Number system ( Logic circuits
Number Systems, Base Conversions, and Computer Data
Representation
Decimal and Binary Numbers
When we write decimal (base 10) numbers, we use a positional notation system. Each digit is multiplied by an appropriate power of 10 depending on its position in the number:
For example:
843 = 8 x 102 + 4 x 101 + 3 x 100
= 8 x 100 + 4 x 10 + 3 x 1
= 800 + 40 + 3
For whole numbers, the rightmost digit position is the one’s position (100 = 1). The numeral in that position indicates how many ones are present in the number. The next position to the left is ten’s, then hundred’s, thousand’s, and so on. Each digit position has a weight that is ten times the weight of the position to its right.
In the decimal number system, there are ten possible values that can appear in each digit position, and so there are ten numerals required to represent the quantity in each digit position.
The decimal numerals are the familiar zero through nine (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
In a positional notation system, the number base is called the radix. Thus, the base ten system that we normally use has a radix of 10. The term radix and base can be used interchangeably.
When writing numbers in a radix other than ten, or where the radix isn’t clear from the context, it is customary to specify the radix using a subscript. Thus, in a case where the radix isn’t understood, decimal numbers would be written like this:
12710
1110
567310
Generally, the radix will be understood from the context and the radix specification is left off.
The binary number system is also a positional notation numbering system, but in this case, the base is not ten, but is instead two. Each digit position in a binary number represents a power of two. So, when we write a binary number, each binary digit is multiplied by an appropriate power of 2 based on the position in the number:
For example:
101101 = 1 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20
= 1 x 32 + 0 x 16 + 1 x 8 + 1 x 4 + 0
Representation
Decimal and Binary Numbers
When we write decimal (base 10) numbers, we use a positional notation system. Each digit is multiplied by an appropriate power of 10 depending on its position in the number:
For example:
843 = 8 x 102 + 4 x 101 + 3 x 100
= 8 x 100 + 4 x 10 + 3 x 1
= 800 + 40 + 3
For whole numbers, the rightmost digit position is the one’s position (100 = 1). The numeral in that position indicates how many ones are present in the number. The next position to the left is ten’s, then hundred’s, thousand’s, and so on. Each digit position has a weight that is ten times the weight of the position to its right.
In the decimal number system, there are ten possible values that can appear in each digit position, and so there are ten numerals required to represent the quantity in each digit position.
The decimal numerals are the familiar zero through nine (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
In a positional notation system, the number base is called the radix. Thus, the base ten system that we normally use has a radix of 10. The term radix and base can be used interchangeably.
When writing numbers in a radix other than ten, or where the radix isn’t clear from the context, it is customary to specify the radix using a subscript. Thus, in a case where the radix isn’t understood, decimal numbers would be written like this:
12710
1110
567310
Generally, the radix will be understood from the context and the radix specification is left off.
The binary number system is also a positional notation numbering system, but in this case, the base is not ten, but is instead two. Each digit position in a binary number represents a power of two. So, when we write a binary number, each binary digit is multiplied by an appropriate power of 2 based on the position in the number:
For example:
101101 = 1 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20
= 1 x 32 + 0 x 16 + 1 x 8 + 1 x 4 + 0