Numeral System: Context and Usage

A numeral system (or system of numeration) is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols “11” to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. Ideally, a numeral system will: * Represent a useful set of numbers (e.

g. ll integers, or rational numbers) * Give every number represented a unique representation (or at least a standard representation) * Reflect the algebraic and arithmetic structure of the numbers. b – numeral system base dn – the n-th digit n – can start from negative number if the number has a fraction part. N+1 – the number of digits Base (Radix):- The total number of distinct symbols used by a number system called base or radix. E. g. : Decimal number system uses ten distinct symbols: 0,1,2,3,4,5,6,7,8,9.

Existing Numeral Systems

Decimal number system:-The decimal numeral system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by modern civilizations. [1][2] Decimal notation often refers to a base-10 positional notation such as the Hindu-Arabic numeral system Decimal number system uses ten distinct symbols: 0,1,2,3,4,5,6,7,8,9.

Binary number system:- In mathematics and computer science, the binary numeral system, or base-2 numeral system, represents numeric values using two symbols: 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2.

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Numbers represented in this system are commonly called binary numbers. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computer-based devices such as mobile phones. E. g. 101012 = 10101B = 1? 24+0? 23+1? 22+0? 21+1? 20 = 16+4+1= 21 101112 = 10111B = 1? 24+0? 23+1? 22+1? 21+1? 20 = 16+4+2+1= 23 1000112 = 100011B = 1? 25+0? 24+0? 23+0? 22+1? 21+1? 20 =32+2+1= 35 Octal number system: The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7.Octal numerals can be made from binary numerals by grouping consecutive binary digits into groups of three (starting from the right). For example, the binary representation for decimal 74 is 1001010, which can be grouped into (00)1 001 010 – so the octal representation is 112. In the decimal system each decimal place is a power of ten. For example: In the octal system each place is a power of eight. For example: By performing the calculation above in the familiar decimal system we see why 112 in octal is equal to 64+8+2 = 74 in decimal. E. g. 278 = 2? 1+7? 80 = 16+7 = 23 308 = 3? 81+0? 80 = 24 43078 = 4? 83+3? 82+0? 81+7? 80 = 2247

Hexadecimal Number System: In mathematics and computer science, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A,? B,? C,? D,? E,? F (or alternatively a–f) to represent values ten to fifteen. For example, the hexadecimal number 2AF3 is equal, in decimal, to (2??? 163) + (10??? 162) + (15??? 161) + (3??? 160), or 10995. Each hexadecimal digit represents four binary digits (bits), and the primary use of hexadecimal notation is a human-friendly representation of binary-coded values in computing and digital electronics. One hexadecimal digit represents a nibble, which is half of an octet or byte (8 bits). For example, byte values can range from 0 to 255 (decimal), but may be more conveniently represented as two hexadecimal digits in the range 00 to FF. Hexadecimal is also commonly used to represent computer memory addresses. Examples: 2816 = 28H = 2? 161+8? 160 = 40 2F16 = 2FH = 2? 161+15? 60 = 47 BC1216 = BC12H = 11? 163+12? 162+1? 161+2? 160 = 48146 Numeral Systems Conversion Table Decimal Base-10| BinaryBase-2| OctalBase-8| HexadecimalBase-16| 0| 0| 0| 0| 1| 1| 1| 1| 2| 10| 2| 2| 3| 11| 3| 3| 4| 100| 4| 4| 5| 101| 5| 5| 6| 110| 6| 6| 7| 111| 7| 7| 8| 1000| 10| 8| 9| 1001| 11| 9| 10| 1010| 12| A| |

ASCII Code: Ascii stands for American standard code for information interchange. This code uses 7 bits to represent any character. These 7 bits lead to128 codes. There are two types of character coded by ACII printable and nonprintable control characters.The printable character include digits from 0 to 9, The upper case and lowercase English letters from A to Z, mathematical operation +,-,*, etc punctuation marks and various special symbols like $, @, #, & etc. Non-printable characters are those which do not provide and display on the screen. These codes are used to provide any control information to the computer system. For example end of line, end of pages, start of text, end of text, carriage return etc are nonprintable control character coded by the ASCII code.These codes are extensively used by printers, terminals and keyboard 7 bit code enters inside the computer circuit. E. g. HI FI This string has five characters: H, I, SPACE, F and I Therefore 5*7=35 bits will needed in computer to represent this character string.

