Abstract:
There is disclosed a method and apparatus for rounding numerals according to the number of significant digits or a rounding interval. The method starts with entering a numerical value x to be rounded and the number of significant digits n or a rounding interval w. The entered numerical value x is stored in a first storage portion of the apparatus. The entered number of significant digits n or rounding interval w is stored in a second storage memory. The numerical value x stored in. the first storage portion is rounded in a digit place determined by the number of significant digits n or rounding interval w stored in the. second storage portion. The numerical value x is rounded while using the number of significant digits n or rounding interval w as it is.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to an apparatus and method used for rounding that is necessary for analyses, measurements, investigations, and so on. More particularly, the invention relates to an apparatus and method for performing rounding processing according to significant digits or a rounding interval. 
     DESCRIPTION OF PRIOR ART 
     Generally, rounding processing typified by rounding off has enjoyed wide acceptance. Examples include reorganization of experimental data and accounting processes. Many rounding processes are prepared as functions in high-level languages and data-processing software. A user can obtain rounded a numerical value by specifying a numerical value and a digit place to be processed. 
     FIG. 14 illustrates processing performed using a conventional rounding function. First, a user enters a numerical value x to be processed and a digit place p to be processed (step  601 ). If the digit place p to be processed is the place of 100, for example, p is −2. If an integer should be derived, p is 0. If the digit place is the third decimal place, p is +3. Then, the digits in the successive digit places of the given numerical value x are stored in turn in memory (step  602 ). A digit in the digit place immediately to the right of the specified digit place p is read from the memory (step  603 ). A decision is made as to whether the digit is equal to or greater than 5 (step  604 ). If the result of the decision is YES (i.e., the numerical value is equal to or greater than 5), rounding up is performed (step  605 ). If the result of the decision is NO, rounding down is performed (step  606 ). In this way, the conventional conditional branching processing using a function takes out only one digit from digits in plural digit places. Then, a conditional branch is effected. 
     The prior art technique described above has the following problems. 
     (1) It is impossible to specify the processed digit place by significant digits. 
     Spreadsheet software programs presently available on the market do not have the concept that the processed digit place is specified by significant digits. If a digit place to be processed is counted from the leftmost digit place and specified, other conditional branching processing is necessary so that the processed digit place is specified by significant digits across a decimal point, as long as the prior art conditional branching processing is performed (i.e., only one digit is taken from plural digit places and a decision is made). For example, where 0.0000234 should be rounded to two significant digits, if this numerical value is counted as it is, then the digit in the third decimal place will be taken and a rounding process will be performed. Therefore, digits are taken in turn from the leftmost digit place and counted. A decision is made as to whether each taken digit is 0 or not. If it is 0, this digit place is not counted. In this way, conditional branching processing is needed. Furthermore, where a rounding process is executed with four significant digits, if the numerical value is 30200, “0” in the fourth digit place, “0” in the second digit place, and “0” in the first digit place all need to be counted. In the case of 0.34502, “0” in the first digit place(the most significant digit) is not counted. However, “0” in the fourth decimal place and so on must be counted. Therefore, more conditional branches are necessary. 
     Where the prior art processing sequence as described above is used, if the digit to be rounded is specified by significant digits, very complicated processing inevitably takes place. For this reason, the prior art function does not make it possible to specify a digit place by significant digits. Furthermore, any spreadsheet software capable of specifying a desired digit place by a significant number is not available. 
     (2) Rounding processing directly using a rounding interval cannot be performed. 
     Rounding using a rounding interval is based on ISO (International Standards Organization) 31-0:1992(E), appendix B, “Guide to the rounding of numbers”. Rounding using a rounding interval is to replace a given numerical value by a numerical value selected from a sequence of integral multiples of a chosen rounding interval. For instance, where the rounding interval is 0.1, 12.849 is rounded to 12.8 and 13.451 is rounded to 13.5. Where the rounding interval is 10, 1284.9 is rounded to 1280, and 1345.1 is rounded to 1350. Accordingly, in rounding processing using a rounding interval, what is entered first is not the processed digit place but a rounding interval. However, spreadsheet software programs currently available on the market do not have the concept that rounding is performed based on a rounding interval. Hence, it has been impossible to perform a rounding process by directly entering a rounding interval. Therefore, in order to perform the prior art rounding process using a rounding interval, additional processing for converting the rounding interval to the processed digit place is necessary. This complicates the processing, and the processing time is prolonged. 
     (3) It is impossible to cope with JIS (Japanese Industrial Standard). 
     Where simple rounding off is used, the result is biased according to the numerical values contained in the data. Therefore, rounding for correcting the biased result and minimizing the round-off error is necessary. JIS stipulates a rounding method for that purpose. Many materials submitted to the public agencies have numerical values that must be rounded in accordance with JIS. 
     FIG. 15 illustrates a rounding method stipulated by JIS, Z8401. Fundamentally, a round-off process is performed, and the remaining 5 is sorted. Where a numerical value is rounded to a numerical value having n significant digits, a digit in the (n+1)th place and less significant digits are reorganized as follows. 
     (a) Where a digit in the (n+1)th digit place or a less significant digit is less than half of 1 unit in the nth digit place, the digit is dropped. 
