Conversions Between Number Bases
To convert to base 10 from bases 2, 8, and 16 use the remainder method.
To convert from base 2 to base 8
To convert from base 2 to base 16
To convert from base 10 to base 2, 8, or 16 use expanded notation
To convert from base 10 to base 2, 8, or 16 use the subtraction method
To convert to Two’s Complement from Binary

Values of Number Bases 10, 2, 8, 16


Base 10

Decimal

 Base 2

Binary

 Base 8

Octal

 Base 16

Hexadecimal
0 0000 0 0
1 0001 1 1
2 0010 2 2
3 0011 3 3
4 0100 4 4
5 0101 5 5
6 0110 6 6
7 0111 7 7
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
16 10000 20 10
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Conversions Between Number Bases

Positional notation means that the value of a digit in a number depends on its location in the number. In the decimal number system the positional value is a power of the base 10.


Thousands Hundreds Tens Units
10 3 10 2 10 1 10 0
1000 100 10 1
The decimal number 1232 is understood to mean:
(1 * 1000) + (2 * 100) + (3 * 10) + 2 or
1000 + 200 + 30 + 2  
This is expanded notation.

You can evaluate a number in another number system in the same manner.

Octal number system positional value is a power of the base 8.


Five-Hundred-Twelves Sixty-fours Eights Units
8 3 8 2 8 1 8 0
512 64 8 1
The octal number 1232 is understood to mean:
(1 * 512) + (2 * 64) + (3 * 8) + 2 or
512 + 128 + 24 + 2 or 666 10

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Hexadecimal number system positional value is a power of the base 16.


  Two-Five-Sixes Sixteens Units
16 3 16 2 16 1 16 0
4096 256 16 1
The Hex number 1232 is understood to mean:
(1 * 4096) + (2 * 256) + (3 * 16) + 2 or
4096 + 512 + 48 + 2 or 4658 10
Binary number system positional value is a power of the base 2.

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Eights Fours Twos Units
2 3 2 2 2 1 2 0
8 4 2 1
    Binary digits can only be 0 or 1.

The Binary number 1101 is understood to mean:

(1 * 8) + (1 * 4) + (0 * 2) + 1 or
8 + 4 + 0 + 1 or 13 10

 
 
 
 
 
 
 
 
To convert to base 10 from bases 2, 8, and 16 use expanded notation using the appropriate positional values for the base you are converting from.
  1. Set up a table of place values. (Remember the right-most column has a value of 1.) 
  2. Write the digits of the number to be converted in each column. 
  3. Multiply the place value of each digit to get its decimal value. 
  4. Sum the decimal values of each position/column.
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To convert from base 10 to base 2, 8, or 16 use the remainder method.


divide by base dividends are quotients of previous division remainders  
8 7146   Remainders in reverse order are the solution
8 893 2  
8 111 5  
8 13 7  
8 1 5  
  0 1 Keep dividing until the quotient is zero
Therefore: 714610 is157528

 
 
 
 

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To convert from base 10 to base 2, 8, or 16 use the subtraction method.
  1. Write the place values (of the base you are converting to) in a table. 
  2. Write down the decimal number to be converted with space belowit for the subtractions. 
  3. Find the largest new place value that you can subtract from the decimal number. 
  4. Subtract that place value from the decimal number. 
  5. Place a 1 in the table column for that place value in the table. 
  6. Repeat until the result of the subtraction is zero. 

    7. Fill in the rest of the table with zeros. 
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To convert to base 2 from base 16, convert each hex digit separately to four binary digits. 2 4 B16 = 

0010 0100 1011 = 0010010010112 To convert from base 2 to base 16, convert each group of four binary digits to an octal digit. (Start at the right end of the binary number to create the groups.) 110100111112 = 110 1001 1111 = 


6 9 F = 69F16


 
 

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To convert to base 2 from base 8, convert each octal digit separately to three binary digits.
  • 1 5 7 5 28 = 

    •  

     

    001 101 111 101 010 = 0011011111010102

         To convert from base 2 to base 8, convert each group of three binary digits to an octal digit. (Start at the right end of the binary number to create the groups.)
  • 110100111112 = 11 010 011 111 = 


    • 3 2 3 7 = 32378
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    To convert to Two’s Complement from Binary, reverse all bits and add 1.

    (Change each 1 to 0 and each 0 to 1, then add 1.)

    11010011111 =

  • 00101100000 

    • +         1

    00101100001
     
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