this definition aids in supporting the statement that the
number one is not a prime number as it is the only number with one divisor/factor
whereas a prime always has two factors.)
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc., are all primes, being evenly divisible by
only 1 and the number itself.
Integers - Any of the positive and negative whole numbers, ..., -3, -2, -1, 0, +1, +2, +3, ... The
positive integers, 1, 2, 3..., are called the natural numbers or counting numbers. The set of
all integers is usually denoted by Z or Z+
Digits - the 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, used to create numbers in the base 10
decimal number system.
Numerals - the symbols used to denote the natural numbers. The Arabic numerals 0, 1, 2, 3,
4, 5, 6, 7, 8, 9 are those used in the Hindu-Arabic number system to define numbers.
Natural Numbers - the set of numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,.....,
that we see and use every day. The natural numbers are often referred to as the counting
numbers and the positive integers.
Whole Numbers - the natural numbers plus the zero.
Rational Numbers - any number that is either an integer "a" or is expressible as the ratio of
two integers, a/b. The numerator, "a", may be any whole number, and the denominator,
"b", may be any positive whole number greater than zero. If the denominator happens to be
unity, b = 1, the ratio is an integer. If "b" is other than 1, a/b is a fraction.
Fractional Numbers - any number expressible by the quotient of two numbers as in a/b, "b"
greater than 1, where "a" is called the numerator and "b" is called the denominator. If "a" is
smaller than "b" it 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.
Irrational Numbers - any number that cannot be expressed by an integer or the ratio of two
integers. Irrational numbers are expressible only as decimal fractions where the digits
continue forever with no repeating pattern. Some examples of irrational numbers
are
.
Transcendental Numbers - any number that cannot be the root of a polynomial equation
with rational coefficients. They are a subset of irrational numbers examples of which are Pi
= 3.14159... and e = 2.7182818..., the base of the natural logarithms.
Real Numbers - the set of real numbers including all the rational and irrational numbers.
Irrational numbers are numbers such as
Rational numbers include the whole numbers (0, 1, 2, 3, ...), the integers (..., - 2, - 1, 0, 1, 2,
...), fractions, and repeating and terminating decimals.
Draw a line.
Put on it all the whole numbers 1,2,3,4,5,6,7.....etc
then put 0
then put all the negatives of the whole numbers to the left of 0
........-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0
Then put in all of the fractions .
Then put in all of the decimals [ some decimals aren't fractions ]
Now you have what is called the "real number line"
The way to get a number that is not "real" is to try to find the square root of - 1
can't be 1 because 1 squared is 1, not -1
can't be -1 because the square of -1 is 1, not -1
So there is no number on your number line to be
be put somewhere.
and new numbers would need to
PALINDROMIC NUMBERS
A palindrome is a number that reads the same backwards as forwards, such as 121, 363,
3883, 37973, to mention a few.
Numbers – Advanced / MACC only!
SQUARE NUMBERS +
A square number, sometimes called a perfect square, is the result of multiplying a number
by itself as in N = nxn = n^2.
The perfect squares are the squares of the counting numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, etc. which produce 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc.
Believe it or not, the perfect squares are simply the successive sums of the odd numbers.
0+1=1
1+3=4
1+3+5=9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25, etc.
CUBIC NUMBERS or CUBES
A cubic number is the third power of a number as in a x a x a = a3. Those familiar with the
evolution of the squares from adding successive odd numbers might not be too surprised to
discover how the cubes evolve from summing odd numbers also. Clearly, the nth cube is
simply n3. Cubes can be derived in other ways also:
The cube of any integer, "n", is the sum of the series of odd numbers beginning with (n2 - n
+ 1) and ending with (n2 + n - 1). Example: For n = 6, (n2 - n + 1) = 31 and 31 + 33 + 35 + 37
+ 39 + 41 = 216 = 63.