     (b) Where a digit in the (n+1)th digit place or a less significant digit is in excess of half of 1 unit in the nth digit place, the digit in the nth digit place is increased by 1 unit. 
     (c) Where the digit in the (n+1)th place or a less significant digit is equal to half of 1 unit in the nth digit place, the following processes (i) and (ii) are carried out. 
     (i) If the digit in the nth digit place is 0, 2, 4, 6, or 8, the digit is dropped. 
     (ii) If the digit in the nth digit place is 1, 3, 5, 7, or 9, the digit in the nth place is increased by 1 unit. 
     Significant digits are counted from the place of the most significant digit that is nonzero. This rounding must be performed in one stage. For example, if 5.346 is rounded to two significant digits by this method, then 5.3 results. If the rounding is done in two stages, the numerical value is rounded to 5.35 in the first stage and to 5.4 in the second stage. 
     The processing is conducted in the sequence in the manner described below. First, the number of digits is judged (step  701 ). A decision is made as to whether the digit in the (n+1)th digit place is 6 or more (step  702 ). If the digit is 6 or more, the digit in the nth digit place is increased by 1. The digit in the (n+1)th place and less significant digits are dropped (step  706 ). For example, if 13.461 is rounded to three significant digits, 13.5 results. 
     Then, a decision is made as to whether the digit in the (n+1)th digit place is 4 or less (step  703 ). If it is 4 or less, the digit in the (n+1)th place and less significant digits are dropped (step  707 ). For example, if 12.849 is rounded to three significant digits, 12.8 is derived. 
     If the digit in the (n+1)th place is neither equal to or greater than 6 nor equal to or less than 4, i.e., the digit in the (n+1)th digit place is 5, a decision is made as to whether the digit in the (n+2)th place and less significant digits are all 0 (step  704 ). If the digit in the (n+2)th place and less significant digits contain nonzero numeral or numerals, the digit in the nth digit place is increased by 1, and the digit in the (n+1)th place and less significant digits are deleted (step  706 ). For example, if 13.451 is rounded to three significant digits on this principle, 13.5 results. 
     If the digit in the (n+2)th digit place and less significant digits are all 0, a decision is made as to whether the digit in the nth place is even or odd (step  705 ). If the result is that the digit is even, then the digit in the (n+1)th place and less significant digits are deleted (step  707 ). For example, if 11.450 is rounded to three significant digits, 11.4 results. Conversely, if the digit in the nth place is odd, the digit in the nth place is increased by 1, and the digit in the (n+1)th place and less significant digits are dropped (step  706 ). For instance, if 12.750 is rounded to three significant digits, 12.8 arises. 
     However, if digits necessary for decision are taken from all the digit places of the digits as in the conditional branching processing using the prior art round-off function, and if the taken digits are treated as one number, numerous conditional branches (steps  804 - 807 ) are necessary in the conditional branching of step  704  of FIG. 15, as illustrated in FIG.  16 . 
     This portion is used to judge that the digit in the (n+1)th digit place is 5 and that the less significant digits are all 0. Where necessary digits are extracted from all digit places and treated as one number, the digits in all the digit places must be judged. 
     For example, where digits located to the right of a rounded digit are 50000001, JIS demands rounding up. If a human makes a decision by his eyes, he can immediately sense that a nonzero digit is present in some less significant digit place. Where a computer is used, the method of decision presents a problem. In the prior art processing where a decision is made for each digit place, a decision is made as to whether the digit in the (n+2)th digit place 0 or not. Then, a decision is made as to whether the digit in the (n+3)th place is 0 or not. Subsequently, a decision is made as to whether the digit in the (n+4)th place is 0 or not. In this way, decisions are made up to the rightmost digit place to judge whether 0 is not contained at all in the (n+2)th and the following places or any nonzero numeral is present in any digit place. Since the digits in the successive digit places are judged in turn, the final judgment cannot be made until a multiplicity of branching operations are carried out. 
     Where the number of digits capable of being calculated by the today&#39;s computer is contemplated, it is expected that the number of the conditional branch operations will be exorbitant. Accordingly, in the conventional procedure, it is necessary to limit the number of treated digits to far below the processing capability of the computer to reduce the number of the necessary conditional branch operations. Therefore, even if the procedure is treated in terms of software, calculation is urged to be performed at a level considerably lower than the computer&#39;s computational capability. This narrows the range that can be processed. As a result, the software would not be viable on the market. In this way, the present situation is that any software capable of performing rounding operations in accordance with JIS Z8401 has not been developed or put into practice. 
     In the rounding processing described above, it is impossible to directly enter a rounding interval for rounding processing. In order to perform the prior art rounding process using a rounding interval, additional processing for converting the rounding interval into a digit place is necessary. Also, in this case, the processing is complicated and the processing time is prolonged. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide an apparatus and method capable of performing a rounding process by rounding a numerical value, using significant digits or a rounding interval as it is. 
     It is another object of the invention to provide an apparatus and method capable of rounding numbers with minimum round-off error as in accordance with JIS (Japanese Industrial Standard), using significant digits or a rounding interval as it is. 
