NUMBER SYSTEM

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• Decimalto Other Base System
• Other Base Systemto Decimal
• Other Base Systemto Non-Decimal
• Shortcut method - Binary to Octal
• Shortcut method - Octalto Binary
• Shortcut method - Binary to Hexadecimal
• Shortcut method - Hexadecimalto Binary
• Decimal to Other Base System

• Steps
• Step 1 - Divide the decimalnumber to be converted by the value of the new base.
• Step 2 - Getthe remainder fromStep 1 as the rightmost dig it(least significant dig it) of new base number.
• Step 3 - Divide the quotient of the previous divide by the new base.
• Step 4 - Record the remainder fromStep 3 as the next dig it(to the left) of the new base number.
• Repeat Steps 3 and 4, g etting remainders fromrightto left, untilthe quotient becomes zero inStep 3.
• The last remainder thus obtained will be the most significant dig it(MSD) of the new base number.

• Example
• DecimalNumber: 2910
• Calculating Binary Equivalent:
• Step Operation Result Remainder
• Step 1 29 / 2 14 1
• Step 2 14 / 2 7 0
• Step 3 7 / 2 3 1
• Step 4 3 / 2 1 1
• Step 5 1 / 2 0 1
• As mentioned inSteps 2 and 4, the remainders have to be arrang ed inthe reverse order so thatthe first
• remainder becomes the least significant dig it(LSD) and the last remainder becomes the most significant dig it
• (MSD).

• DecimalNumber: 2910 = Binary Number: 111012.Other base system to Decimal System
• Steps
• Step 1 - Determine the column(positional) value of eachdig it(this depends onthe positionof the dig it and
• the base of the number system).
• Step 2 - Multiply the obtained columnvalues (inStep 1) by the dig its inthe corresponding columns.
• Step 3 - Sumthe products calculated inStep 2. The totalis the equivalent value indecimal.

• Example
• Binary Number: 111012
• Calculating DecimalEquivalent:
• Step Binary Number Decimal Number
• Step 1 111012 ((1 x 2
• 4) + (1 x 23) + (1 x 22) + (0 x 21) + (1 x 20))10
• Step 2 111012 (16 + 8 + 4 + 0 + 1)10
• Step 3 111012 2910

• Binary Number: 111012 = DecimalNumber: 2910
• Other Base System to Non-Decimal System
• Steps
• Step 1 - Convertthe orig inalnumber to a decimalnumber (base 10).
• Step 2 - Convertthe decimalnumber so obtained to the new base number.

• Example
• OctalNumber: 258
• Calculating Binary Equivalent:
• Step 1: Convert to Decimal
• Step Octal Number Decimal Number
• Step 1 258 ((2 x 8
• 1) + (5 x 8
• 0))10
• Step 2 258 (16 + 5 )10
• Step 3 258 2110
• OctalNumber: 258 = DecimalNumber: 2110
• Step 2: Convert Decimal to BinaryStep Operation Result Remainder
• Step 1 21 / 2 10 1
• Step 2 10 / 2 5 0
• Step 3 5 / 2 2 1
• Step 4 2 / 2 1 0
• Step 5 1 / 2 0 1
• DecimalNumber: 2110 = Binary Number: 101012
• OctalNumber: 258 = Binary Number: 101012

• Shortcut method - Binary to Octal
• Steps
• Step 1 - Divide the binary dig its into g roups of three (starting fromthe right).
• Step 2 - Convert eachg roup of three binary dig its to one octal dig it.

• Example
• Binary Number: 101012
• Calculating OctalEquivalent:
• Step Binary Number Octal Number
• Step 1 101012 010 101
• Step 2 101012 28 58
• Step 3 101012 258
• Binary Number: 101012 = OctalNumber: 258
• Shortcut method - Octal to Binary
• Steps
• Step 1 - Convert eachoctal dig itto a 3 dig it binary number (the octal dig its may be treated as decimalfor
• this conversion).
• Step 2 - Combine allthe resulting binary g roups (of 3 dig its each) into a sing le binary number.

• Example
• OctalNumber: 258
• Calculating Binary Equivalent:
• Step Octal Number Binary Number
• Step 1 258 210 510Step 2 258 0102 1012
• Step 3 258 0101012
• OctalNumber: 258 = Binary Number: 101012
• Shortcut method - Binary to Hexadecimal
• Steps
• Step 1 - Divide the binary dig its into g roups of four (starting fromthe right).
• Step 2 - Convert eachg roup of four binary dig its to one hexadecimal symbol.

• Example
• Binary Number: 101012
• Calculating hexadecimalEquivalent:
• Step Binary Number Hexadecimal Number
• Step 1 101012 0001 0101
• Step 2 101012 110 510
• Step 3 101012 1516
• Binary Number: 101012 = HexadecimalNumber: 1516
• Shortcut method - Hexadecimal to Binary
• Steps
• Step 1 - Convert eachhexadecimal dig itto a 4 dig it binary number (the hexadecimal dig its may be treated
• as decimalfor this conversion).
• Step 2 - Combine allthe resulting binary g roups (of 4 dig its each) into a sing le binary number.

• Example
• HexadecimalNumber: 1516
• Calculating Binary Equivalent:
• Step Hexadecimal Number Binary Number
• Step 1 1516 110 510
• Step 2 1516 00012 01012
• Step 3 1516 000101012
• HexadecimalNumber: 1516 = Binary Number: 101012