Decimal Number System

Courses : Digital Logic Lecturer : Czachary Semester : 4thSemester, 2022/2023 Session Decimal Number System The Decimal Number System is the most commonly used number system in everyday life. In this system, numbers are represented using 10 symbols known as decimal digits, i.e. the numbers 0 through 9. The decimal number system is based on the concept of position or place value. Each digit in the decimal number is placed in a specific position that represents a multiple of 10. The rightmost position is the lowest position and represents a multiple of 10 to the power of 0 (1), while the next position moves to the left and represents a multiple of 10 to the power of 1 (10), 10 to the power of 2 (100), 10 to the power of 3 (1000), and so on. For example, the number 356 in the decimal system can be interpreted as 6 units (3x10^0), 50 (5x10^1), and 3 hundred (3x10^2). Thus, 356 in the decimal system has the same overall value as 300 + 50 + 6 = 356. The advantages of the decimal number system are its ability to represent numbers with high precision and the ability to perform accurate and intuitive mathematical calculations. Because of this, the decimal number system is widely used in various aspects of life, including financial calculations, measurements, and many other fields of science and technology. However, it should be noted that in the decimal system, the number of digits used to represent a particular number can vary depending on the scale used. For example, in the usual decimal system, we use the digits from 0 to 9, but in restricted decimal notation, we might only use a subset of decimal digits to represent numbers with lower precision. In the context of computing, decimal numbers are often converted to binary or hexadecimal representations, because computers work on a binary basis. However, the decimal number system remains the most recognized and commonly used system in everyday life. In the decimal number system, numbers are calculated using basic arithmetic operations such as addition, subtraction, multiplication, and division. These operations are performed taking into account the position of the digits in the number. For example, if we want to add two decimal numbers such as 45 and 27, we add the digits in the appropriate positions. First, we add up the digits in the ones position (1's place), which is 5 + 7 = 12. Since 12 is two digits, we write down its last digit (which is 2) and lift the first digit to the next position. Next, we add up the digits in the tens position (10's place), which is 4 + 2 + 1 (the number we lifted from the previous position) = 7. Thus, the sum is 45 + 27 = 72. Likewise, the operations of subtraction, multiplication and division on decimal numbers are carried out by paying attention to the position of the digits. These operations involve manipulating digits and moving numbers to their appropriate positions. For example, if we wanted to multiply 8 by 3, we multiplied the digits in the ones position (8) by the ones digit (3), which results in 24. Then, we shifted the 2 digit to the tens position, so the result is 24. The decimal number system also allows the representation of fractions using the decimal point. The digits to the left of the decimal point represent the whole number portion, while the digits to the right of the decimal point represent the fractional portion. For example, in the number 3.75, the digit 3 represents a whole number, while the digits 7 and 5 represent a fraction. The digits to the right of the decimal point represent multiples of 1/10, 1/100, 1/1000, and so on. The decimal number system has wide applications in everyday life, such as counting money, measuring time, and determining geographic coordinates. The ability to understand and use the decimal number system effectively is essential in many fields, including mathematics, science, finance, and technology. In the decimal number system, we can also perform conversions between decimal and other number systems, such as binary, octal, or hexadecimal. This conversion allows us to represent numbers in different formats. For example, to convert a decimal number to the binary number system, we use the method of repeated division by base 2. First, we divide the decimal number by 2 and record the remainder of the quotient. Then, we continue to divide the previous quotient by 2 until we reach 0. The remaining digits of this quotient, written from the bottom up, form the binary representation of the decimal number. For example, we will convert the decimal number 25 to binary. The process is as follows: 25 ÷ 2 = 12 remainder 1 12 ÷ 2 = 6 remainder 0 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 with a remainder of 1 1 ÷ 2 = 0 remainder 1 In binary representation, the remaining digits of the quotient will form the binary number from bottom to top. So 25 in the binary number system is 11001. Conversion from the binary to decimal number system is also possible using the opposite method. For example, to convert the binary number 11001 to decimal, we multiply each binary digit by the multiple of 2 according to the position of that digit, and add them up. In this case: 1 × 2^4 + 1 × 2^3 + 0 × 2^2 + 0 × 2^1 + 1 × 2^0 = 16 + 8 + 0 + 0 + 1 = 25 So, the binary number 11001 in decimal is 25. Conversions between decimals and other number systems also involve similar principles, using the appropriate base for each number system. With an understanding of the decimal number system and the ability to perform conversions with other number systems, we can use it in a variety of applications, including computing, programming, and data processing. The decimal number system is the most commonly used number system in everyday life. In this system, numbers are represented using 10 symbols, namely the decimal digits 0 through 9. The decimal number system is based on the concept of position or place value, where each digit is placed in a position that represents a multiple of 10. An advantage of the decimal number system is its ability to represent numbers with high precision and perform accurate and intuitive mathematical calculations. The decimal number system also allows the representation of fractions using the decimal point. An understanding of the decimal number system is essential in many fields, including mathematics, science, finance, and technology. In computing, conversions between decimal and other number systems, such as binary, octal, or hexadecimal, are possible to represent numbers in different formats. With the ability to understand, use, and convert with the decimal number system, we can use it in a variety of applications and problem solving involving numbers and calculations.