     A rounding apparatus in accordance with the present invention comprises an input portion for entering a numerical value x to be rounded and the number of significant digits n or a rounding interval w, a first storage portion for storing the entered numerical value x, a second storage portion for storing the entered number of significant digits n or rounding interval w, and a rounding processing portion for rounding the numerical value x stored in the first storage portion in a digit place specified by the number of significant digits n or rounding interval w stored in the second storage portion. This apparatus can round the numerical value while using the number of significant digits or rounding interval as it is. 
     Other objects and features of the invention will appear in the course of the description thereof, which follows. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram of a rounding processor in accordance with a first embodiment of the present invention; 
     FIG. 2 is a block diagram of one example of the hardware of the rounding processor shown in FIG. 1, realized using a computer; 
     FIG. 3 is a flowchart illustrating rounding processing performed by the rounding processor shown in FIG. 1; 
     FIG. 4 is a block diagram of a rounding processor in accordance with a second embodiment of the invention; 
     FIG. 5 is a flowchart illustrating rounding processing performed by the rounding processor shown in FIG. 4; 
     FIG. 6 is a block diagram of a rounding processor in accordance with a third embodiment of the invention; 
     FIG. 7 is a flowchart illustrating rounding processing performed by the rounding processor shown in FIG. 6; 
     FIG. 8 is a block diagram of a rounding processor in accordance with a fourth embodiment of the invention; 
     FIG. 9 is a flowchart illustrating rounding processing performed by the rounding processor shown in FIG. 8; 
     FIG. 10 is a block diagram of a rounding processor in accordance with a fifth embodiment of the invention; 
     FIG. 11 is a flowchart illustrating rounding processing performed by the rounding processor shown in FIG. 10; 
     FIG. 12 is a block diagram of a rounding processor in accordance with a sixth embodiment of the invention; 
     FIG. 13 is a flowchart illustrating rounding processing performed by the rounding processor shown in FIG. 12; 
     FIG. 14 is a flowchart illustrating the prior art processing procedure for rounding; 
     FIG. 15 is a flowchart illustrating the prior art processing procedure for performing rounding specified by JIS, using a computer; and 
     FIG. 16 is a flowchart illustrating the prior art processing procedure for performing rounding specified by JIS, using a computer, by detailed branching processing. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The preferred embodiments of the present invention are hereinafter described by referring to the drawings. A rounding processor in accordance with a first embodiment of the present invention is shown in FIG. 1 in block-diagram form. First and second embodiments described below pertain to rounding processors for performing rounding processes using a rounding interval. 
     Referring to FIG. 1, the rounding processor comprises an input portion  1 , a storage portion  2 , an extraction portion  3 , a rounding processing portion  4 , and an output portion  5 . The storage portion  2  has a numeral storage portion  21  and a rounding interval storage portion  22 . The extraction portion  3  has a less significant numeral extraction portion  31  and a rounded numeral extraction portion  32 . The rounding processing portion  4  has a remainder calculation portion  41 , an adder portion  42 , and a dropping portion  43 . 
     The input portion  1  is used to enter a numerical value x to be rounded and a rounding interval w. The numeral storage portion  21  stores digits in the successive digit places of the entered numerical value x. The rounding interval storage portion  22  stores the entered rounding interval w. The less significant numeral extraction portion  31  extracts a numerical value y from the numerical value x stored in the numeral storage portion  21 , the numerical value y being formed by digits in digit places to the right of a digit place corresponding to the rounding interval w stored in the rounding interval storage portion  22 . The rounded numeral extraction portion  32  extracts a rounded digit R from the numerical value x stored in the numeral storage portion  21 , the digit R being placed in a digit place corresponding to the rounding interval w stored in the rounding interval storage portion  22 . The remainder calculation portion  41  calculates the remainder b when the rounded digit R extracted by the rounded numeral extraction portion  32  is divided by 2. If the rounding interval w stored in the rounding interval storage portion  22  is less than twice of the less significant numerical value y extracted by the less significant numeral extraction portion  31 , or if the rounding interval w stored in the rounding interval storage portion  22  is equal to twice of the less significant numerical value y extracted by the less significant numeral extraction portion  31  and, at the same time, the remainder b calculated by the remainder calculation portion  41  is 1, then the adder portion  42  adds 1 to the rounded digit R extracted by the rounded numeral extraction portion  32  and rewrites the rounded digit R stored in the numeral storage portion  21 . The dropping portion  43  deletes the less significant numerical value y extracted by the less significant numeral extraction portion  31  and changes the digits forming the less significant numerical value y in the numeral storage portion  21  to 0. The output portion  5  produces a rounded numerical value formed by the rewritten digits stored in the numeral storage portion  21 . A rounded numeral storage portion for storing a rounded numerical value may be formed in the storage portion  2  independent of the numeral storage portion  21 . 
     On example of hardware of the aforementioned rounding processor realized using a computer is shown in FIG.  2 . 
