In the decimal number systems each of the ten digits, 0 through 9, represents a certain quantity. The position of each digit in a decimal number indicates the magnitude of the quantity represented and can be assigned a weight. The weights for whole numbers are positive powers of ten that increases from right to left, beginning with 10º = 1 that is 10³ 10² 10¹ 10º

For fractional numbers, the weights are negative powers of ten that decrease from left to right beginningwith 10¯¹ that is 10² 10¹ 10º. 10¯¹ 10¯² 10¯³

The value of a decimal number is the sum of digits after each digit has been multiplied by its weights asin following examples

**Ex****p****re****s****s the decimal number 87 as a sum of the values of each digit**.

The digit 8 has a weight of10 which is 10 as indicated by its position. The digit 7 has a weight of 1 which is 10º as indicated by its position.

**8****7 = (8 x 10****1****) ****+ (7 x 10****0****)**

**Ex****p****re****s****s the decimal number 725.45 as a sum of the values of each digit.**

**725****. 45 = (7 x 10²) + (2 x 10¹) + (5 x 10º) + (4 x 10¯¹) + (5 x 10¯²) = 700 + 20 + 5 + 0.4 + 0.05**

__BI____NAR____Y NUMBERS__

The binary system is less complicated than the decimal system because it has only two digits, it is a basetwo system. The two binary digits (bits) are 1 and 0. The position of a 1 or 0 in a binary number indicates its weight, or value within the number, just as the position of a decimal digit determines the value of that digit. The weights in a binary number are based on power of two as:

**….****. 2****4 ****2****³ 2****2 ****2****1 ****2****0****. 2****-****1 ****2****-****2 ****….**

With 4 digits position we can count from zero to 15.In general, with n bits we can count up to a number equal to Ķ - 1. Largest decimal number = Ķ - 1.A binary number is a weighted number. The right-most bit is the least significant bit (LSB) in a binary whole number and has a weight of

2º =1. The weights increase from right to left by a power of two for each bit. The left-most bit is the most significant bit (MSB); its weight depends on the size of the binary number.

__BI____NARY-____T____O____-D____E____C____I____M____A____L CONVERSION__

The decimal value of any binary number can be found by adding the weights of all bits that are 1 and discarding the weights of all bits that are 0

Example

Let‘s convert the binary whole number 101101 to decimal

**W****e****ig****h****t****:****2****5 ****2****4 ****2****3 ****2****2 ****2****1 ****2****0**

**X**

**Bin****a****r****y no: 1 0 1 1 0 1**

**Va****lu****e 32 0 8 4 0 1**

**S****u****m = 45**