Introduction to Computer-2_removed.pdf

Definition of Number Systems A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner. The value of any digit in a number can be determined by a digit, its position in the number, and the base of the number system. The number systems also h...

Definition of Number Systems A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner. The value of any digit in a number can be determined by a digit, its position in the number, and the base of the number system. The number systems also help in converting one number system to another. Number systems helps in representing the numbers in a small symbol set. Types of Number Systems There are different types of number systems in which the four main types are: 1. Binary number system (Base - 2) 2. Octal number system (Base - 8) 3. Decimal number system (Base - 10) 4. Hexadecimal number system (Base - 16) We will study each of these systems one by one in detail. Octal Number System The octal number system uses eight digits: 0,1,2,3,4,5,6,7 with the base of 8. The advantage of this system is that it has lesser digits when compared to several other systems, hence, there would be fewer computational errors For example: 358, 9238, 14183588 are some examples of numbers in the octal number system. Decimal Number System The decimal number system uses ten digits: 0,1,2,3,4,5,6,7,8 and 9 with the base number as 10 The decimal number system is the system that we generally use to represent numbers in real life. If any number is represented without a base, it means that its base is 10. For example: 72310, 524710, 3210, 983710 are some examples of numbers in the decimal number system.  Conversion of Binary Number Systems to Decimal Number Systems To convert a number from the binary system to the decimal system, we use the following steps. Example:1 Convert (100111)2 into the decimal system. Solution: Step 1: Identify the base of the given number. Here, the base of (1001112100111)2 is 2. Step 2: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base. The exponents should start with 0 and increase by 1 every time as we move from right to left. Since the base here is 2, we multiply the digits of the given number by 20, 21, 22 , and so on from right to left. Step 3: We just simplify each of the above products and add them. Here, the sum is the equivalent number in the decimal number system of the given number. Or, we can use the following steps to make this process simplified. 100111=(1×25)+(0×24)+(0×23)+(1×22)+(1×21)+(1×20) =(1×32)+(0×16)+(0×8)+(1×4)+(1×2)+(1×1) =32+0+0+4+2+1=39 Example:2 Convert (111111)2 into the decimal system. 111111=(1×25)+(1×24)+(1×23)+(1×22)+(1×21)+(1×20) =(1×32)+(1×16)+(1×8)+(1×4)+(1×2) +(1×1) =32+16+8+4+2+1=63  Conversion of Decimal Number System to Binary Number System To convert a number from the decimal number system to binary number system, we use the following steps. Example: Convert (4320104320)10 into the binary system. Solution: Step 1: Identify the base of the required number. Since we have to convert the given number into the binary system, the base of the required number is 2. Step 2: Divide the given number by the base of the required number and note down the quotient and the remainder in the quotient-remainder form. Repeat this process (dividing the quotient again by the base) until we get the quotient to be less than the base. Example : Convert (17)10 into the binary system base- 2. Solution: We divide 17 by 2 and note down the result and the remainder. We will repeat this process for every quotient until we get a quotient that is less than 2. Thus, (160)10=(10100000)2 Binary Addition It is a key for binary subtraction, multiplication, division. There are four rules of binary addition. Examples Perform binary addition on the following numbers