     This rounding processor comprises an input device  11 , a display device  12 , an external storage  13 , a recording medium  14 , a RAM  15 , a CPU  16 , a ROM  17 , a printer  18 , and a bus line  19 . The input device  11  comprises a keyboard having a ten-key pad, a mouse, etc., and is used by the user to select desired processing or to enter a numerical value x rounded and a rounding interval w. The entered data are sent to the CPU  16  via the bus line  19 . If necessary, the data are stored in the RAM  15  or on the recording medium  14  in the external storage  13 . The display device  12  comprises a liquid crystal display, a CRT, or the like, and displays a desired frame of image. The external storage  13  comprises a hard disk drive, a floppy disk drive, a CD-ROM drive, a magneto-optical drive, or the like. The external storage  13  records data processed by the CPU  16  on the recording medium  14 , for example, a hard disk, a floppy disk, a CD-ROM, a magneto-optical disk, or the like, or sends data recorded on the recording medium  14  to the CPU  16 . The data processed by the CPU  16  is stored in the RAM  15 . If necessary, the data are read out by the CPU  16 . A program for performing the functions of the rounding processor shown in FIG. 1 is stored in the ROM  17  or in the recording medium  14 . The CPU  16  controls various components via the bus line  19  in accordance with this program. The printer  18  comprises a laser printer or the like, and prints out data processed by the CPU  16 . 
     The rounding processor constructed in this way and shown in FIG. 2 corresponds to the rounding processor shown in FIG. 1 as follows, for example. The input portion  1  corresponds to the input device  11 , the RAM  15 , and/or the external storage  13 . The storage portion  2  corresponds to the RAM  15  and/or the external storage  13 . The extraction portion  3  and the rounding processing portion  4  correspond to the CPU  16 . The output portion  5  corresponds to the display device  12  and/or the printer  18 . The program corresponding to the rounding processing described below may be previously stored in the ROM  17 . Alternatively, the program may be installed from the recording medium  14  such as a CD-ROM, a floppy disk, a magneto-optical disk, or a semiconductor memory, using the external storage  13 . 
     No limitations are imposed on the computer described above. An ordinary computer can be used. For example, a server, a desktop computer, a laptop computer, a portable terminal, or the like may be used. Furthermore, it can be an analytical instrument or a measuring instrument equipped with a microcomputer or the like. In addition, no limitations are placed on the used programming language. The program can be written in terms of various languages. In the case of an analytical instrument or measuring instrument, a calculation/measurement result storage portion for storing numerical data of the measurement or experiment results and of the calculation results data may be incorporated in the recording medium  14  of the external storage  13  or in the RAM  15 . These principles apply similarly to the descriptions given below. 
     The rounding processing executed by the rounding processor constructed as described above is next described in detail. FIG. 3 is a flowchart illustrating the rounding performed by the rounding processor shown in FIG.  1 . 
     As shown in FIG. 3, in step  1 , a numerical value x to be rounded and a rounding interval w are entered from the input portion  1 . Digits in the successive digit places of the numerical value x are stored in the numeral storage portion  21 . The rounding interval w is stored in the rounding interval storage portion  22 . A human operator may directly enter the numerical value x by the use of the input portion  1 . In the case of a measuring instrument and the like, numerical values stored in the calculation/measurement result storage portion of the storage portion  2  may be entered into the numeral storage portion  21  and into the rounding interval storage portion  22 . 
     In the next step  2 , the rounded numeral extraction portion  32  extracts a rounded digit R from the numeral storage portion  21 , the digit R being placed in a digit place corresponding to the rounding interval w. For example, if 12.354 is entered as the numerical value x and 0.1 is entered as the rounding interval w, digit “3” in the first decimal place corresponding to the rounding interval of 0.1 of digits “1”, “2”, “3”, “5”, and “4” stored in the numeral storage portion  21  is extracted. 
     In the next step  3 , the rounded digit R is divided by 2 by the remainder calculation portion  41 , resulting in a remainder of b which is either 0 or 1. If the remainder b is 0, the rounded digit R is even. If the remainder is 1, the rounded digit R is odd. 
     In the next step  4 , the less significant numeral extraction portion  31  extracts a less significant numerical value y composed of digits in successive digit places to the right of the digit place corresponding to the rounding interval w. For example, if 12.354 is entered as the numerical value x and 0.1 is entered as the rounding interval w, a less significant numeral 0.054 composed of digits “5” and “4” in digit places to the right of the digit place corresponding to the rounding interval of 0.1 is extracted from “1”, “2”, “3”, “5”, and “4” stored in the numeral storage portion  21 . 
     In the next step  5 , the adder portion  42  makes a decision as to whether the rounded interval w is smaller than the twice of the less significant numerical value y or whether the rounded interval w is equal to the twice of the less significant numerical value y and, at the same time, the remainder b is 1. If the rounding interval w is smaller than the twice of the less significant numerical value y or if the rounded interval w is equal to the twice of the less significant numerical value y and, at the same time, the remainder b is 1, control goes to step  6 . If the result of the decision is NO (i.e., the rounding interval w is greater than twice of the less significant numeral or the rounding interval w is equal to twice of the less significant numerical value y and, at the same time, the remainder b is 0), control proceeds to step  7 . 
