Number Systems

Is it a number?

2005

No!! No! It is not a number!! It is a representation of a number. Then, what information is restored in the representation 2005?

What is the definition of a number? In ordinary English, the word "number" has two different meanings (at least). One meaning is the concept, the other meaning is how the concept is represented. For example, think about "the number of eggs in an egg carton" You know the number I mean. Now write it down on paper.

Unit and NumberThe terms unit and number when used with the decimal system are almost self-explanatory. By definition the unit is a single object; that is, an apple, a dollar, a day. A number is a symbol representing a unit or a quantity. The figures 0, 1, 2, and 3 through 9 are the symbols used in the decimal system. These symbols are called Arabic numerals or figures. Other symbols may be used for different number systems. For example, the symbols used with the Roman numeral system are letters V is the symbol for 5, X for 10, M for 1,000, and so forth. We will use Arabic numerals and letters in the number system discussions in this chapter.

Another definition of numberThe idea of number is a mathematical abstraction. The word "number" is a general term which refers to a member of a given (possibly ordered) set. The meaning of "number" is often clear from context (i.e., does it refer to a complex number, integer, real number, etc.?). Wherever possible in this work, the word "number" is used to refer to quantities which are integers, and "constant" is reserved for nonintegral numbers which have a fixed value. Because terms such as real number, Bernoulli number, and irrational number are commonly used to refer to nonintegral quantities, however, it is not possible to be entirely consistent in nomenclature.

Number representations There are many types of representations. In some representations position is not strictly important. In some representation position is strictly important. These types of representation are called positional notations.

Positional notation

In positional notation each digit of a multidigit number has a fixed value (weight) determined by its position.

200510 is a decimal representation of a number. This number could also be represented as two thousand and five or, two thousand-five. MMV (Roman Notation) and so on.

In how many ways a number can be represented?

XII, ///// ///// //, 11002 and 12 are different ways to represent the same number.

Early number representation.1 Egyptian hieroglyphics 2500 B.C. How to represent 1999

Early number representation.2 How do you represent 1999? This representation is additive. Position is not strictly important.

Early number representation.3 Similar to Roman Numerals.. 1999 MDCCCCLXXXXVIIII VLD 5, 50, 500

Subtractive notation Position becomes important MCMXCIX M = 1000 CM = M-C = 1000 100 = 900 XC = C-X = 500 10 = 490 C = 50 IX = X-I = 9

Early number representation.4 Based on powers of 10 up to 106 How many of each symbol is needed?

Numerals We use Arabic numerals in decimal system. These are 0,1,2,3,4,5,6,7,8 & 9. These are also called digits. A number x is said to have "n figures" if it takes n digits to express it. In systems with a radix larger than 10 A,B,C,D, are used to represent 10,11,12,13, .

Number System The idea of number is a mathematical abstraction. To use numbers, we must represent them in some way, whether by piles of pebbles or in some other way. It is common to create a code for representing numbers. Such codes are called number systems. Every number system uses symbols to convey information about the value of a number.

A positional (or radix) number system is one in which the value that a symbol contributes to a number is determined by the both symbol itself and the position of the symbol within the number. That's just a fancy way of saying that 300 is far different from 3. Compare the idea of a positional number system with Roman numbers, where X means 10 no matter where in a number it appears. This is often referred to as a weighted numbering system because each position is weighted by a power of the radix.

Positional Number Systems

Numbers Positional Notation Decimal Every positive integer has a unique representation in the form ak ak-1 a0 Where ak ak-1 a0 = ak x 10k + ak-1 x 10k-1 + + a0 x 100 with 0 < ak < 10 and 0 ai < 10 for 0 i < k

Numbers Positional Notation Number representation d31d30 ... d2d1d0 is a 32 digit number value = d31x B31 + d30 x B30 + ... + d2 x B2 + d1 x B1 + d0 x B0

Binary Example1011010 = 1x26 + 0x25 + 1x24 + 1x23 + 0x22 + 1x2 + 0x1 = 64 + 16 + 8 + 2 = 90

What is a base that converts to binary easily?

What is radix? The base of a number system, i.e., 2 for binary, 8 for octal, 10 for decimal, and 16 for hexadecimal. The radix is sometimes called the base or scale.

More about Base.1 A real number x can be represented using any integer number b as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base b, the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ...are generally used as symbols representing the decimal numbers 10, 11, 12, ...).

