12-01-2013, 03:48 PM
Number System
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In the binary system, there can be only two choices for this number -- either a "0" or a "1". In the octal system, there can be eight possibilities:
"0", "1", "2", "3", "4", "5", "6", "7".
In the decimal system, there are ten different numbers that can enter the digit box:
"0", "1", "2", "3", "4", "5", "6", "7", "8", "9".
In the hexadecimal system, we allow 16 numbers:
"0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "A", "B", "C", "D", "E", and "F".
As demonstrated by the following table, there is a direct correspondence between the binary system and the octal system, with three binary digits corresponding to one octal digit. Likewise, four binary digits translate directly into one hexadecimal digit. In computer usage, hexadecimal notation is especially common because it easily replaces the binary notation, which is too long and human mistakes in transcribing the binary numbers are too easily made.
Convert From Any Base To Decimal
Let's think more carefully what a decimal number means. For example, 1234 means that there are four boxes (digits); and there are 4 one's in the right-most box (least significant digit), 3 ten's in the next box, 2 hundred's in the next box, and finally 1 thousand's in the left-most box (most significant digit). The total is 1234:
Convert From Decimal to Any Base
Again, let's think about what you do to obtain each digit. As an example, let's start with a decimal number 1234 and convert it to decimal notation. To extract the last digit, you move the decimal point left by one digit, which means that you divide the given number by its base 10.
1234/10 = 123 + 4/10
The remainder of 4 is the last digit. To extract the next last digit, you again move the decimal point left by one digit and see what drops out.
Addition and Multiplication Tables
You generate the addition tables in bases other then 10 by following the same rule you do in base 10. The resulting tables have the appearance of shifting the columns to the left by one in each subsequent rows. Note how simple the addition and multiplication tables are for the binary system; addition operation is simply the bit-wise XOR operation with carry, and multiplication is simply the logical AND operation.
Decimal Addition Table:
[attachment=46803]
In the binary system, there can be only two choices for this number -- either a "0" or a "1". In the octal system, there can be eight possibilities:
"0", "1", "2", "3", "4", "5", "6", "7".
In the decimal system, there are ten different numbers that can enter the digit box:
"0", "1", "2", "3", "4", "5", "6", "7", "8", "9".
In the hexadecimal system, we allow 16 numbers:
"0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "A", "B", "C", "D", "E", and "F".
As demonstrated by the following table, there is a direct correspondence between the binary system and the octal system, with three binary digits corresponding to one octal digit. Likewise, four binary digits translate directly into one hexadecimal digit. In computer usage, hexadecimal notation is especially common because it easily replaces the binary notation, which is too long and human mistakes in transcribing the binary numbers are too easily made.
Convert From Any Base To Decimal
Let's think more carefully what a decimal number means. For example, 1234 means that there are four boxes (digits); and there are 4 one's in the right-most box (least significant digit), 3 ten's in the next box, 2 hundred's in the next box, and finally 1 thousand's in the left-most box (most significant digit). The total is 1234:
Convert From Decimal to Any Base
Again, let's think about what you do to obtain each digit. As an example, let's start with a decimal number 1234 and convert it to decimal notation. To extract the last digit, you move the decimal point left by one digit, which means that you divide the given number by its base 10.
1234/10 = 123 + 4/10
The remainder of 4 is the last digit. To extract the next last digit, you again move the decimal point left by one digit and see what drops out.
Addition and Multiplication Tables
You generate the addition tables in bases other then 10 by following the same rule you do in base 10. The resulting tables have the appearance of shifting the columns to the left by one in each subsequent rows. Note how simple the addition and multiplication tables are for the binary system; addition operation is simply the bit-wise XOR operation with carry, and multiplication is simply the logical AND operation.
Decimal Addition Table: