"Base" value Numbering Systems

Throughout our lives, we have used a Base10 numbering system - called "Decimal".  Computers use a Base2 numbering system, called "Binary".  Any base can be used for any given numbering system, such as Base27, Base5, etc.  However, the most common are Base10 (Decimal), Base2 (Binary), Base16 (Hexadecimal) and occasionally Base8 (octal).

In any numbering system, the value of each placeholder digit is found by multiplying that digit times the "base", raised to the power of that particular place.  Sounds complicated !!

Actually it's not.  You can find the weighted value of any place in a number, by counting from right to left ( count 0, 1, 2, 3, etc). 

Ancient Egyptians used special symbols to represent a specified grouping value with 10 as the common denominator - the same as modern-day decimal base10 numbering systems !!  They used the following symbols:

eg_m1.gif (1953 bytes) =Millions     eg_hth1.gif (1671 bytes)=Hundred Thousands    eg_tth1.gif (1513 bytes)=Ten Thousands    eg_th1.gif (1516 bytes)=Thousands    eg_h1.gif (1707 bytes)=Hundreds     eg_t1.gif (1607 bytes)=Tens    eg_o1.gif (1252 bytes)=Ones

For example, the number 1,475,268 is represented in the Egyptian numbering system as follows:

eg_m1.GIF (1953 bytes)eg_hth4.GIF (2507 bytes)eg_tth7.GIF (1832 bytes).eg_th5.GIF (2623 bytes)eg_h2.GIF (1998 bytes)eg_t6.GIF (2109 bytes)eg_o8.GIF (1604 bytes)

Each of the magnitudes of 10 were represented in the above number, for example the 4 frogs represent 4 hundred thousands and the 5 lotus flowers represent 5 thousands, etc. In this numbering system, only the magnitudes of 10 that are used are expressed in the written number. The number 5,060 is thus represented as:

eg_th5.GIF (2623 bytes)eg_t6.GIF (2109 bytes)

 

So the number 384 has a 4 in the 0 place, an 8 in the 1 place, and a 3 in the 2 place.  Each place has a "weighted value" - for example, the Base 10 numbering system, if we just look at a number with four digits, from right to left we can calculate the weight :

Position of Digit

2   

1   

0
Number

3   

8   

4
Weight
(10 raised to the power of the position)

102 = 100

101 = 10   

100 = 1
value
(number x weight)		 300 80 4

Base10  -  Decimal

384 in Decimal (Base 10) = sum of all values = 300+80+4

First Place :         100 = 10 to the power of 0  = 1  (anything raised to the 0 power is 1)
Second Place :    101 = 10 to the power of 1 = 10 
Third Place:         102 = 10 to the power of 2 = 10x10 = 100
Fourth Place:       103 = 10 to the power of 3 = 10x10x10 = 1000 

From our previous number, 384, it uses three places, weighted  .  .  .  100, 10, and 1 - and to find the total value of a number, you multiply each digit by the weight of it's place, and add them up :

384 = 3x100   +   8x10   +   4x1   =   300 + 80 + 4   =   384

Similarly the other systems have a different Base, and the weights of each place change.  Also, the allowable digits that can be used changes.  Base 10 has 10 allowable digits (0,1,2,3,4,5,6,7,8,9), Base 8 has 8 allowable digits (0,1,2,3,4,5,6,7), Base 4 has 4 allowable digits (0,1,2,3) and Base 2 has 2 allowable digits (0,1).

Base 2, also called "Binary"  - is the numbering system that computers use.

 

Powers

To understand how any numbering system works, you need to understand powers.  Raising something to a power, is the mathematical equivalent of "multiplying it by itself that many times".  So, 7 raised to the power of 4 would be 7 multiplied by itself 4 times, or 7x7x7x7.  shorthand 

The notation used is to place the power right after the number raised up high (superscript text, which in this case would be 74.  The on special case that you simply need to memorize is the power of zero.  It seems that if you multiply something by itself 0 times, the result would be 0, but instead it is 1.  Therefore, anything raised to the 0 power is always 1, and therefore, no matter what base numbering system you are using - the rightmost digit is always weighted as 1.