Decimal and binary numbers and their interconversion

Decimal and binary numbers and their interconversion

Number of systems

There are four system of arithmetic which are often used in digital circuits. These system are

• Decimal: It has a base of 10 different symbols to represent numbers, they are 0 and 1.
• Binary: It has a base of 2, i.e. it uses only two different symbols, they are 0 and 1.
• Octal: It has a base of 8, i.e. it uses the eight different symbols, they are 0, 1, 2, 3, 4, 5, 6, 7.
• Hexadecimal : It has base of 16, i.e. it uses sixteen different symbols, they are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.

All these systems use the same type positional notation expect that

• Decimal system uses the powers of 10
• Binary system uses the power of 2
• Octal system powers of 8
• Hexadecimal system uses powers of 16

Decimal numbers are used to represent quantities which are outside the digital system. Binary system is extensively used by digital systems like digital computer which operate on binary information. Octal system has certain advantage in digital work because it requires less circuitary to get information into and out of a digital system. Moreover, it is clear to read, record and print out octal numbers than binary numbers. Hexadecimal number system is particularly suited for microcompurters.

THE DECIMAL NUMBER SYSTEM

we will briefly recount some important characteristics of this more familiar system before taking up other systems. This system has a base of 10 and is position-value system meaning that the value of a digit depends on its position. It has the following characteristics:

1. Base or Radix

It is defined as the number of different digits which can occur in each position in the number system.

The decimal number system has a base of 10 meaning that it contains ten unique symbols (or digits). These are 0, 1, 2, 3, 4, 5, 6, 7, 7, 8, 9. Any one of these may be used in each position of the number. Incidentally, it may be noted that we call it a decimal (10’s)n system although it does not have a distinct symbol of 10. As well-known, it represents 10 and any number above 10 as a combination of its ten unique symbols.

1. Position value

The absolute value of each digit is fixed but its position value is determined by its position in the overall number

The number 2573 can be broken down as follow:

2573=\(2\times 10^3+5\times 10^2+7\times 10^1+3\times 10^\circ\)

It will be noted that in this number, 3 is the least significance digit (LSD) whereas 2 is the most significant digit (MSD).

Again the number 2573.469 can be written as,

2573.469=\(2\times 10^3+5\times 10^2+7\times 10^1+3\times 10^\circ+4\times 10^{-1}+6\times 10^{-2}+9\times 10^{-2}\).

It is seen that position values are found by raising the base of the number system (i.e. 10 in this case ) to the power of the position. Also, powers are numbered to the left of the decimal point starting with 0 and to the right of decimal point starting with -1.

BINARY NUMBER

Like decimal (or denary) system, it has a radix and it also uses the same type of position value system.

Its base or radix is two because it uses only two digits 0 and 1. It may be noted that binary numbers need more places for counting because their base is small.

Binary numbers are used extensively by all digital system primarily due to the nature of electronics itself. The bit 1 may be represented by a saturated transistor, a light turned ON, a relay energized or a magnet magnetized in a particular direction.

The bit O, n the other hand, can be represented a cut off transistor, light turned off, a relay de-energized or a magnet magnetized in the opposite direction.

Binary to Decimal conversion

Following procedure should be adopted for converting a given binary integer ( whole number ) into its equivalent decimal number.

Step 1: Write the binary number, i.e. all bits in a row.

Step 2: Directly under the bits, write 1, 2, 4, 8, 16,…start from right to left.

Step 3: Write zero for the decimal weights which lie under o bits.

Step 4: Add the remaining weights to get the decimal equivalent.

Decimal to binary conversion

1. Integers

Such conversion can be achieved by using the so- called double-double method. It is also known as divided-by-two methods. In this method, we progressively divide the given decimal number by 2 and write down the remainders after each division. These remainders taken in the reverse order (i.e. from bottom to top) form the required binary number.

2.Fraction

In this case, Multiply-by-two rule is used, i.e. we multiply each bit by 2 and the integer position. These carries taken in the forward (top-bottom) direction give the required binary fraction.

References:

(1)Theraja, B.L. Basic Electronics. N.p.: S.Chand, n.d. Print.

(2)C.L.Arora. Refresher Course in Physics. Vol. II and III. N.p.: S.Chand, 2006. Print.

(3)Malvino. Electronic Principles. N.p.: Tata McGraw-Hill, n.d. Print.

(4)N.Nelkon and P.Parker. Advanced Level Physics. 5th ed. N.p.: Arnold Heinemann, n.d. Print.

(5)Priti Bhakta Adhikari,Diya Nidhi Chaatkuli, Ishowr Prasad Koirala. A Textbook of Physics (2nd Year). N.p.: Sukunda Pustak Bhawan, 2070. Print.