     This method is compared with the rounding method in accordance with JIS (Japanese Industrial Standard) described above. Where the rounding interval w is smaller than the twice of the less significant numerical value y, a numerical value composed of digits in successive digit places to the right of the (n+1)th digit place is in excess of half of 1 unit in the nth digit place, i.e., the digit in the nth place is increased by 1 unit. Where the rounded interval w is equal to the twice of the less significant numerical value y and, at the same time, the remainder b is 1, a numerical value composed of digits in successive digit places to the right of the (n+1)th digit place is half of 1 unit in the nth place, and the digit in the nth place is 1, 3, 5, 7, or 9, i.e., the digit in the nth place is increased by 1 unit. Where the rounding interval w is greater than the twice of the less significant numerical value y, a numerical value composed of digits in successive digits to the right of the (n+1)th place is smaller than 1 unit in the nth place, i.e., rounding off is performed. Where the rounding interval w is equal to the twice of the less significant numerical value y and, at the same time, the remainder b is 0, the numerical value composed of digits in digit places to the right of the (n+1)th place is half of 1 unit in the nth digit place, and the digit in the nth digit place is 0, 2, 4, 6, or 8, i.e., rounding off is performed. Therefore, the decision processing described above is adapted for decision processing in rounding of numerals in accordance with JIS. 
     In step  5 , if the result of the decision is that the rounding interval w is smaller than the twice of the less significant numerical value y, or if the rounding interval w is equal to the twice of the less significant numerical value y and, at the same time, the remainder b is 1, then control goes to step  6 . One is added to the rounded digit R, rewriting the rounded digit R stored in the numeral storage portion  21  (step  6 ). If a carry occurs in an addition, the rounded digit R is rewritten as 0. One is added to the digit in the digit place immediately to the left of the digit place of the rounded digit R, thus rewriting the numerical value. 
     In the next step  7 , the less significant numerical value y is deleted, and the digits forming the less significant numerical value y stored in the numeral storage portion  21  are changed to 0. 
     Finally, in step  8 , a numerical value (i.e., rounded numerical value x) composed of the rewritten digits stored in the numeral storage portion  21  is output. 
     As a result of the steps  5 - 7 , if the rounding interval w is smaller than the twice of the less significant numerical value y (i.e., the numerical value formed by digits in digit places to the right of the (n+1)th digit place is in excess of half of 1 unit in the nth digit place, or if the rounding interval w is equal to the twice of the less significant numerical value y and, at the same time, the remainder b is 1 (i.e., the numerical value formed by digits in digit places to the right of the (n+1)th digit place is half of 1 unit of the nth digit place and, at the same time, the digit in the nth digit place is 1, 3, 5, 7, or 9), then 1 is added to the rounded digit R, and the less significant numerical value y is dropped. Thus, rounding up is performed. 
     Meanwhile, if the rounding interval w is greater than the twice of the less significant numerical value y (i.e., the numerical value formed by digits in digit places to the right of the (n+1)th digit place is less than half of 1 unit in the nth digit place), or if the rounding interval w is equal to the twice of the less significant numerical value y and, at the same time, the remainder b is 0 (i.e., the numerical value formed by digits in digit places to the right of the (n+1)th place is half of 1 unit in the nth place and the numeral in the nth place is 0, 2, 4, 6, or 8), the rounded digit R is left as it is, and the less significant numerical value y is dropped. Thus, a round-down operation is carried out. 
     Therefore, because of the processing described above, the rounding interval can be used as it is for comparisons without converting the rounding interval into a digit place. Also, a rounding operation for minimizing rounding-off error as in JIS can be performed. In the processing described above, the multiplier and the divisor are 2 and so in the case of a binary system, multiplications and divisions can be done simply by means of shifting operations. Hence, a rounding operation can be performed with less process steps. Consequently, it is easy to prepare the program. Also, the processing time can be shortened. 
     A rounding processor in accordance with a second embodiment of the invention is next described. FIG. 4 is a block diagram of this rounding processor, which performs round-off operations using a rounding interval as it is. This processor shown in FIG. 4 is similar to the rounding processor shown in FIG. 1 except that the remainder calculation portion  41  of the rounding processor shown in FIG. 1 is omitted and that the adder portion  42  is replaced by an adder portion  42   a  which makes decisions based on different conditions. Therefore, like components are indicated by like reference numerals and those components which have been already described will not be described below. Hardware of the rounding processor shown in FIG. 4 that is realized using a computer is similar to the hardware shown in FIG.  2  and thus will not be illustrated. 
     As shown in FIG. 4, a rounding processing portion  4 a comprises the adder portion  42   a  and the dropping portion  43 . If the rounding interval w stored in the rounding interval storage portion  22  is equal to or smaller than the twice of the less significant numerical value y extracted by the less significant numeral extraction portion  31 , then the adder portion  42   a  adds 1 to the rounded digit R extracted by the rounded value extraction portion  32 , and the rounded digit R stored in the numerical storage portion  21  is rewritten. 
     The rounding processing executed by the rounding processor constructed as described above is next described. FIG. 5 is a flowchart illustrating the rounding processing performed by the rounding processor shown in FIG.  4 . The flowchart of FIG. 5 is similar to the flowchart of FIG. 3 except that step  3  of the flowchart of FIG. 3 is omitted and that step  5  is changed to step  9 . Therefore, only their differences are described. 
