DECIMAL NUMBERS

DECIMAL NUMBERS

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

Express 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.

87 = (8 x 101) + (7 x 100)

Express 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

BINARY 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:

….. 24 2³ 22 21 20. 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.

BINARY-TO-DECIMAL 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

Weight:25 24 23 22 21 20

X

Binary no: 1 0 1 1 0 1

Value 32 0 8 4 0 1

Sum = 45