I want some explanation on number system
Conversion
Binary
Hexadecimal
Octal
Decimal......
Sry I wrote the chapter algebraic expressions because number system was not there....
I want some explanation on number system
Conversion
Binary
Hexadecimal
Octal
Decimal......
Sry I wrote the chapter algebraic expressions because number system was not there....
Decimal to Other Base System Steps
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Step 1 − Divide the decimal number to be converted by the value of the new base.
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Step 2 − Get the remainder from Step 1 as the rightmost digit (least significant digit) of new base number.
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Step 3 − Divide the quotient of the previous divide by the new base.
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Step 4 − Record the remainder from Step 3 as the next digit (to the left) of the new base number.
Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes zero in Step 3.
The last remainder thus obtained will be the Most Significant Digit (MSD) of the new base number.
Example − Decimal Number: 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 in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the Least Significant Digit (LSD) and the last remainder becomes the Most Significant Digit (MSD).
Decimal Number − 2910 = Binary Number − 111012.
Other Base System to Decimal System Steps
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Step 1 − Determine the column (positional) value of each digit (this depends on the position of the digit and the base of the number system).
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Step 2 − Multiply the obtained column values (in Step 1) by the digits in the corresponding columns.
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Step 3 − Sum the products calculated in Step 2. The total is the equivalent value in decimal.
Example Binary Number − 111012
Calculating Decimal Equivalent −
Step Binary Number Decimal Number Step 1 111012 ((1 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20))10 Step 2 111012 (16 + 8 + 4 + 0 + 1)10 Step 3 111012 2910
Binary Number − 111012 = Decimal Number − 2910
Other Base System to Non-Decimal System Steps
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Step 1 − Convert the original number to a decimal number (base 10).
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Step 2 − Convert the decimal number so obtained to the new base number.
Example Octal Number − 258
Calculating Binary Equivalent −
Step 1 − Convert to Decimal Step Octal Number Decimal Number Step 1 258 ((2 × 81) + (5 × 80))10 Step 2 258 (16 + 5 )10 Step 3 258 2110
Octal Number − 258 = Decimal Number − 2110
Step 2 − Convert Decimal to Binary Step 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
Decimal Number − 2110 = Binary Number − 101012
Octal Number − 258 = Binary Number − 101012
Shortcut method - Binary to Octal Steps
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Step 1 − Divide the binary digits into groups of three (starting from the right).
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Step 2 − Convert each group of three binary digits to one octal digit.
Example Binary Number − 101012
Calculating Octal Equivalent −
Step Binary Number Octal Number Step 1 101012 010 101 Step 2 101012 28 58 Step 3 101012 258
Binary Number − 101012 = Octal Number − 258
Shortcut method - Octal to Binary Steps
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Step 1 − Convert each octal digit to a 3 digit binary number (the octal digits may be treated as decimal for this conversion).
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Step 2 − Combine all the resulting binary groups (of 3 digits each) into a single binary number.
Example Octal Number − 258
Calculating Binary Equivalent −
Step Octal Number Binary Number Step 1 258 210 510 Step 2 258 0102 1012 Step 3 258 0101012
Octal Number − 258 = Binary Number − 101012
Shortcut method - Binary to Hexadecimal Steps
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Step 1 − Divide the binary digits into groups of four (starting from the right).
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Step 2 − Convert each group of four binary digits to one hexadecimal symbol.
Example Binary Number − 101012
Calculating hexadecimal Equivalent −
Step Binary Number Hexadecimal Number Step 1 101012 0001 0101 Step 2 101012 110 510 Step 3 101012 1516
Binary Number − 101012 = Hexadecimal Number − 1516
Shortcut method - Hexadecimal to Binary Steps
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Step 1 − Convert each hexadecimal digit to a 4 digit binary number (the hexadecimal digits may be treated as decimal for this conversion).
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Step 2 − Combine all the resulting binary groups (of 4 digits each) into a single binary number.
Example Hexadecimal Number − 1516
Calculating Binary Equivalent −
Step Hexadecimal Number Binary Number Step 1 1516 110 510 Step 2 1516 00012 01012 Step 3 1516 000101012
Hexadecimal Number − 1516 = Binary Number − 101012
Steps
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Step 1 − Divide the decimal number to be converted by the value of the new base.
