Number Systems - www.TheMathWebSite.com

### Standard Numbers

Natural 1, 2, 3, 4, 5, 6, 7, ... 0,1, 2, 3, 4, 5, 6, 7, ... ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ..., Rational numbers look like fractions, a/b where a and b are integers and b is not zero Real numbers include rational numbers, such as 4 and -2/9, irrational numbers, such as pi and the square root of two, and infinite decimals such as 3.465453339..., where the digits continue indefinitely. The real numbers are sometimes thought of as points on an infinitely long number line. An imaginary number is a number that when squared gives a negative result. The form bi where b is a non-zero, real number and i, defined by i2 = - 1, is known as the imaginary unit. Imaginary numbers can be thought of as complex numbers where the real part is zero. A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = -1.
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### INFINITY.....

Infinity is from a Latin word which literally means that which is unlimited or unbounded. Originally it was applied to things that were unmeasureably large. It has come to mean the largest number imaginable. The origin of the theory of limits has also led to the need for a word that expressed the idea of things growing smaller and smaller without bound.

The symbol we now use for infinity, was first used by John Wallis (1616-1703) in 1655. Why he used it seems lost to history. The Late Romans used a symbol like two hooked together zeros, 00, for the number 1000. Or perhaps, it is a variant of the lowercase symbol for Omega, the last letter in the Greek alphabet, to symbolize the "final number" in a sense.

There was a young fellow from Trinity
Who took the square root of infinity
But the number of digits,
Gave him the fidgets;
He dropped Math and took up Divinity.
-- George Gamow

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### Pi = 3.14159...

That the ratio of the circumference to the diameter of a circle is constant (namely, pi) has been recognized for as long as we have written records.

A ratio of 3:1 appears in the following biblical verse: And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23; II Chronicles 4, 2.)

The ancient Babylonians generally calculated the area of a circle by taking 3 times the square of its radius (=3), but one Old Babylonian tablet (from ca. 1900-1680 BCE) indicates a value of 3.125 for pi.

The first theoretical calculation of a value of pi was that of Archimedes of Syracuse (287-212 BCE), one of the most brilliant mathematicians of the ancient world. Archimedes worked out that 223/71 < pi < 22/7. Archimedes's results rested upon approximating the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed. Beginning with a hexagon, he worked all the way up to a ploygon with 96 sides!

European mathematicians in the early modern period developed new arithmetical formulae to approximate the value of pi, such as that of James Gregory (1638-1675), which was taken up by Leibniz as :
pi/4 = 1 - 1/3 + 1/5 - 1/7 . . . .

The symbol for pi was introduced by the English mathematician William Jones in 1706. This symbol was adopted by Euler in 1737 and became the standard symbol for pi.

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### Roman Numerals

The big difference between Roman and the Arabic numerals used today is that Romans did not have a symbol for zero.
 I The easiest way to note down a number is to make that many marks. Thus I is 1, II is 2, III is 3. However, four strokes was too many. V V is 5. Placing I in front of the V indicates subtraction. Thus IV is 4. Placing any smaller number after the V indicates addition. Thus VI is 6, VII is 7, VIII is 8. X X is 10. Thus IX is 9, XI is 11, XXXI is 31, XXIV is 24. L L is 50. Thus XL is 40, LX is 60, LXX is 70, LXXX is 80. C C stands for centum, the Latin word for 100. A centurion led 100 men. We still use this in words like "century" and "cent." The subtraction rule means XC is 90. Thus by their method CCCLXIX is 369. D D is 500. Thus CD is 400, CDXLVIII is 448. M M is 1,000. Roman numerals are used a lot to indicate dates. Thus 1998 is MCMXCVIII.
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### Tautonymic Numbers

A Tautonymic number is one which can be broken into two equal non-palindromic halves and with each part having at least two different digits.
Examples of tautonymic numbers are 5656, 2525, 3737, 4141, 165165, 34723472 etc..

How many bygone tautonymic years can you write down?

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A Tetradic Number is one which is both strobogrammatic and palindromic in nature. It is the same when viewed from left to right, right to left, top to bottom or upside down. This four-way symmetry explains the name, tetra- being the greek prefix for four.
The only digits that can be found in a tetradic number are 0, 1 and 8, since although matched pairs of 6 and 9 can be used in strobogrammatic numbers, they won't yield a palindrome. Thus the first few tetradic numbers are 0,1,8,11,88,101.
Given a tetradic number, a larger one can always be generated by adding another tetradic number to each end, retaining the symmetry. There are tetradic primes, the first half dozen being 11, 101, 181, 18181, 1008001, and 1180811.

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### Strobogrammatic Numbers

A Strobogrammatic Number is a number that appears the same whether viewed normally or upside down. In base 10, given that 0, 1 and 8 are symmetrical around the horizontal axis, and 6 and 9 are the same as each other upside down, the first few strobogrammatic numbers are:
1, 8, 11, 69, 88, 96, 101,
111, 181, 609, 619, 689, 808,
818, 888, 906, 916, 986, 1001

What years are strobogrammatic?