     As illustrated in FIG. 5, in steps  1 ,  2 , and  4 , processing similar to that performed in steps  1 ,  2 , and  4  of FIG. 3 is performed. In step  9 , a decision is made as to whether the rounding interval w is equal to or smaller than the twice of the less significant numerical value y. If the result of this decision is YES, control goes to step  6 . If the result of the decision is NO (i.e., the rounding interval w is greater than the twice of the less significant numerical value y), control proceeds to step  7 . That the rounding interval w is equal to or smaller than the twice of the less significant numerical value y means that the most significant digit of the less significant number y is equal to or greater than 5, i.e., a round-up operation is performed. That the rounding interval w is greater than the twice of the less significant numerical value y means that the most significant digit of the less significant numerical value y is 4 or less, i.e., a round-down operation is performed. 
     In the next steps  6 - 8 , processing similar to the processing of steps  6 - 8  of FIG. 3 is performed. Finally, a numerical value composed of rewritten digits stored in the numeral storage portion  21 , i.e., a numerical value obtained by rounding off the numerical value x, is produced. Accordingly, in the present embodiment, the rounding interval can be used as it is for comparisons in rounding processing without converting the rounding interval into a digit place. In the processing described above, 2 is used as the multiplier. Therefore, in the binary system, multiplications are performed simply by means of shifting operations. In consequence, rounding processing can be performed with less process steps. Hence, it is easy to prepare the program. Also, the processing time can be shortened. 
     A rounding processor in accordance with a third embodiment of the present invention is next described. FIG. 6 is a block diagram showing the structure of this rounding processor that specifies significant digits and performs a round-off operation. This round-off operation can be carried out by combining a round-down procedure illustrated in FIG. 9 with a round-up procedure illustrated in FIG.  11 . 
     As shown in FIG. 6, the rounding processor comprises an input portion  1 , a storage portion  2 , an arithmetic portion  3 , a rounding processor portion  4 , and an output portion  5 . The storage portion  2  comprises a significant digit storage portion  21 , an unrounded numeral storage portion  22 , an integer part storage portion  23 , an index part storage portion  24 , and a rounded numeral storage portion  25 . The arithmetic portion  3  comprises an integer/index calculation portion  31  and a digit place restoration portion  32 . The rounding processor  4  comprises a five-judging portion  41 , an odd/even discrimination portion  42 , a round-off judging portion  43 , a round-down portion  44 , and a round-up portion  45 . 
     The input portion  1  is used to enter a numerical value x to be rounded and the number of significant digits n. The unrounded numeral storage portion  22  stores all the digits in successive digit places of the entered numerical value x. The significant digit storage portion  21  stores the entered number of significant digits n. The integer/index calculation portion  31  calculates an integer K and an index M where the numerical value x stored in the unrounded numeral storage portion  22  is represented in K×10 M  form. That is, the original numeral is represented by the product of K and 10 M , where K is a numeral whose least significant digit is not zero, and nonzero values are not present in decimal places. The index part storage portion  24  stores the index M calculated by the integer/index calculation portion  31 . The integer part storage portion  23  stores the integer K calculated by the integer/index calculation portion  31 . The five-judging portion  41  takes a numerical value formed by digits in digit places of the integer K stored in the integer part storage portion  23  which are located in the (n+1)th and following digit places, and makes a decision as to whether this taken numerical value is equal to 5. If the result of the decision is YES (i.e., it is equal to 5), the odd/even discrimination portion  42  divides the digit in the nth digit place of the integer K by 2, and makes a decision as to whether the remainder is 0. If the decision made by the five-judging portion  41  is that the taken one numeral is not equal to 5, the round-off judging portion  43  makes a decision as to whether the digit in the (n+1)th digit place of the integer K is equal to or greater than 5. If the odd/even discrimination portion  42  judges that the remainder is 0, or if the round-off judging portion  43  judges that the digit in the (n+1)th digit place is not equal to or greater than 5, the round-down portion  44  deletes digits in the (n+1)th and the following digit places, and rewrites the integer K in the integer part storage portion  23 . The round-up portion  45  increases the digit in the nth digit place of the integer K by 1, deletes the digits in the (n+1)th and following digit places, and rewrites the integer K in the integer part storage portion  23 , if the odd/even discrimination portion  42  judges that the remainder is nonzero or if the round-off judging portion  43  judges that the digit in the (n+1)th place is 5 or more. Because of the processing described above, the integer K is rounded with n significant digits using the integer K stored in the integer part storage portion  23  as well as the number of significant digits n stored in the significant digit storage portion  21 . The rounded integer L can be stored in the integer part storage portion  23 . The digit place restoration portion  32  calculates L×10 M , using the integer L stored in the integer part storage portion  23  and the index M stored in the index part storage portion  24 , restores the digit place, and stores the result of the calculation in the rounded numeral storage portion  25 . The output portion  5  delivers the numerical value with n significant digits stored in the rounded value storage portion  25 . Instead of providing the rounded value storage portion  25 , the numeral in the unrounded numeral storage portion  22  may be directly rewritten, and the rewritten numeral in the unrounded numeral storage portion  22  may be delivered from the output portion  5 . 
     This rounding processor can also be constructed using the hardware shown in FIG.  2  and so the relation between the hardware and the rounding processor using a computer is described by referring to FIG.  2 . 
