To convert between Binary, Octal and Hexadecimal systems, it will be easier if you understand our Base 10 system first, as follows:

BASE 10:

Use the number 345.26 Base 10.

Starting at the DECIMAL point (from here on the DECIMAL point will  be referred to as the BASE point).

Going from the Base point to the left, the first space (5) is referred to as UNITS.  In Base 10 it equals five units.

The next digit to the left is referred to as the X(BASE).  The 4 is four times the Base (10) or 40.

The next digit to the left is referred to as X(BASE SQUARED) or 100.  The 3 is 3 times 100, or 300.

The next digit to the left is referred to as X(BASE TO THE THIRD POWER) or 1000.

The next digit to the left is referred to as X(BASE TO THE FOURTH POWER) or 10000.

So, as you continue to the left, increase the power by one.

Going from the BASE point to the RIGHT; the first space s referred to as 1 over Base. Or, in this case, one tenth.  So the 2 would equal .2 or two tenths.

The next position to the right is one over Base squared, the next would be Base to the third power, and so on.

BINARY BASE:

The Binary base system is the same as above, in Base ten, in that the first digit to the left of the Base point is UNITS.

In this case, it is either ZERO or ONE.

The next would be X(BASE) or the number in that position times two.

The next position to the left would be X(BASE SQUARED) or in this case, two squared, or four.

The next position to the left is X(BASE) to the third power, or in this is EIGHT.  Note: It is important that you do not ADD 3 and get Six.

Continuing to the left, just increase each position to the NEXT POWER.

Starting at the BASE point and going right, works the same as BASE TEN, in that the first position would be ONE over TWO.
Remember, you can only have a ONE or a ZERO.  So, it would be equal ZERO or equal 1/2.

The next position to the right would be 1/4.  The next would be 1/8 and so on.

The number 1011.01 would be equal  to the following:

Going from the BASE point to the left is Units or 1.  The next would be X(BASE), or 1 times two equals two.  The next position, in this case, ZERO, is only used as a placeholder.  The next position is X(BASE) squared, or 0.  And, the next position is X(BASE) to the third power or 2x2x2.  Remember, its eight not six.

So, the binary 1011 is equal to (add each position), eight, place holder, plus 2, plus 1, or all together...ELEVEN.  Each position to the right is figured the same as BASE TEN.  Position number 1
is 1/2, the next is 1/4, and so on.

The above number.01 is equal to.0(placeholder), the next .01 equal to 1/4 or one fourth.
Therefore, the above binary number 1011.01 = 11.25 in base ten.

Note: I should have mentioned earlier that the only way to understand what the BASE number we are working with equals, is to convert it to BASE 10.  Because, that is what we learned in school and have used most of our lives.

Each BASE system follows the same rules that we started with
in BASE 10.

Note:  The largest number that you may have in any Base System is always ONE LESS THAN THE BASE.

In BASE 10, it is NINE. In BINARY or BASE 2, it is ONE. In BASE 8 (Octal) it is SEVEN.  And, in BASE 16 (Hexadecimal) it is 15.

Computers work on the BINARY system.  OCTAL and HEXADECIMAL systems are also used because they are easier than Binary to work with.  Example: The hexadecimal fifteen is 1111 in the Binary system.  In Hexadecimal fifteen is F. I will explain this later.

The OCTAL number 46 would equal (4XBase or 4X8 = 32. And the 6 would = 6.  Added together = 38  So, 46 in OCTAL equals 38 in BASE 10.

In HEXADECIMAL, the largest number is 15. If we use 15 in the Hex system, 10 thru 15 would be confusing because they need two places.  You may use any system, such as circle, square,
triangle, and so on.  Or you could use any six symbols, or colors. However, most of the time we use A through F.  The following applies: 10 = A, 11 = B, 12 = C, 13 = D, 14 = E, and 15 = F.

HEXADECIMAL is calculated the same way as BASE 10, BINARY AND OCTAL.

Therefore, 734 Base 16 would = 7 times Base squared, or 156.
The 3 = 3XBase, or 16, or 48 and 4 units, or 4.
Added together 156 + 48 + 4  = 208 in BASE 10

How to count in BASE 10, BINARY, OCTAL and HEXADECIMAL:
 

1	001	001	0001
2	010	010	0010
3	011	011	0011
4	100	100	0100
5	101	101	0101
6	110	110	0110
7	111	111	0111
8	1000		1000
9	1001		1001
10	1010		A
11	1011		B
12	1100		C
13	1101		D
14	1110		E
15	1111		F

The nice thing about the computer system is that the 1s and 0s can be grouped into threes, for OCTAL and groups of four for HEXADECIMAL.

Therefore, Binary 111011011110 for OCTAL would be 111 011 011 110 or 7336.

Grouping into fours for HEXADECIMAL would be 1110 1101 1110 or E D E.

I hope by now that you can see how to convert between the BASES and how it is easier to read the numbers on the computer when they are grouped as described above.

Feel free to contact me via email at chucke@zianet.com if you have questions.

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