More about Base.2 The base, or radix, of a number system tells you the number of symbols used in that system. The base of any system is always expressed in decimal numbers. The base of a number system is indicated by a subscript (decimal number) following the value of the number. The following are examples of numerical values in different bases with the subscript to indicate the base: 759210 2145 or, (123)4 (656)7

Number 10

= = = = = = = = = =

Representation 10102 1013 224 205 146 137 128 119 1010 A11

Base 2 3 4 5 6 7 8 9 10 11

Names of some number systems with different base Binary(2) Ternary(3) Quaternary(4) Quinary(5) Senary(6) Septenary(7) Octal(8) Nonary(9) Decimal(10) Undenary(11) Duodecimal(12) Hexadecimal(16) Vigesimal(20) Sexagesimal(60)

Sys. Name Binary Ternary Octal Decimal

Base Digits Employed 2 3 8 10 0,1 0,1,2 0,1,2,3,4,5,6,7 0,1,2,3,4,5,6,7,8,9 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, F 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, F,G,H,I,J

Hexadecimal 16 Vigesimal 20

Some Facts About Num. Systems Binary is used in representing data and in operational activities of computers. Ternary numbers arise in a number of problems in mathematics, including some problems of weighing. A ternary representation can be used to uniquely identify totalistic cellular automaton rules, where the three colors (white, gray, and black) correspond to the three numbers 0, 1 and 2 (Wolfram 2002, pp. 60-70 and 886).

Some Facts About Num. Systems We use decimal system in real life. Some aspects of a base-12 system are preserved in the terms dozen and gross.

Some Facts About Num. Systems The hexadecimal system is particularly important in computer programming, since four bits (each consisting of a one or zero) can be succinctly expressed using a single hexadecimal digit. Two hexadecimal digits represent numbers from 0 to 255, a common range used, for example, to specify colors. Thus, in the HTML language of the web, colors are specified using three pairs of hexadecimal digits RRGGBB, where RR is the amount of red, GG the amount of green, and BB the amount of blue.

Some Facts About Num. Systems A base-20 number system was used by the Aztecs and Mayans. The Mayans compiled extensive observations of planetary positions in base-20 notation. A base-60 number system was used by the Babylonians and is preserved in the modern measurement of time (hours, minutes, and seconds) and angles (degrees, arc minutes, and arc seconds).

Mixture of Bases Some number systems use a mixture of bases for counting. Examples include the Mayan calendar and the old British monetary system (in which ha'pennies, pennies, threepence, sixpence, shillings, half crowns, pounds, and guineas corresponded to units of 1/2, 1, 3, 6, 12, 30, 240, and 252, respectively).

Irrational Base Bergman (1957/58) considered an irrational base, and Knuth (1998) considered transcendental bases. This leads to some rather unfamiliar results, such as equating to 1 in "base ," =10. Even more unexpectedly, the representation of a given integer in an irrational base may be nonunique, for example

Num. 10 = =

Rep. 10100.010010101011 10100.0101

Base 18 19

Where is the golden ratio.

Index of the leading digit The index of the leading digit needed to represent the number is

where

is the floor function.

Now, recursively compute the successive digits

Successive Digits

where and for i = n, n-1, ..., 1, 0, .... This gives the base b representation of x. Note that if x is an integer, then i need only run through 0, and that if x has a fractional part, then the expansion may or may not terminate. For example, the hexadecimal representation of 0.1 (which terminates in decimal notation) is the infinite expression

Number length.1 The length of a number n in base b is the number of digits in the base-b numeral for n, given by the formula

where is the floor function. The multiplicative persistence of an n-digit is sometimes also called its length.

Number Length.2 If L is the number length, R is the radix & N is the number, then the maximum number that could be represented is, N=RL 1 By simplifying this equation we get,

This is a ceiling function.

Significant Digits & Insignificant DigitsDigits whose removal changes the value of the number are significant. Digits whose removal dont changes the value of the number are insignificant.

Most Significant Bit Most Significant Bit (MSB) is also known as high-order bit. Left most, Highest order Bit n-1 d31d30 ... d2d1d0

Note : Bit. Not digit.

Least Significant Bit Least Significant Bit (LSB) is also known as low-order bit. Right most, Lowest-order Bit 0 d31d30 ... d2d1d0

What is an integer, a fraction or a mixed number ?

Radix Point The period placed to the right of the units place in a decimal number is called the decimal point (or, in the case that a comma is instead used for this purpose as is the case in continental Europe, the decimal comma).

Fixed Point Number

Floating Point Number

What we use? In real life we use decimal number system. Its easy for human. Easy to start with. Easy to get skilled in. Especially when doing arithmetic.

Which is more convenient for computers? For computers binary system is used. Most information-processing systems are constructed from switches, which are binary devices. The basic decision-making processes required of digital systems are binary. Binary signals are more reliable than those formed by more than two quantization levels.

Why Binary not Hexadecimal?(1) Hexadecimal notation is clearly the most compact of the three ways to represent numbers. Here are those larger numbers: 110,0100 binary is100 decimal and 64 hexadecimal 1111,1111,1111,1111,1111,1111 binary is 16,777,215 decimal and FFFFFF hexadecimal. It is these six hexadecimal digits that are used to represent the 16+million colors in HTML.

Why Binary not Hexadecimal?(2)In everyday life, we use the base 10 decimal number system. But the basic number system in computers is base two, or binary. This is because of mathematician John V. Atanasoff's decision in 1938 to try building an all-electronic computing device using the simplest numbering system available. Base two has only two digits: 0 and 1. These can easily be represented in many devices, and from Atanasoff's successful experiment using vacuum tubes has come our entire world of digital electronics.

Decimal Expansion & fraction