     The hardware shown in FIG.  2  and the rounding processor shown in FIG. 6 have the following relationship. The input portion  1  corresponds to the input device  11 , the RAM  15 , and/or the external storage  13 . The storage portion  2  corresponds to the RAM  15  and/or the external storage  23 . The arithmetic portion  3  and the rounding processing portion  4  correspond to the CPU  16 . The output portion  5  corresponds to the display device  12  and/or the printer  18 . 
     The rounding processing executed by the rounding processor constructed as described above is next described in detail. FIG. 7 is a flowchart illustrating the rounding processing performed by the rounding processor shown in FIG.  6 . 
     Referring to FIG. 7, in step  11 , a numerical value x to be rounded and the number of significant digits n are first entered from the input portion  1 . The digits in the successive digit places of the numerical value x are stored in the unrounded numeral storage portion  22 . The number of significant digits n is stored in the significant digit storage portion  21 . The numerical value x may be entered directly by the operator through the input portion  1 . In the case of a measuring instrument and the like, numerals stored in the calculation/measurement result storage portion of the storage portion  2  may be loaded into the unrounded numeral storage portion  22  and into the significant digit storage portion  21 . 
     In the next step  12 , the numerical value x yet to be processed is divided into an integer part and an index part. The integer/index calculation portion  31  calculates integer K and index M where x=K×10 M , in which the least significant digit of K is nonzero, and the original numeral is represented in terms of 10 M  such that nonzero digits are not present in decimal places. For example, if x is 0.00021, it can be written as 21×10 −5 . Therefore, K is  21  and M is −5. If the numerical value x is 320100, it can be written as 3201×10 2 . Therefore, the integer K is 3201 and the index M is 2. 
     In the next step  13 , the calculated index M is stored in the index part storage portion  24 . In step  14 , the digits in successive digit places of the calculated integer K are stored in the integer part storage portion  23 . For example, if the integer K is 3201, the digits 3, 2, 0, and 1 in the successive digit places are separately stored. 
     In the next step  15 , a digit in the (n+1)th digit place and digits in the following digit places of the integer K stored in the integer part storage portion  23  are returned to one numeral by the five-judging portion  41 . Thus, the numeral is divided into a first part between the leftmost digit place and the nth digit place and a second part in the (n+1)th and the following digit places. The digits in the (n+1)th and the following digit places are taken as one numeral. In the next step  16 , the five-judging portion  41  divides this taken numeral by 5, and makes a decision as to whether the quotient is 1 or not. 
     In the present embodiment, the numerical value x yet to be rounded is divided into an integer part and an index part, and the number formed by digits in the (n+1)th and the following digit places of the integer K is divided by 5 because of the processing described above. A decision substantially the same as the decision made as to whether the digits in the (n+2)th and the following places are all zero where the digit in the (n+1)th place is 5 can be made. This decision has been regarded as being difficult to make where rounding is done in accordance with JIS. In particular, where a number formed by digits in the (n+1)th place and in the following digit places is 5 (i.e., the least significant nonzero digit of the numeral is 5), the most serious problem takes place. In the present invention, the unrounded numerical value x is represented in terms of an index, and its integer part is used. Therefore, in such a case, the digit in the first digit place located to the left of the decimal point of the integer K is always 5. Since the nth digit place is the second place located to the left of the decimal point of the integer K, if the processing of step  15  is undergone, 5 is taken as a number formed by digits in the (n+1)th and the following places. This taken numeral 5 is divided by 5, and a decision is made as to whether the quotient is 1. Therefore a case in which the least significant digit of the unrounded numerical value x is 5 and this digit place corresponds to the (n+1)th digit place can be found. 
     Where 0.0125 is rounded with two significant digits in accordance with JIS, if the numeral is divided into an integer part and an index part, the numeral is given by 125×10 −4 . Therefore, the integer K is 125 and the index M is −4. If 125 is divided into a first portion up to the nth digit part (in this case, up to the second digit part) and a second portion in the (n+1)th digit place and following digit places (in this case, the third digit place), the numeral is divided into 12 and 5. If the latter numeral 5 is divided by 5, then 1 results. On the other hand, where 0.0125001 is rounded with two significant digits in accordance with JIS, if the numeral is divided into an integer part and an index part, 125001×10 −7  results. Therefore, the integer K is 125001 and the index M is −7. If 125001 is divided into a first portion up to the second digit place and a second portion to the right of the second digit place, the numeral is divided into 12 and 5001. However, if 5001 is divided by 5, the quotient is not equal to 5. In this case, therefore, an ordinary round-off operation should be performed. 
     If the result of the decision made in step  16  is that the quotient (i.e., the numeral formed by digits in the (n+1)th and the following digit places divided by 5) is 1, control goes to step  17 , where the digit in the nth digit place of the integer part K stored in the integer part storage portion  23  is read. A decision is made as to whether the remainder occurring when the digit in the nth place is divided by 2 is 0 or not. If the remainder is judged to be 0 (i.e., the digit in the nth place is even), the round-down portion  44  deletes digits in the (n+1)th and the following digit places. If the result of the decision is that the remainder is not 0 (i.e., the numeral in the nth digit place is odd), the round-up portion  45  increases the digit in the nth place by 1 and deletes the digits in the (n+1)th and the following digit places, thus performing rounding up. 
