MT1 Number Systems. In general, the number a 3 a 2 a 1 a 0 in a base b number system represents the following number:

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1 MT1 Number Systems MT1.1 Introduction A number system is a well defined structured way of representing or expressing numbers as a combination of the elements of a finite set of mathematical symbols (i.e., digits). The functions of a number system are: 1. Represent a useful set of numbers. For example, all integers or rational numbers. 2. Give each number a standard representation. F There are two major types of number systems. These are, 1. Positional number systems This uses the same symbol for different orders of magnitude (e.g., ones place, thousands place etc). This greatly simplifies arithmetic. Examples include, decimal number system, binary number system, hexadecimal number systems. 2. Non positional number system It combines digits to signify their sums or their differences. The sum/difference represents a number. Example: Roman number systems. MMX represents 2010, IX represents 9. Base is the number of unique digits including zero. For example, in decimal number system there are 10 digits (0, 1 9). Therefore, the base is 10 for decimal number systems. In general, the number a 3 a 2 a 1 a 0 in a base b number system represents the following number: a 3 a 2 a 1 a 0 = a 3 * b 3 + a 2 * b 2 + a 1 * b 1 + a 0 * b 0 (1234) 10 = 1 * * * * 10 0 = Most frequently used number systems are as follows. Decimal Binary Hexadecimal Octal

2 The digits used under these number systems are tabulated below. Counting from 0 to 15 under these number systems are tabulated as follows. MT1.2 Binary number system The binary number system represents numerical values using two symbols, 0 and 1. It is a positional number system. It has a base of 2 with a radix of 2. Since it is a straightforward to implement, it is often used in digital circuitry, logic gates and in almost all modern computers. In binary the number system we use the digits 0, 1 to represent a number. In general a number could be represented as a binary number as follows,

3 a n a 3 a 2 a 1 a 0 = a n * 2 n +a 3 * a 2 * a 1 * a 0 * 2 0 A binary number could be divided into two parts. These are called the Most Significant Bit (MSB) and the Least Significant Bit (LSB). These MSB is the bit position in a binary number that carries the most value. The bit at the extreme left is the MSB. The bit position in a binary number that gives the units value (whether the number is even or odd) is called the LSB. It is the right most bit. In positional notation less significant digits are located further to the right. Eg: In binary number , is the MSB and is the LSB. The MSB can also correspond to the sign of a signed binary number in one or two's complement notation. The LSB indicates whether it is odd or even. An n-bit number is composed of n binary digits. Often leading zeros are concatenated in front of the MSB to make a binary number of certain bits. For example, 0111 is a 4-bit number. 8 bits constitute a byte. The decimal ranges for some of the commonly used bit values are shown in the table. Bit values Decimal Range , ,294,967, MT1.2.1 Binary to Decimal Conversion Since binary is a base-2 system, each digit represents an increasing power of 2, with the rightmost digit representing 2 0, the next representing 2 1, then 2 0, and so on. To determine the decimal representation of a binary number simply take the sum of the products of the binary digits and the powers of 2 which they represent. Eg: Convert (1101) 2 to a decimal number. (1101) 2 = 1 x x x x 2 0 = = 1310 Eg: Convert 8 bit ( ) 2 to a decimal number. What is the MSB and LSB of the number? ( ) 2 = 1 x x x x x x x x 2 0 = = 217

4 MSB = 1, LSB = 1 MT1.2.2 Decimal to Binary Conversion There are two main methods to convert a decimal number to a binary number. These are namely, power method and remainder method. Power Method In the power method, the highest whole powers of a decimal number are subtracted until there is a remainder of 1 or 0. The Binary number corresponds to the Step 1: Find the highest whole power of 2 contained in the decimal number. Step 2: Subtract the highest whole power of 2 from the decimal number. Step 3: Repeat Step 1 with the remainder. Step 4: Construct the binary number by placing a 1 in the position which corresponds to power value of the highest whole power of 2. Ex: Convert (1971) 10 to a binary number using the power method. Remainder Method Step 1: Divide the decimal number by the base (in the case of binary, divide by 2). Step 2: Indicate the remainder to the right. Step 3: Continue dividing into each quotient (and indicating the remainder) until the divide operation produces a zero quotient. Ex: Convert (1971) 10 to a binary number using the remainder method.

5 The base 2 number is the numeric remainder reading from the last division to the rest (if you start at the bottom, the answer will read from top to bottom). (1971) 10 = ( ) 2 MT1.3 Hexadecimal Number System Hexadecimal number system is a human-friendly binary number representation that is frequently used in computer science and in digital electronics. Counting in binary from 0 to 15 in decimals will be as 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. Each hexadecimal digit represents four bits. Therefore, an eight bit binary number can be represented by two hexadecimal digits. For example, the ( ) 2 can be represented by FE 16. Ex: Convert ABCDEF in hexadecimal into a decimal number. ABCDEF 16 = A x B x C x D x E x F x 16 0 = 10 x x x x x x 16 0 = = For converting decimal into binary both power method and remainder method applies. Whole powers of = = = = = = = Example: Convert ( ) 2 into a hexadecimal number.

6 = 7B3 16 MT1.4 Representing Negative Numbers in Binary There are different techniques used to represent negative numbers in binary. These are, Signed magnitude method One's complement method Two's complement method These three representations produce different results. Unless the method of conversion has been specified the value of the number cannot be figured out. Signed magnitude method This approach is directly similar to the most common way of constructing a sign (placing a + or - next to the number's magnitude). In signed magnitude, the left-most bit is not actually part of the number, but is just the equivalent of a + or 1 sign. In the left most bit 0 indicates that the number is positive, 1 indicates that the number is negative. The 8-bit, ( ) 2 would be +12 in decimal. To indicate -12, we would simply put a 1 rather than a 0 as the first bit: Some of the early computers (eg. IBM 7090) used this method. One's complement method In one's complement method, the positive numbers are represented as a regular binary numbers. However, the negative numbers are represented differently. Under this method in order to negate a number, it replaces all zeros with ones, and ones with zeros. Thus, 12 would be , and -12 would be As in signed magnitude, the leftmost bit indicates the sign (1 is negative, 0 is positive). One drawback of this method is that it had two different representations of zero. Ex: Representation of 4 bit binary in one s compliment method.

7 Ex: = (0111) = (1000) 2. (Note: The MSB is actually the sign bit.) Two's complement method Begin with the number in one's complement. Add 1 if the number is negative. Twelve would be represented as , and -12 as To verify this, let's subtract 1 from , to get If we flip the bits, we get , or 12 in decimal. In practice the representation most generally used in current computing devices is the two's complement method. For Positive Numbers: 1. Convert the magnitude of the number to binary. 2. Add zeros to make the binary number an n-bit number. Example: If your number consists of 5 bits but the goal is to get a number consisting 8-bits, pad the 5-bit binary number by adding three zeros to the left. For Negative Numbers: 1. Convert the magnitude of the number to binary. 2. Add zeros to the left to make the binary number n-bit. 3. Complement the number (i.e., invert the bits). 4. Add 1 to the inverted binary number to get the n-bit 2's complement notation. n-bit 2's Complement Binary to Decimal 1. Look at the leftmost bit to determine whether the number is positive or negative. If the leftmost bit is 0, the number is positive. If the leftmost bit is 1, the number is negative. 2. If positive, convert the number from binary to decimal. 3. If negative, determine the magnitude by: Invert the bits of the binary number. Add 1 to the inverted number.

8 Convert the result of the addition operation to decimal to get the magnitude of the corresponding decimal number. The actual decimal number is the negative of this number.