ࡱ> bdav] (bjbj11 ;[,\[,\ F!F!..J2J2J2^2^2^282Z3\^2n4j ;(H;H;p;bTbTbTy{{{{{{$hJ2T"T@bTTT..H;p;\\\T.H;(J2p;y\Ty\\FL1}p;`YV%Teʖ0yW}J2}bTvT\T TXbTbTbT[jbTbTbTTTTTbTbTbTbTbTbTbTbTbTF!> -:  Chapter 01.04 Binary Representation of Numbers After reading this chapter, you should be able to: convert a base-10 real number to its binary representation, convert a binary number to an equivalent base-10 number. In everyday life, we use a number system with a base of 10. For example, look at the number 257.56. Each digit in 257.56 has a value of 0 through 9 and has a place value. It can be written as  EMBED Equation.3  In a binary system, we have a similar system where the base is made of only two digits 0 and 1. So it is a base 2 system. A number like (1011.0011) in base-2 represents the decimal number as  EMBED Equation.3  in the decimal system. To understand the binary system, we need to be able to convert binary numbers to decimal numbers and vice-versa. We have already seen an example of how binary numbers are converted to decimal numbers. Let us see how we can convert a decimal number to a binary number. For example take the decimal number 11.1875. First, look at the integer part: 11. Divide 11 by 2. This gives a quotient of 5 and a remainder of 1. Since the remainder is 1,  EMBED Equation.3 . Divide the quotient 5 by 2. This gives a quotient of 2 and a remainder of 1. Since the remainder is 1,  EMBED Equation.3 . Divide the quotient 2 by 2. This gives a quotient of 1 and a remainder of 0. Since the remainder is 0,  EMBED Equation.3 . Divide the quotient 1 by 2. This gives a quotient of 0 and a remainder of 1. Since the remainder is ,  EMBED Equation.3 . Since the quotient now is 0, the process is stopped. The above steps are summarized in Table 1. Table 1 Converting a base-10 integer to binary representation. QuotientRemainder11/25 EMBED Equation.3 5/22 EMBED Equation.3 2/21 EMBED Equation.3 1/20 EMBED Equation.3 Hence  EMBED Equation.3  For any integer, the algorithm for finding the binary equivalent is given in the flow chart on the next page. Now let us look at the decimal part, that is, 0.1875. Multiply 0.1875 by 2. This gives 0.375. The number before the decimal is 0 and the number after the decimal is 0.375. Since the number before the decimal is 0,  EMBED Equation.3 . Multiply the number after the decimal, that is, 0.375 by 2. This gives 0.75. The number before the decimal is 0 and the number after the decimal is 0.75. Since the number before the decimal is 0,  EMBED Equation.3 . Multiply the number after the decimal, that is, 0.75 by 2. This gives 1.5. The number before the decimal is 1 and the number after the decimal is 0.5. Since the number before the decimal is 1,  EMBED Equation.3 . Multiply the number after the decimal, that is, 0.5 by 2. This gives 1.0. The number before the decimal is 1 and the number after the decimal is 0. Since the number before the decimal is 1,  EMBED Equation.3 . Since the number after the decimal is 0, the conversion is complete. The above steps are summarized in Table 2. Table 2. Converting a base-10 fraction to binary representation. NumberNumber after decimalNumber before decimal0.1875 EMBED Equation.3 20.3750.375 EMBED Equation.3 0.375 EMBED Equation.3 20.750.75 EMBED Equation.3 0.75 EMBED Equation.3 21.50.5 EMBED Equation.3 0.5 EMBED Equation.3 21.00.0 EMBED Equation.3  SHAPE \* MERGEFORMAT Hence  EMBED Equation.3  The algorithm for any fraction is given in a flowchart on the next page. Having calculated  EMBED Equation.3  and  EMBED Equation.3 , we have  EMBED Equation.3 . In the above example, when we were converting the