Numeral Systems Rubel Biswas

Sexagesimal (base 60) It is possible to count to 12 with your thumb acting as a pointer, touching each finger bone in turn. In this system, the right hand counts repeatedly to 12 (which is called a DOZEN) The left hand counts the number of iterations until five dozens This gives 60 as full range.

Sexagesimal (base 60)

Sexagesimal (base 60) It was originated with the ancient Sumerians in the 3000 years BC. It is still used (in a modified form) for measuring time, angles, and the geographic coordinates that are angles. The number 60 has twelve factors, namely { 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 } Fractions involving sexagesimal numbers are simplified. For example, one hour can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, six minutes, five minutes, etc.

Interesting facts

Babylonian numerals Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.

Babylonian numerals

The decimal (base 10) 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. Decimal notation often refers to a base-10 positional notation such as the Hindu-Arabic numeral system. When we write numbers, the position (or "place") of each number is important. In the number 327: "7" is in the Units position, meaning just 7 "2" is in the Tens position meaning 2 tens (or twenty), "3" is in the Hundreds position, meaning 3 hundreds 327 = 3x100+ 2x10 + 7x1 = 3x102 + 2x101+ 7x100

Binary numeral system The system is base-2 number system, It 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. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers. In this system ON = 1 OFF = 0

Counting in Binary (1) Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Decimal counting uses the symbols 0 through 9, while binary only uses the symbols 0 and 1. 000, 001, 002, ... 007, 008, 009, 010, 011, 012, ...    ... 090, 091, 092, ... 097, 098, 099 100, 101, 102, ...

Counting in Binary (2) 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 10 1010 11 1011 12 1100 13 1101 14 1110 15 1111 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111 00 1 01 2 10 3 11

Converting Binary into Decimal 2 1 21 20 22 21 20 4 2 1 23 22 21 20 8 4 2 1 8 4 2 1 1 1101 = 1x23 + 1x22+ 0x21 +1x20 = = 1x8 + 1x4 + 0x2 + 1x1= 8 + 4 + 0 + 1 = 13

Converting Decimal into Binary Position 9 8 7 6 5 4 3 2 1 Value 29 28 27 26 25 24 23 22 21 20 Value 512 256 128 64 32 16 8 4 2 1 Example Suppose that we want to convert 500 into binary Start from the left, find the first number in figure above which is less that 500 Put 1 in the corresponding field 1 3. Do the following subtraction 500 – 256 = 244 4. If the result is greater than the value of the field to the right then put 1 in this field otherwise put 0 and move one step to the right. 5. Repeat this operation to the end of all fields 1

Converting Decimal into Binary Another simple method is as follows Divide the number by 2 If the remainder = 0 put 0 at the right most else put 1 Go to step 1. Example: 500 / 2 = 250 remainder = 0 250 / 2 = 125 remainder = 0 125 / 2 = 62 remainder = 1 1 62 / 2 = 31 remainder = 0 1

Binary Addition Since there two number the following is valid 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 with a carry of 1 3 + 7 = 10 In Binary carry 1 1 1 1 1 1

Binary Subtraction Following the same criteria of the addition 0 - 0 = 0 0 - 1 = 1, borrow 1 1 - 0 = 1 1 - 1 = 0 10 - 7 = 3 In Binary 1 1 1 borrow 1 1 1

Binary Multiplication Multiplication in the binary system works the same way as in the decimal system: 0 x 0 = 0 1 x 0 = 0 0 x 1 = 0 1 x 1 = 1 101 * 11 ____ _____ 1111

Binary Division Binary division is again similar to its decimal counterpart Example 27/5 = (11011)/(101) 1 ___________ 1 0 1 ) 1 1 0 1 1 − 1 0 1 ----- 0 1 1 1 0 1 ___________ 1 0 1 ) 1 1 0 1 1 − 1 0 1 ------- 0 1 1 1 - 1 0 1 --------- 1 0 Not enough, put 0 in the result and get the next number. remainder

Octal (base 8) The octal is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three (starting from the right) Binary Decimal Octal 000 001 1 010 2 011 3 100 4 101 5 110 6 111 7 1000 8 10

Binary to Octal and vice versa 1 74 Octal 1 1 2 Binary 1 1 1

Converting Decimal to Octal Converting any decimal into octal can be done as follows Divide the number by 8 Take the remainder Go to step 1. Read the number from the bottom to the top. Example convert 60110 to octal Divide 601 by 8 = 75 and the remainder is 1 Divide 75 by 8 = 9 and the remainder is 3 Divide 9 by 8 =1 and the remainder is 1 Divide 1 by 8 = 0 and the remainder is 1 Reading from bottom to top 60110 is 11318

Converting Octal to Decimal 1. Keep the right number 2. Multiply all other digits by the base number times itself the same number of times that it is to the left of your right number. Converting Octal into Decimal 112 = 1 x 82 + 1 x 81 + 2 x 80 = 64 + 8 + 2 = 74 Which means 7410 = 1128

Hexadecimal (base 16) Binary Decimal Hex 0 000 0 001 1 0 010 2 0 011 3 0 100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 10 A 1011 11 B 1100 12 C 1101 13 D 1110 14 E 1111 15 F The hexadecimal is the base-16 number system, and uses the digits 0 to 15. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of four (starting from the right)

Binary to Hex and vice versa 1 330 4 A Hexadecimal 1 1 1 1 Binary

Converting Decimal to Hex Converting any decimal into octal can be done as follows Divide the number by 16 Take the remainder Go to step 1. Read the number from the bottom to the top. Example convert 60110 to Hex Divide 601 by 16 = 37 and the remainder is 9 Divide 37 by 16 = 2 and the remainder is 5 Divide 2 by 16 = 0 and the remainder is 2 Reading from bottom to top 60110 is 25916

Converting Hex to Decimal 1. Keep the right number 2. Multiply all other digits by the base number times itself the same number of times that it is to the left of your right number. Converting Hex into Decimal 25916 = 2 x 162 + 5 x 161 + 9 x 160 = 512 + 80 + 9 = 60110 Which means 60110 = 25916