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### Kaprekar Number

The Kaprekar Number is 6174.
(1) Take any four-digit number.
(2) Form the largest and the smallest numbers from these four digits.
(3) Find the difference between those digits. Maybe this is 6174.
If it is not, form the largest and the smallest number from the difference and subtract these numbers again. You may have to repeat this procedure.
The end result is always 6174, with no more than 7 steps.

Take the number 3546.
1st step: 6543 - 3456 = 3087
2nd step: 8730 - 0378 = 8352
3rd step: 8532 - 2358 = 6174

Take the number 5184.
1st step: 8541 - 1458 = 7083
2nd step: 8730 - 0378 = 8532
3rd step: 8532 - 2358 = 6174

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### Dudeney Number

A Dudeney number is a positive integer that is a perfect cube such that the sum of its decimal digits is the cube root of the number. The name derives from Henry Dudeney, who noted the existence of these numbers in one of his puzzles.
There are exactly six Dudeney numbers.

1 = 1 x 1 x 1 ; 1 = 1
512 = 8 x 8 x 8 ; 8 = 5 + 1 + 2
4913 = 17 x 17 x 17 ; 17 = 4 + 9 + 1 + 3
5832 = 18 x 18 x 18 ; 18 = 5 + 8 + 3 + 2
17576 = 26 x 26 x 26 ; 26 = 1 + 7 + 5 + 7 + 6
19683 = 27 x 27 x 27 ; 27 = 1 + 9 + 6 + 8 + 3

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### Fibonacci Numbers

The Fibonacci Number Sequence was first presented in Leonardo Pisano's book, "Liber abaci" or "Book of Calculating". It is a sequence that I find to be very fascinating, and suprisingly it is a part of every day nature.

The Fibonacci sequence can be found in sea shell spirals, branching plants, petals on flowers, and in pine cones.

The Basic Sequence
The first twenty numbers of the sequence are as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181

The numbers are obtained by adding two numbers to get the next.
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### Square Numbers

Square array of dots, probably formed with pebbles, led the Greeks to numbers that were perfect squares- that is to numbers which, when expressed in a various of ways as the products of two numbers, would have two equal factors.
The most complete discussion of square numbers was given by a Greek, Nicomachus of Gerasa (c. A.D. 100) in Introdictio Arithmetica, the earliest extant manuscript, dating back to the tenth century. Nicomachus was not the original mathematician, but he did organize previous generations of mathematics in a clear and precise manner. The first 10 square numbers-

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Each is a result of multiplying a number by itself-

1*1, 2*2, 3*3, 4*4, 5*5 ...

which can also be written -

12 , 22 , 32 , 42 ...

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### Triangular Numbers

A triangular number is the number of dots in an equilateral triangle evenly filled with dots. For example, three dots can be arranged in a triangle; thus three is a triangle number. The nth triangle number is the number of dots in a triangle with n dots on a side. A triangle number is, equivalently, the sum of the n natural numbers from 1 to n.

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
...

So the nth triangular number can be obtained as Tn = n*(n+1)/2, where n is any natural number.
In other words triangular numbers form the series 1,3,6,10,15,21,28.....

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### Prime Numbers

A natural number that possesses only two factors, itself and 1, is called a prime number. It was an early Greek mathematician that is most famous for his work with prime numbers. His name was Eratosthenes. It was this Greek mathematician that created what is known as the Sieve of Eratosthenes. This Sieve allows anyone to find all of the prime numbers quite easily. There are infinitely many primes. We know that there is an infinite number of primes because if you were to multiply all of the known primes together and add 1, then you would get a number that must be divisible by at least one new prime number. Prime numbers are considered the "building blocks" of the natural numbers because every single natural number, excluding the number 1, is either a prime number or a product of prime numbers.
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### Palindromic

A palindromic number is a 'symmetrical' number like 16761, that remains the same when its digits are reversed. The first palindromic numbers are -
11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, …

Some numbers possess a certain property and are palindromic. For instance,

• the palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, …
• the palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, …
Special Numbers - www.TheMathWebSite.com

### Amicable Numbers

Throughout history there have been many different interesting numbers or types of numbers. One of these types is amicable numbers. Amicable numbers are a pair of numbers with the following property: the sum of all of the proper divisors of the first number (not including itself) exactly equals the second number while the sum of all of the proper divisors of the second number (not including itself) likewise equals the first number.

For example let's show that 220 & 284 are amicable numbers:
First we find the proper divisors of 220:

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

If you add up all of these numbers you will see that they sum to 284.

Now find the proper divisors of 284:

1, 2, 4, 71, 142

These sum to 220, and therefore 220 & 284 are amicable numbers.

The set of 220 and 284 was the first known set of amicable numbers. Pythagoras discovered the relationship and coined the term amicable because he considered the numbers to be a symbol of friendship. No other pairs were known until 1636 when Fermat discovered 17,296 and 18,416 as a second pair. This pair was actually discovered over three hundred years earlier by the Arab mathematician al-Banna, but it was never known in the West until Fermat's findings. Then in 1638, Descartes discovered a third pair of 9,363,584 and 9,437,056.