fractional part of the number, we were left with 0 after the decimal number and used that as a place to stop. In many cases, we are never left with a 0 after the decimal number. For example, finding the binary equivalent of 0.3 is summarized in Table 3. Table 3. Converting a base-10 fraction to approximate binary representation. NumberNumber after decimalNumber before decimal0.3 EMBED Equation.3 20.60.6 EMBED Equation.3 0.6 EMBED Equation.3 21.20.2 EMBED Equation.3 0.2 EMBED Equation.3 20.40.4 EMBED Equation.3 0.4 EMBED Equation.3 20.80.8 EMBED Equation.3 0.8 EMBED Equation.3 21.60.6 EMBED Equation.3  As you can see the process will never end. In this case, the number can only be approximated in binary format, that is,  EMBED Equation.3  Q: But what is the mathematics behinds this process of converting a decimal number to binary format? A: Let  EMBED Equation.3  be the decimal number written as  EMBED Equation.3  where  EMBED Equation.3  is the integer part and  EMBED Equation.3  is the fractional part. We want to find the binary equivalent of  EMBED Equation.3 . So we can write  SHAPE \* MERGEFORMAT   EMBED Equation.3  If we can now find  EMBED Equation.3 in the above equation then  EMBED Equation.3  We now want to find the binary equivalent of  EMBED Equation.3 . So we can write  EMBED Equation.3  If we can now find  EMBED Equation.3 in the above equation then  EMBED Equation.3  Let us look at this using the same example as before. Example 1 Convert  EMBED Equation.3  to base 2. Solution To convert  EMBED Equation.3  to base 2, what is the highest power of 2 that is part of 11. That power is 3, as  EMBED Equation.3  to give  EMBED Equation.3  What is the highest power of 2 that is part of 3. That power is 1, as  EMBED Equation.3  to give  EMBED Equation.3  So  EMBED Equation.3  What is the highest power of 2 that is part of 1. That power is 0, as  EMBED Equation.3  to give  EMBED Equation.3  Hence  EMBED Equation.3  To convert  EMBED Equation.3  to the base 2, we proceed as follows. What is the smallest negative power of 2 that is less than or equal to 0.1875. That power is  EMBED Equation.3  as  EMBED Equation.3 . So  EMBED Equation.3  What is the next smallest negative power of 2 that is less than or equal to 0.0625. That power is  EMBED Equation.3  as  EMBED Equation.3 . So  EMBED Equation.3  Hence  EMBED Equation.3  Since  EMBED Equation.3  and  EMBED Equation.3  we get  EMBED Equation.3  Can you show this algebraically for any general number? Example 2 Convert  EMBED Equation.3  to base 2. Solution For  EMBED Equation.3 , conversion to binary format is shown in Table 4. Table 4. Conversion of base-10 integer to binary format. QuotientRemainder13/26 EMBED Equation.3 6/23 EMBED Equation.3 3/21 EMBED Equation.3 1/20 EMBED Equation.3  So  EMBED Equation.3 . Conversion of  EMBED Equation.3  to binary format is shown in Table 5. Table 5. Converting a base-10 fraction to binary representation. NumberNumber after decimalNumber before decimal0.875 EMBED Equation.3 21.750.75 EMBED Equation.3 0.75 EMBED Equation.3 21.50.5 EMBED Equation.3 0.5 EMBED Equation.3 21.00.0 EMBED Equation.3  So  EMBED Equation.3  Hence  EMBED Equation.3  INTRODUCTION TO NUMERICAL METHODSTopicBinary representation of numberSummaryTextbook notes on binary representation of numbersMajorGeneral EngineeringAuthorsAutar KawDate DATE \@ "MMMM d, yyyy" \* MERGEFORMAT September 4, 2014Web Site HYPERLINK "http://numericalmethods.eng.usf.edu" http://numericalmethods.eng.usf.edu     01.04. PAGE 6 Chapter 01.04 Binary Representation of Numbers 01.04. 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