PritilataBiswas_DE

BENGAL SCHOOL OF TECHNOLOGY AND MANAGEMENT CA1 Number System University Roll Number: 24 Course Name: Bachelor of Computer Science First Year(1st Semester) Subject Name: Digital Electronics Subject Code: BCA101 Submitted By PRITILATA BISWAS.

[Audio] Number systems have been used since ancient times to represent and manipulate numbers. These early systems were based on counting and tallying, with symbols representing small numbers. As mathematics developed, new concepts such as place value, zero, and negative numbers were introduced, leading to more sophisticated number systems. This evolution of number systems has allowed us to express and operate on numbers in increasingly complex ways..

Structure Of Number System.

[Audio] The decimal number system is based on ten digits from 0 to 9 and uses powers of 10 to represent numbers. It is easy to understand and is used in many areas, including finance, commerce, and education..

[Audio] The hexadecimal number system is a base 16 system that uses 16 symbols, including the digits 0 through 9 and the letters A through F. These symbols are equivalent in both hexadecimal and decimal representations. The letters A through F represent numbers starting from 10, with A equal to 10, B equal to 11, and so on. This system is widely used in computer programming and digital electronics due to its ease of conversion between binary and decimal numbers. For example, the hexadecimal number 3F represents the decimal number 63. Conversion between number systems can be achieved through multiplication and summation, or repeated division and remainder recording. These conversions are crucial in computer programming and digital electronics, where binary and decimal numbers are used extensively..

[Audio] =====. Converting 13 to binary : 13/2=6, remainder 1 6/2=3, remainder 0 3/2=1, remainder 1 1/2=0, remainder 1 So, the binary equivalent of 13 is 1101 . 3. Decimal To Octal : To convert a decimal number to octal, we have to divide the number by 8 and note the remainders, until the quotient is zero. The octal number is read from the last remainder to the first. Example : Converting 65 (decimal) to octal : 65/8=8, remainder 1 8/8=1 , remainder 0 1/8=0 , remainder 1 So, the octal equivalent of 65 is 101..

[Audio] The process of converting octal numbers to binary involves replacing each octal digit with its three-bit binary equivalent. For instance, the octal number 27 becomes 0101112 when converted to binary. Similarly, hexadecimal numbers can be converted to binary by representing each digit as a four-bit binary equivalent. For example, the hexadecimal number 2F becomes 00101111 when converted to binary. When converting octal numbers to binary and then to hexadecimal, the binary digits are grouped in sets of four, starting from the right, and each group is converted to the corresponding hexadecimal digit. For instance, the binary number 111011010 becomes 1DA when converted to hexadecimal..

[Audio] The process of converting between number systems involves replacing each digit in the original number with its equivalent value in the target system. This is done by grouping the binary digits into sets of three for octal or four for hexadecimal, and then converting each group to the corresponding digit. For instance, when converting 732 from octal to hexadecimal, seven becomes 111, three becomes 011, and two becomes 010. These binary digits are then grouped together to form the hexadecimal equivalent, which is 111011010. Breaking this down into groups of four yields the final hexadecimal result, 1DA. Similarly, when converting 1A3 from hexadecimal to octal, one becomes 0001, A becomes 1010, and three becomes 0011. These binary digits are then grouped together to form the octal equivalent, which is 000110100011. By following this process, conversions between different number systems can be performed effortlessly..

[Audio] When converting binary numbers to octal and hexadecimal, we need to group the binary digits into sets of three or four, respectively, starting from the right. Each group is then converted to its corresponding digit. This process is demonstrated in the example provided, where the binary number 1011012 is converted to both octal and hexadecimal. The resulting values are 558 in octal and 2D16 in hexadecimal..

[Audio] Number systems play a crucial role in various aspects of our lives, including computer science, digital electronics, mathematics, engineering, physics, and even music. In computer science, binary numbers are used for memory, communication, and processing purposes. Digital circuits employ binary, octal, and hexadecimal numbering systems to perform operations. Mathematics relies heavily on number systems, particularly in arithmetic, algebra, and cryptography. Moreover, different base systems are utilized in various fields, such as engineering, physics, and music. The importance of number systems lies in their ability to facilitate the representation and manipulation of numbers in diverse contexts, ultimately enabling better comprehension of computing, data storage, and mathematical problem-solving..

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