The trick to understanding alternate number systems has more to do with what you must UNLEARN than with what you must LEARN.

To understand alternate base number systems you must forget the idea that '10' represents any specific quantity, or that our decimal number system is in any way a 'natural' or 'universal' number system.

"10" is an arbitrary combination of symbols that WE use to represent an amount equal to our (again arbitrary) grouping size.

In our number system, 10 is this many: **********

but that is only because we happen to have that many fingers.

When our primitive ancestors ran out of fingers while counting, they may have placed a small rock to indicate that they had 'grouped' once. Three small rocks plus 4 fingers equals 34 and so the grouping number system is born :)

If we had happened to evolve with more (or fewer) fingers, we would have a grouping system based on that quantity... let's say we had 8 fingers for example. We would count 1,2,3,4,5,6,7,10, again, placing that small rock as a marker that we had 'grouped' once. For an 8-fingered primitive race, 3 small rocks plus 4 fingers would equal a number that THEY would still call 34, but we would call 28... the size of the groups used is different.

Computers, figuratively speaking, have only 2 'fingers' that is, they can recognize and generate two electronic states... on and off.

The binary number system therefore "runs out of fingers" rather quickly when counting..1,10,11,100 etc. (100 in binary represents what we would call 4)

Every number system has the same number of symbols as the absolute size of it's base... that is, we have a decimal system and we have the symbols 0,1,2,3,4,5,6,7,8,9

The binary number system has only 0 and 1.

Every number system combines the 1 and zeros to symbolize the different grouping levels or powers of the base.... 10, 100, 1000 etc, but these mean different quantities according to the size of the groups (the 'base' of the system).

This little table illustrates the comparison

REMEMBER... '10' IS A RELATIVE AMOUNT, AND REPRESENTS A DIFFERENT ABSOLUTE QUANTITY WHEN THE BASE OF THE SYSTEM (THE GROUP SIZE) IS DIFFERENT

System: decimal Symbols: 0,1,2,3,4,5,6,7,8,9 Counting this many ********** 1,2,3,4,5,6,7,8,9,10 value of 100(in decimal terms): 10X10=100

System: octal Symbols: 0,1,2,3,4,5,6,7 Counting this many ********** 1,2,3,4,5,6,7,10,11,12 octal 100 = decimal 8X8 = decimal 64 octal 1000 = decimal 8X8X8 = decimal 512

System: binary Symbols: 0,1 Counting this many ********** 1,10,11,100,101,110,111,1000,1001,1010

binary 100 = decimal 2X2 = decimal 4 binary 1000 = decimal 2X2X2 = decimal 8

System: hexadecimal Symbols: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Counting this many ********** 1,2,3,4,5,6,7,8,9,A

hex 100 = dec 16X16 = dec 256

The actual conversion is mechanical... to convert decimal to anything else, divide the decimal number by the decimal equivalent of the alternate base, listing the remainders as the new, alternate base digits from right to left, and repeating the division upon the quotient until zero is reached.

For example, to convert decimal 9 to binary:

9/2 = 4, remainder 1 4/2=2, remainder 0 2/2=1, remainder 0 1/2=0, remainder 1

The binary equivalent of (decimal) 9 is 1001.

To convert any other system to decimal, multiply each of the digits by the decimal equivalent of their position value and then add up the results.

For example, to convert binary 1001 to decimal:

1* 2 to the zero power = 1*1=1 0* 2 to the first power=0*2=0 0*2 to the second power=0*4=0 1*2 to the third power=1*8=8

Add them up and they equal 9.