【正文】 2 3 4 5 6 7 One’s Complement +0 +1 +2 +3 +4 +5 +6 +7 7 6 5 4 3 2 1 0 Two’s Complement +0 +1 +2 +3 +4 +5 +6 +7 8 7 6 5 4 3 2 1 計(jì)算機(jī)導(dǎo)論 EXCESS SYSTEM Another representation that allows you to store both positive and negative numbers in a puter is called the Excess system. In this system, it is easy to transform a number from decimal to binary, and vice versa. However, operations on the numbers are very plicated. The only application in use today is in storing the exponential value of a fraction. This is discussed in the next section. In an excess conversion, a positive number, called the magic number, is used in the conversion process. The magic number is normally (2N1) or (2N11), where N is the bit allocation. For example, if N is 8, the magic number is either 128 or 127. In the first case, we call the representation Excess_128, and in the second case, it is Excess_127. exponential ：指數(shù)的 magic number：幻數(shù) 計(jì)算機(jī)導(dǎo)論 Representation To represent a number in Excess, use the following procedure: 1. Add the magic number to the integer. 2. Change the result to binary and add 0s so that there is a total of N bits. 4位二進(jìn)制數(shù)表示帶符號(hào)整數(shù) 余 7系統(tǒng) 加 7 計(jì)算機(jī)導(dǎo)論 Represent –25 in Excess_127 using an 8bit allocation. First add 127 to get 102. This number in binary is 1100110. Add one bit to make it 8 bits in length. The representation is 01100110. Solution Example 15 計(jì)算機(jī)導(dǎo)論 Interpretation To interpret a number in excess, use the following procedure: 1. Change the number to decimal. 2. Subtract the magic number from the integer. Interpret 11111110 if the representation is Excess_127. First change the number to decimal. It is 254. Then subtract 127 from the number. The result is decimal 127. Solution Example 16 計(jì)算機(jī)導(dǎo)論 FLOATINGPOINT REPRESENTATION To represent a floatingpoint number (a number containing an integer and a fraction), the number is divided into two parts: the integer and the fraction. For example, the floatingpoint number has an integer of 14 and a fraction of . CONVERTING TO BINARY To convert a floatingpoint number to binary, use the following procedure: 1. Convert the integer part to binary. 2. Convert the fraction to binary. 3. Put a decimal point between the two parts. floatingpoint number：浮點(diǎn)數(shù) 計(jì)算機(jī)導(dǎo)論 ? Converting the integer part（整數(shù)部分） ? Converting the fraction part（小數(shù)部分） To convert a fraction to binary, use repetitive multiplication. 計(jì)算機(jī)導(dǎo)論 Transform the fraction to binary Write the fraction at the left corner. Multiply the number continuously by 2 and extract the integer part as the binary digit. Stop when the number is . ? ? ? ? 0 . 1 1 1 Solution Example 17 計(jì)算機(jī)導(dǎo)論 Transform the fraction to a binary of 6 bits. Write the fraction at the left corner. Multiply the number continuously by 2 and extract the integer part as the binary digit. You can never get the exact binary representation. Stop when you have 6 bits. ? ? ? ? ? ? 0 . 0 1 1 0 0 1 Solution Example 18 計(jì)算機(jī)導(dǎo)論 ? 例 → → 計(jì)算機(jī)導(dǎo)論 Original Number + + 001110011 Move ? 6 ? 2 6 ? 3 ? Normalized +26 x 22 x +26 x 23 x NORMALIZATION（標(biāo)準(zhǔn)化，規(guī)格化） (+)， (+)， () A standard representation for floatingpoint numbers. Normalization is the moving of the decimal point so that there is only one 1 to the left of the decimal point. To indicate the original value of the number, multiply the number by 2e, where e is the number of bits that the decimal points moved: positive for left movement, negative for right movement. A positive or negative sign is then added depending on the sign of the original number. . 計(jì)算機(jī)導(dǎo)論 SIGN, EXPONENT, AND MANTISSA After a number is normalized, you store only three pieces of information about the number: sign, exponent, and mantissa (the bits to the right of the decimal point). For example, + bees + 26 Sign: + Exponent: 6 Mantissa: 0001110101 Sign The sign of the number can be stored using 1 bit (0 or 1). Exponent The exponent (power of 2) defines the movement of the decimal point. Excess representation is the method used to store the exponent. Mantissa It defines the precision of the number. The mantissa is stored as an unsigned integer. SIGN：符號(hào) , EXPONENT ：指數(shù)， mantissa：尾數(shù) 計(jì)算機(jī)導(dǎo)論 IEEE STANDARDS The institute of electrical and electronics engineers (IEEE) has defined three standards for storing floatingpoint number。 two are used to store numbers in memory: single precision and double precision. IEEE ：電氣和電子工程師協(xié)會(huì) single precision ：?jiǎn)尉? double precision：雙精度 計(jì)算機(jī)導(dǎo)論 SinglePrecision Representation The procedure for storing a normalized floatingpoint number in memory using singleprecision format is as follows: 1. Store the sign as 0 (positive) or 1 (negative). 2. Store the exponent (power of 2) as excess_127(8 bits). 3. Store the mantissa as an unsigned integer(23 bits). 計(jì)算機(jī)導(dǎo)論 Show the representation of the normalized number + 26 x The sign is positive. The Excess_127 representation of the exponent is 133. In binary, this is 10000101. The mantissa is add extra 0s on the right to make it 23 bits. The number in memory is stored as: 0 10000101 01000111001000000000000 Solution Example 19 計(jì)算機(jī)導(dǎo)論 Floatingpoint interpretation for single precision