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Step 2 − Get the remainder from Step 1 as the rightmost digit (least significant digit) of new base number.
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Step 3 − Divide the quotient of the previous divide by the new base.
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Step 4 − Record the remainder from Step 3 as the next digit (to the left) of the new base number.
Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes zero in Step 3.
The last remainder thus obtained will be the Most Significant Digit (MSD) of the new base number.
Example −Decimal Number: 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 in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the Least Significant Digit (LSD) and the last remainder becomes the Most Significant Digit (MSD).
Decimal Number − 2910 = Binary Number − 111012.
Other Base System to Decimal SystemSteps
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Step 1 − Determine the column (positional) value of each digit (this depends on the position of the digit and the base of the number system).
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Step 2 − Multiply the obtained column values (in Step 1) by the digits in the corresponding columns.
-
Step 3 − Sum the products calculated in Step 2. The total is the equivalent value in decimal.
Binary Number − 111012
Calculating Decimal Equivalent −
Step Binary Number Decimal Number| Step 1 | 111012 | ((1 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20))10 |
| Step 2 | 111012 | (16 + 8 + 4 + 0 + 1)10 |
| Step 3 | 111012 | 2910 |
Binary Number − 111012 = Decimal Number − 2910
Other Base System to Non-Decimal SystemSteps
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Step 1 − Convert the original number to a decimal number (base 10).
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Step 2 − Convert the decimal number so obtained to the new base number.
Octal Number − 258
Calculating Binary Equivalent −
Step 1 − Convert to Decimal Step Octal Number Decimal Number| Step 1 | 258 | ((2 × 81) + (5 × 80))10 |
| Step 2 | 258 | (16 + 5 )10 |
| Step 3 | 258 | 2110 |
Octal Number − 258 = Decimal Number − 2110
Step 2 − Convert Decimal to Binary Step 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 |
Decimal Number − 2110 = Binary Number − 101012
Octal Number − 258 = Binary Number − 101012
Shortcut method - Binary to OctalSteps
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Step 1 − Divide the binary digits into groups of three (starting from the right).
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Step 2 − Convert each group of three binary digits to one octal digit.
Binary Number − 101012
Calculating Octal Equivalent −
Step Binary Number Octal Number| Step 1 | 101012 | 010 101 |
| Step 2 | 101012 | 28 58 |
| Step 3 | 101012 | 258 |
Binary Number − 101012 = Octal Number − 258
Shortcut method - Octal to BinarySteps
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Step 1 − Convert each octal digit to a 3 digit binary number (the octal digits may be treated as decimal for this conversion).
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Step 2 − Combine all the resulting binary groups (of 3 digits each) into a single binary number.
Octal Number − 258
Calculating Binary Equivalent −
Step Octal Number Binary Number| Step 1 | 258 | 210 510 |
| Step 2 | 258 | 0102 1012 |
| Step 3 | 258 | 0101012 |
Octal Number − 258 = Binary Number − 101012
Shortcut method - Binary to HexadecimalSteps
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Step 1 − Divide the binary digits into groups of four (starting from the right).
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Step 2 − Convert each group of four binary digits to one hexadecimal symbol.
Binary Number − 101012
Calculating hexadecimal Equivalent −
Step Binary Number Hexadecimal Number| Step 1 | 101012 | 0001 0101 |
| Step 2 | 101012 | 110 510 |
| Step 3 | 101012 | 1516 |
Binary Number − 101012 = Hexadecimal Number − 1516
Shortcut method - Hexadecimal to BinarySteps
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Step 1 − Convert each hexadecimal digit to a 4 digit binary number (the hexadecimal digits may be treated as decimal for this conversion).
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Step 2 − Combine all the resulting binary groups (of 4 digits each) into a single binary number.
Hexadecimal Number − 1516
Calculating Binary Equivalent −
Step Hexadecimal Number Binary Number| Step 1 | 1516 | 110 510 |
| Step 2 | 1516 | 00012 01012 |
| Step 3 | 1516 | 000101012 |
Hexadecimal Number − 1516 = Binary Number − 101012
