The decimal system

The numbering scheme in everyday use is the decimal (or denary) system and numbers are represented by 10 digits i.e. the numbers 0 to 9.

Counting starts at zero, (see counting wheels below). Rotating the wheel anticlockwise adds 1 to the current count until the number reaches 9 whereupon the count is reset to 0.

To count past nine, a second counting wheel is needed. Each time the first wheel goes from 9 to 0 the second wheel is incremented by 1 until the count reaches 99. At this point a third wheel must be added and so on. As the second wheel goes from 9 to 0 the third wheel is incremented by one.

Digits are weighted according to their column position (which is important) as shown in the table below. The table corresponds to the three counting wheels above. The number of segments on each wheel are the same and correspond to the number base orweighting. In the decimal number system the weighing is base 10.

For example the decimal number 764 can be written as

weight	102	101	100
	hundreds	tens	units
digit	7	6	4

I.e. 7 * 100 + 6 * 10 + 4 * 1 = 764.

This representation of a decimal number provides the formula for converting a number to a different base by a process of repeated division. The generalised formula is shown on the right. The % sign (or modulus) is the remainder when a number is divided by the base number

  764 / 10 = 76 remainder 4 
  
   76 / 10 =  7 remainder 6
    7 / 10 =  0 remainder 7 
    0           stop when zero.


  number % base = remainder (or modulus)
  number = number / base

The binary system

Well the binary numbering scheme uses only 2 digits for arithmetic, i.e. 1 and 0. The weighting is to the base 2.

• The values 1 and 0 can be represented in terms of boolean logic where True = 1 and False = 0.
• The binary number system is important in electrical terms, because true and false can also represent ON and OFF. This is all any digital computer can understand.
  • One may be used to represent ON while 0 may represent OFF.
  • The value ON usually being represented by a positive voltage level, while OFF is usually represented 0 volts.

Decimal to binary conversion

The formula developed above will now be used to convert the decimal number 764 to its binary equivalent by repeatedly dividing it by two.

  764 / 2 = 382 remainder 0 LSB
  382 / 2 = 191 remainder 0
  191 / 2 =  95 remainder 1
   95 / 2 =  47 remainder 1
   47 / 2 =  23 remainder 1
   23 / 2 =  11 remainder 1
   11 / 2 =   5 remainder 1
    5 / 2 =   2 remainder 1
    2 / 2 =   1 remainder 0
    1 / 2 =   0 remainder 1 MSB
    0           stop when zero.

I.e. 764 = 1* 512 + 0* 256 + 1*128 + 1*64 + 1*32 + 1*16 +1*8 + 1*4 +0*2+0*1

The leftmost digit is the Most Significant Bit (MSB) the rightmost digit being the Least Significant Bit. (LSB).

Now 10 positions or bits, are needed to represent our decimal number 764.

Hexadecimal numbering

One other number scheme which is of interest is hexadecimal. This is counting to base 16. In the hexadecimal system the digits 0-9 are used along with the letters A - F. So in decimal, A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15.

This is typically used to represent RGB color values. Two hexadecimal digits representing each of the color components. For example FF0000 is red ; i.e. the red component has a decimal value of 255

The hexadecimal equivalent of 764 decimal is

  764 / 16 = 47 remainder 12 (i.e. C)
   47 / 16 =  2 remainder 15 (i.e. F)
    2 / 16 =  0 remainder  2 
    0         step when zero

weight	162	161	160
decimal	256	16	1
hexadecimal	2	F	C

I.e. 764 = 2*256 + 15*16 + 12*1

function test()
{
   //change number base up to base 36
   var alphabet = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ";
   var output=0; 
   base = prompt("enter number base ");
   number = prompt("enter number to convert");
   number = number.toUpperCase();
   pw = number.length -1;
   for (i=0;i<number.length>0;i++)
   {
        digit = number.charAt(i);
        num = alphabet.indexOf(digit); 
        output = Math.pow(base,pw-i)*num + output;
   } 
   alert(number + " base " + base + " is\n" + output + " base 10");
}