GRAY CODES: Gray code is used in shaft encoder which indicates the angular position of shaft in digital form. Binary bits are arranged in such a way that only one binary bit changes at a time when we make a change from any number to the next. Its use reduced the error in the reading shaft position.The largest possible error will be one least significant digit. Suppose that the shaft changes it position from 7 to 8, the gray code will change from 0100 to 1100. If change in the binary bit is not picked up by the sensor, the encoder will show the previous position that is 7. But in case of standard binary code 7 is represented by 0111 and by 1000. If sensor picks up the least significant bits that is 000, but fails to pick up the most significant bit 1, the output will be 0000 instead of 1000 . there will be large error if standard binary code is used for shaft encoding.In this case an error of the most significant bit may occur . the Gray code is often used in computer controlled machines such as lathes etc. Addition The circuit diagram for a binary half adder, which adds two bits together, producing sum and carry bits. The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying: 0 + 0 > 0 0 + 1 > 1 1 + 0 > 1 1 + 1 > 0, carry 1 (since 1 + 1 = 0 + 1 ? binary 10) Adding two “1” digits produces a digit “0”, while 1 will have to be added to the next column.This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented: 5 + 5 > 0, carry 1 (since 5 + 5 = 10 carry 1) 7 + 9 > 6, carry 1 (since 7 + 9 = 16 carry 1) This is known as carrying. When the result of an addition exceeds the value of a digit, the procedure is to “carry” the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix.Carrying works the same way in binary: 1 1 1 1 1 (carried digits) 0 1 1 0 1 + 1 0 1 1 1 ————- = 1 0 0 1 0 0 = 36 In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 102. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 102 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 112. This time, a 1 is carried, and a 1 is written in the bottom row.Proceeding like this gives the final answer 1001002 (36 decimal). When computers must add two numbers, the rule that: x xor y = (x + y) mod 2 for any two bits x and y allows for very fast calculation, as well. A simplification for many binary addition problems is the Long Carry Method or Brookhouse Method of Binary Addition. This method is generally useful in any binary addition where one of the numbers has a long string of “1” digits. For example the following large binary numbers can be added in two simple steps without multiple carries from one place to the next. 1 1 1 1 1 1 1 1 (carried digits) (Long Carry Method) 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 + 1 0 1 0 1 1 0 0 1 1 Versus: + 1 0 1 0 1 1 0 0 1 1 add crossed out digits first ———————– + 1 0 0 0 1 0 0 0 0 0 0 = sum of crossed out digits = 1 1 0 0 1 1 1 0 0 0 1 ———————– now add remaining digits 1 1 0 0 1 1 1 0 0 0 1 In this example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 02 (95810) and 1 0 1 0 1 1 0 0 1 12 (69110). The top row shows the carry bits used.Instead of the standard carry from one column to the next, the lowest place-valued “1” with a “1” in the corresponding place value beneath it may be added and a “1” may be carried to one digit past the end of the series. These numbers must be crossed off since they are already added. Then simply add that result to the uncanceled digits in the second row. Proceeding like this gives the final answer 1 1 0 0 1 1 1 0 0 0 12 (164910). Addition table | 0| 1| 0| 0| 1| 1| 1| 10| The binary addition table is similar, but not the same, as the Truth table of the Logical disjunction operation .The difference is that , while . Subtraction Subtraction works in much the same way: 0 ? 0 > 0 0 ? 1 > 1, borrow 1 1 ? 0 > 1 1 ? 1 > 0 Subtracting a “1” digit from a “0” digit produces the digit “1”, while 1 will have to be subtracted from the next column. This is known as borrowing. The principle is the same as for carrying. When the result of a subtraction is less than 0, the least possible value of a digit, the procedure is to “borrow” the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value. * * * (starred columns are borrowed from) 1 1 0 1 1 1 0 ? 1 0 1 1 1 —————- = 1 0 1 0 1 1 1 Subtracting a positive number is equivalent to adding a negative number of equal absolute value; computers typically use two’s complement notation to represent negative values. This notation eliminates the need for a separate “subtract” operation. Using two’s complement notation subtraction can be summarized by the following formula: A ? B = A + not B + 1 For further details, see two’s complement. MultiplicationMultiplication in binary is similar to its decimal counterpart. Two numbers A and B can be multiplied by partial products: for each digit in B, the product of that digit in A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used. The sum of all these partial products gives the final result. Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication: * If the digit in B is 0, the partial product is also 0 If the digit in B is 1, the partial product is equal to A For example, the binary numbers 1011 and 1010 are multiplied as follows: 1 0 1 1 (A) ? 1 0 1 0 (B) ——— 0 0 0 0 + 1 0 1 1 + 0 0 0 0 + 1 0 1 1 ————— = 1 1 0 1 1 1 0 Binary numbers can also be multiplied with bits after a binary point: 1 0 1. 1 0 1 (A) (5. 625 in decimal) ? 1 1 0. 0 1 (B) (6. 25 in decimal) ————- 1. 0 1 1 0 1 + 1 0 1 1. 0 1 + 1 0 1 1 0. 1 ———————– = 1 0 0 0 1 1. 0 0 1 0 1 (35. 15625 in decimal)See also Booth’s multiplication algorithm. Multiplication table | 0| 1| 0| 0| 0| 1| 0| 1| The binary multiplication table is the same as the Truth table of the Logical conjunction operation . Division See also: Division algorithm Binary division is again similar to its decimal counterpart: Here, the divisor is 1012, or 5 decimal, while the dividend is 110112, or 27 decimal. The procedure is the same as that of decimal long division; here, the divisor 1012 goes into the first three digits 1102 of the dividend one time, so a “1” is written on the top line.This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a “1”) is included to obtain a new three-digit sequence: 1 ___________ 1 0 1 ) 1 1 0 1 1 ? 1 0 1 —– 0 1 1 The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted: 1 0 1 ___________ 1 0 1 ) 1 1 0 1 1 ? 1 0 1 —– 0 1 1 ? 0 0 0 —– 1 1 1 ? 1 0 1 —– 1 0 Thus, the quotient of 110112 divided by 1012 is 1012, as shown on the top line, while the remainder, shown on the bottom line, is 102. In decimal, 27 divided by 5 is 5, with a remainder of 2.