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### Narcissistic Numbers

A narcissistic number is a number that is the sum of its own digits each raised to the power of the number of digits.
There are just four numbers, after unity, which are the sums of the cubes of their digits:
153 = 13 + 53 + 33
370 = 33 + 73 + 03
371 = 33 + 73 + 13
407 = 43 + 03 + 73
.........
abc = a3 + b3 + c3
abcd = a4 + b4 + c4 + d4.

 Base-10 Narcissistic Numbers 1-digit number ... 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 2-digit number ... none 3-digit number ... 153, 370, 371, 407 4-digit number ... 1634, 8208, 9474 5-digit number ... 54748, 92727, 93084
Special Numbers - www.TheMathWebSite.com

### ZERO

It's hard to believe that most ancient number systems didn't include zero. The Mayan civilization in Mexico about 1300 years ago may have been among the first to have a symbol for zero. It is considered one of their cultures greatest achievements.

The ancient Egyptians, Romans, and Greeks also had no symbol for zero. The Greeks made great strides in mathematics, but it was all done with a number system without zero. The Greek astronomer Ptolemy (ca. A.D. 150) was the first to write a zero at the end of a number. For this he used a circular symbol.

In ancient Babylonian history there was no use of the zero. Up until the time of Aristotle, there seems to be no evidence that the Babylonians ever regarded zero as a number.

Throughout the Dark Ages, Western mathematics was held back by the Roman's traditional numbering system. The first to think differently was Leonardo Fibonacci.

In the sixth century, mathematicians in India developed a place-value system. They introduced the concept of zero to keep their symbols in their proper places. In the seventh century, Hindu scholars introduced to Islam the ideas of zero and place-value. These ideas spread rapidly throughout the Arabic world. By the fifteenth century, the numerals were showing up on coins and gravestones. Western mathematics had emerged from the Dark Ages, and was flourishing into a new number system with a zero, the Hindu-Arabic numerals. The immediate advances in mathematics after that time are proof of the importance of, the zero.

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### Chinese Numbers

The ancient numeral system is the written numbers system. It is still in use when writing numbers in long form, such as on cheques. This character system is roughly analogous to spelling out a number in English text. The Chinese character system can be classified as part of the language, but it still counts as a number system. Most people in China now use the Arabic system for convenience.

Special Numbers - www.TheMathWebSite.com

### Binary Numbers

The Binary number system presents its values using only two symbols, 0 & 1. Since binary is a base-2 system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. To determine the decimal representation of a binary number simply take the sum of the products of the binary digits and the powers of 2 which they represent.

For example, the binary number 100101 is converted to decimal form by adding
[(1) × 25] + [(0) × 24] + [(0) × 23] + [(1) × 22] + [(0) × 21] + [(1) × 20]
= [1 × 32] + [0 × 16] + [0 × 8] + [1 × 4] + [0 × 2] + [1 × 1] = 37

Similarly the number 101011012 has
1 x 27 = 128
0 x 26 = 0
1 x 25 = 32
0 x 24 = 0
1 x 23 = 8
1 x 22 = 4
0 x 21 = 0
1 x 20 = 1
which sums to 173 in decimal notation.   Thus, 101011012 = 17310

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### Octal Numbers

Octal Numbers are similar in principle to the hexadecimal numbering system except that in Octal a binary number is divided up into groups of only 3 bits, with each group or set of numbers having a distinct value of between 000 and 111 giving a range of just 8, (0, 1, 2, 3, 4, 5, 6, 7).

To count above 7 in octal we add another column and start over again in a similar way to hexadecimal. 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21....etc.

Convert the Binary number 11010101110011112 into its Octal equivalent by first dividing into groups of three bits, 001 101 010 111 001 111 and then replacing each three bit
group with its Octal equivalent, 1 5 2 7 1 78.

Thus, 0011010101110011112 in Binary form
is equivalent to 1527178 in Octal form
or 54,735 in Decimal form.
Special Numbers - www.TheMathWebSite.com

Hexadecimal is a numeral system with a base of 16.
It uses sixteen distinct symbols, the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen. Each hexadecimal digit represents four binary digits (bits). Large binary numbers are split up into groups of four bits to make them easier to write down and understandable. Thus 1101 0101 1100 1111 is much easier to read and understand than 1101010111001111.

To convert Binary 1110 1010 into its Hex equivalent just convert each group of four bits into its Hex equivalent.   1110 1010 = E A

Hex number 2AF3 is equal to Decimal number 10,995
= (2 × 163) + (10 × 162) + (15 × 161) + (3 × 160).

The binary number 1101 0101 1100 11112 is equivalent to the hexadecimal number of D5CF16.

Convert Hex number 3FA716 into its Binary and Decimal equivalent
(using subscripts to identify each numbering system).
3FA716 = 0011 1111 1010 01112
= (8192 + 4096 + 2048 + 1024 + 512 + 256 + 128 + 32 + 4 + 2 + 1)
= 16,29510
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