However, it is worth noticing that decimal, binary, octal and hex have different bases. The bases tell you how many digits each have to represent the numbers. For example has decimal 10 numbers (from 0₁₀ through 9₁₀) while octal only has 8 (from 0₈ through 7₈).
(The subscript indicates the number's base. In everyday language we assume that the number is given to us is in decimal/base 10. We would be rather surprised if the cashier at the local grocery store gave us the price in octal...)

When decimal has counted up to 9₁₀, it has run out of available digits and must increase to two digits, from 9₁₀ to 10₁₀, setting the lowest based digit to 0. Also, when counted up to 19₁₀ decimal must use the next available digit after 1 (here 2) in its 10-column, again setting the lowest based digit to 0.

Where one digit is added to the number, the number is given in bold in the table above.

The Link Between Decimal and Binary Numbers

Decimal numbers are the numbers that people usually use to communicate with in everyday life.  The sequence 1,2,3,4 could be considered a listing of decimal numbers.  Numbers used in decimal numbering are considered to have a base 10.  One reason we find the decimal system easy to communicate with is that people have ten fingers and can easily keep track of figures in a base 10 format.  A sample showing how a number can be expressed in decimal format is presented below:

The decimal number 942 can be broken out as followed to show its base 10 properties:

9x 10²  = 900
4 x 10¹ =   40
2 x 10⁰ =     2
                942

?10⁴ 10³ 10² 10¹ 10⁰ (decimal point) 10^-0 10^-1 10^-2 10^-3 10^-4?
 

Computers do not have 10 fingers (or any fingers for that matter) and can only communicate using the two bits 1 and 0.  As such, they can more easily represent figures using a base 2 format.  Numbers expressed in a base 2 format are called binary numbers.  A sample binary number looks like the following:  10011.  In decimal form, this binary number is equivalent to the number 19.  Conversion will be discussed more thoroughly at the end of this section, but you can begin to see how this conversion may take place by examining the base 2 structure of the number:

1 x 2⁴    =  16
0 x 2³    =    0
0 x 2²    =    0
1 x 2¹    =    2
1 x 2⁰    =    1
                19

?2⁴ 2³ 2² 2¹ 2⁰ (decimal point) 2^-0 2^-1 2^-2 2^-3 2^-4?
 

Counting in Binary

Remember when we count in decimal that when we run of available digits from 0 to 9, that we ?carry? the 1 to the tens column and reset the ones column to 0?  Well, the same technique can be used in binary counting, except we will be ?carrying? a lot more often as we only have 2 digits to work with!  See the table below for information on counting from 0 to 20 in binary along with relevant notes on the ?carrying process.?
 
 

Converting from Decimal to Binary and Binary to Decimal

When a chart is not handy and you don?t want to count all day, you can perform conversions between decimal and binary (and vice versa).  Each method is documented below.

Decimal to Binary Conversion Process

Basically we need to repeatedly divide our decimal number by two (as we are converting to a base 2 format).  The remainders given in this process will yield our binary equivalent number.

For example, to convert the decimal number 19 to binary:
 
 

 Perform calculations in ascending order
 starting with the 19.

                                     1 / 2 = 0 R1
                              2 / 2 = 1 R0
                         4 / 2 = 2 R0
                9 / 2 = 4 R1
        19 / 2  = 9 R1

      Take the remainders in descending order and 10011 yields the binary equivalent to 19 decimal.
 

Binary to Decimal Conversion Process

Our Binary to Decimal conversion process was actually illustrated earlier, but we will explain it here.  Basically, remembering that each digit in the binary sequence can be represented in the following format:

?2⁴ 2³ 2² 2¹ 2⁰ (decimal point) 2^-0 2^-1 2^-2 2^-3 2^-4?

we can then determine the value of each 1 and 0 in the binary string by applying base two raised to the appropriate power.  The appropriate power is determined by the digits location in the string.  Thus if we wish to take the binary number previously obtained above, 10011, it would convert to decimal as follows:

1 x 2⁴   =  16
0 x 2³   =    0
0 x 2²   =    0
1 x 2¹   =    2
1 x 2⁰   =    1
                19

Since binary is the way computers "see the world", the above chart shows how Octal lines up with it's binary equivalent.  Aside from binary, To make some sense of the above table, a practical use would be to see how the real world, who uses decimal (or base 10), can be derived from an octal number representaion.

But firse, here are some general mathematical rules and notes:

• X⁰ = 1
• Exponents of significant go from right to left and follow exponential precedence.  8ⁿ ...8³ ,8²  , 8¹ , 8⁰
• Any conversion to decimal of a non-decimal number will use it's base to derive a decimal number.  In octal for example, we use base 8 to get a base 10 result.

Here is a conversion from 176 octal (base 8), to a decimal (base 10) representation.
 

    6 X 8⁰  = 6    +
    7 X 8¹  = 56  +
    1 X 8²  = 64  +
                       126 base 10 decimal or 126₁₀

Here is another conversion 78 octal (base 8), to a decimal (base 10) representation.
 

    8 X 8⁰  = 8    +
    7 X 8¹  = 56  +
                        62 base 10 decimal or 62₁₀

Here is a conversion from decimal 89, to octal.  By using simple division by 8 (octal base 8), we derive the decimal base 10 equivalent from the remainders of the divison.

    1/8  =    0 R 1
    11/8 =   1 R 3
    89/8 = 11 R 2

Therefore, 89₁₀ = 132₈

Here is a conversion from decimal 5, to octal.

    5/8 = 0 R 5

Therefore, 5₁₀ = 5₈
What's interesting about this result, is that numbers in Octal from 0 through 7, it will have an equivalent decimal number (base 10).  Becaues Octal is base 8, it begins to devert from the decimal pattern after 7 (i.e., 10, 11,...17, 20, 21, 22,...).
 

When you want to make conversion between binary to octal, octal to binar, binary to hex or hex to binary, you don't need to worry about division and remainder.You may only need to remember the first few numbers in binary, octal, hex.Then you can do the conversion freely. Here is the way.

• Binary to Octal: look at the binary number by group of every 3 digits.

Example: converting binary number 010  110 011  101   into octal number, for 010 it is 2 in octal, 110 it is 6 in octal,
               011 it is 3 , 101 it's 5 in octal, so the number should be 2635 ₈
More example
 
                Binary                                                Octal

        100 110 010 001₂                                        4621₈
        001 101 000 010 ₂                                      1502₈
 
 

• Octal to Binary: same method, but reverse direction

• Binary to Hex: Look at the number by group of 4 digits.

                        Binary                                                Hex

                    0101 1100 1010 0001 ₂                        5CA1
                    1111 1101 1010 0101 ₂                        FDA5

• Hex to Binary: same method, but reverse direction

• Octal to Hex: you may first convert into binary and use the method we discuss before to convert it into hex

• Hex to Octal: first convert the number into binary and then convert into octal