Change a number from
any base
back to base 10:

The
computer is representing a number typed at the keyboard by a pattern of
ON and OFF signals represented by the binary 1011. What number was typed
at the keyboard to produce this pattern? 
Let's
see how we can quickly convert numbers back to base 10 so that they
will have more meaning to us humans. 
Example 1: Convert
235_{8} into base 10.

The Process:
Above each of the digits in your number, list the power of the base
that the digit represents. See the example on the left.
It is now a simple process of multiplication and addition to
determine your base 10 number. In this example you have
5 x 8^{0 } = 5
3 x 8^{1} = 24
2 x 8^{2} = 128 
Now simply add these values together.
5 + 24 + 128 = 157
Answer: 235_{8} = 157_{10} 
** Remember: any number to the zero power equals one. 
Example 2: Let's do the problem mentioned at the top of this lesson.
Convert 1011_{2} to base 10.
The process is the same as in Example 1.
1 x 2^{0 }= 1
1 x 2^{1} = 2
0 x 2^{2} = 0
1 x 2^{3} = 8 
1 + 2 + 0 + 8
= 11
Answer: 1011_{2} = 11_{10}_{} 

Example 3: Convert 1C4_{16} to base 10.
4 x 16^{0 }=
4
C x 16^{1} = 12 x 16^{1} = 192
1 x 16^{2} = 256 
4 + 192 + 256 = 452
Answer: 1C4_{16} = 452_{10} 