n3 (n)3
n3
(n2 - n + 1)
(n2 + n - 1)
1
13
= 1
1
=
1(1)
2
23
= 8
3+5
=
2(1 + 2 + 1)
3
33
= 27
7 + 9 + 11
=
3(1 + 2 + 3 + 2 + 1)
4
43
= 64
13 + 15 + 17 + 19
=
4(1 + 2 + 3 + 4 + 3 + 2 + 1)
5
53
= 125 21 + 23 + 25 + 27 + 29 = 5(1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1)
Starting with
1....2....3....4....5....6....7....8....9....10....11....12....13....14....15....16....17....18....19....20....21
Cross out every third number giving us
1....2..........4....5.........7.....8..........10....11............13....14............16....17............19....20
Summing
1...3...........7...12.......19...27.........37....48............61....75...........91...108...........127..147
Cross out every second number giving us
1................7..............19................37....................61...................91....................127
Summing
1................8..............27................64..................125.................216....................343
The last line of numbers are the perfect cubes.
The first ten cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.
The sum of the first n cubes starting with 1, is 13 + 23 + 33 +........+ n3.
which, quite surprisingly, is the square of the nth triangular number, defined by Tn =
n(n+1)/2.
The sum of the cubes of the first n odd numbers is 2n4 - n2 = n2(2n2 - 1).
The sum of the cubes of the first n even numbers is 2n4 + 4n3 + 2n2 = 2n2(n + 1)2.
The sum of the first n cubes, 13 + 23 + 33 + 43 +........+ n3 is equal to the square of the sum of
the first n integers. Thus, 13 + 23 + 33 + 43 +........+ n3 = (1 + 2 + 3 + 4 +........+n)2.
The cube of any integer is the difference of the squares of two other integers.
Every cube is either a multiple of 9 or right next to one.
A perfect cube can end in any of the digits 0 through 9.
Three digit numbers that are the sum of the cubes of their digits: 153, 370, 371, 407.
The smallest number that is the sum of 2 cubes in two different ways. 1729 = 13 + 123 =
103 + 93.
What are the dimensions of two cubes with integral sides that have their combined volume
equal to the combined length of their edges. What are the dimensions of the cubes?
x
= 2 and y = 4.
The sum of any two cubes can never be a cube.
The sum of a series of three or more cubes can equal a cube.
Examples:
33 + 43 + 53 = 63
113 + 123 + 133 + 143 = 203
11343 + 11353 + .........21333 = 16,8303
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.
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.
TRIANGULAR NUMBERS
Triangular numbers, members of the class of figurate, or 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. The
number of dots, circles, etc., that can be arranged in a line is called a linear number. The
number of dots, circles, etc., that can be arranged in an equilateral or right triangular
pattern is called a triangular number. The 10 bowling pins form a triangular number as do
the 15 balls racked up on a pool table. Upon further inspection, it becomes immediately
clear that the triangular numbers, T1, T2, T3, T4,.....Tn, are simply the sum of the
consecutive integers 1-2-3-4-.....n or Tn = n(n + 1)/2, namely, 1, 3, 6, 10, 15, 21, 28, 36, 45,
55, 66,78, 91, etc.
FULL LIST OF NUMBERS
Abundant, Algebraic, Amicable, Arrangement, Automorphic, Binary, Cardinal, Catalan,
Complex, Composite, Congruent, Counting, Cubic, Decimal, Deficient, Even, Factor,
Factorial, Fermat, Fibonacci, Figurate, Fractional, Friendly, Generating, Gnomon, Golden,
Gyrating, Happy, Hardy-Ramanujan, Heronian, Imaginary, Infinite, Integers, Irrational,
Mersenne, Monodigit, Narcissistic, Natural, Oblong, Octahedral, Odd, Ordinal, Parasite, Pell,
Pentatope, Perfect , Persistent, Polygonal, Pronic, Pyramidal , Pythagorean, Quasiperfect,
Random, Rational, Real, Rectangular, Relatively Prime, Semi-perfect, Sequence, Sociable,
Square, Superabundant, Tag, Tetrahedral, Transcendental, Triangular, Unit Fraction,
Whole.