     If the result of the decision made in step  16  is that the quotient (the numeral formed by the digits in the (n+1)th and the following places divided by 5) is not 1, an ordinary round-off operation is performed (steps  20 ,  19 , and  21 ). That is, in step  16 , a case in which the digit in the (n+1)th digit place is 5 and all digits in the (n+2)th and following digit places are zero is detected. In other cases, if an ordinary round-off operation is performed, it is possible to cope with JIS. First, in step  20 , the round-off judging portion  43  reads the digit in the (n+1)th digit place and makes a decision as to whether it is equal to or greater than 5. At this time, if the digit in the (n+1)th digit place is equal to or greater than 5, control proceeds to step  19 , where rounding up is performed. If the digit in the (n+1)th digit place is not equal to or greater than 5, i.e., equal to or less than 4, control goes to step  21 , where a round-down operation is effected. The integer K is rounded by the processing described above. 
     The digit place restoration portion  32  reads the index M from the index part storage portion  24  (step  22 ), and multiplies the rounded-down integer L by the index part 10 M  and stores the product in the rounded value storage portion  25  (step  23 ). Finally, in step  24 , the numeral stored in the rounded value storage portion  25  is output through the output portion  5  (step  209 ). 
     Where the processing illustrated in FIG. 7 is performed in the example described above in which 0.0125 and 0.0125001 are rounded with two significant digits, the numeral is varied from the input to the output in the manner described below. 
     In the case of 0.0125, it is given by 125×10 −4  and so the integer K is 125 and the index is −4. Accordingly, the index −4 is written to the index part storage portion  24  (step  13 ). With respect to the integer K, digits 1, 2, and 5 in the successive digit places are written to the integer part storage portion  23  (step  14 ). 
     If the digits in the (n+1)th and following digit places (i.e., the third and the following digit places.) of the integer K are returned to one numeral, 5 is extracted (step  15 ). If this 5 is divided by 5 (step  16 ), a quotient of 1 is obtained. Therefore, processing of step  17  is performed. In this case, the digit in the nth place, i.e., the second place, is 2. This is divided by 2, resulting in a remainder of 0. Control goes to step  18 , where digits in the (n+1)th and the following places, i.e., the third and the following places, are deleted. The result of the processing is 120. 
     The index −4 stored in the index part storage portion  24  is read (step  22 ). Calculation 120×10 −4  is performed (step  23 ). The result of the calculation is 0.012, which is output (step  24 ). 
     In the case of 0.0125001, it is given by 125001×10 −7 . Therefore, the integer K is 125001, and the index M is −7. Thus, the index −7 is written to the index part storage portion  24  (step  13 ). With respect to the integer K, digits 1, 2, 5, 0, 0, and 1 in the successive digit places are stored in the integer part storage portion  23  (step  14 ). 
     If the digits in the (n+1)th and the following digit places, i.e., the third and the following places, of the integer K are returned to one numeral, 5001 is taken (step  15 ). If this 5001 is divided by 5 (step  16 ), the quotient is not 1 and so the processing of step  20  is performed. In this case, the digit in the (n+1)th place, i.e., the third place, is 5. Therefore, control goes to step  19 , where rounding up is performed. The result of the processing is 130000. 
     The index of −7 is read from the. index part storage portion  24  (step  22 ), and calculation 130000×10 −7  is performed (step  23 ). A product of 0.013 is output (step  24 ). 
     In the description provided above, to avoid misunderstanding of digit places, one numeral formed by the digits in the (n+1)th and following digit places is divided by 5, and a decision is made as to whether the quotient is 1or not (step  16 ). Alternatively, a decision as to whether this one numeral is 5 or not may be immediately made. 
     A rounding processor in accordance with a fourth embodiment of the present invention is next described. FIG. 8 is a block diagram showing the structure of the rounding processor in accordance with the fourth embodiment. This rounding processor performs rounding processing by making use of rounding off using significant digits. This rounding processor shown in FIG. 8 is similar to the rounding processor described already in connection. with FIG. 6 except that the five-judging portion  41 , the odd/even discrimination portion  42 , the round-off judging portion  43 , and the round-up portion  45  of the processor shown in FIG. 6 are deleted and that only rounding down is performed by a round-down portion  44 . Note that like components are indicated by like reference numerals in various figures and that those components which have been already described will not be described in detail below. The structure of hardware of the rounding processor shown in FIG. 8 realized using a computer is similar to the structure shown in FIG.  2  and so the structure of hardware will not be illustrated. 
     Rounding processing is performed by the rounding processor in the manner described below. FIG. 9 is a flowchart illustrating rounding processing performed by the rounding processor shown in FIG.  8 . The flowchart of FIG. 9 is similar to the flowchart of FIG. 7 except that steps  15 - 21  are replaced by steps  31  and  32 . Therefore, only the differences are described. 
     Referring to FIG. 9, in steps  11 - 14 , processing similar to the processing of steps  11 - 14  shown in FIG. 7 is performed. In step  31 , the round-down portion  44  change digits in the (n+1)th and the following digit places (as counted from the leftmost digit place) of the integer K stored in the integer part storage portion  23  into 0. In. this stage, the digits forming the rounded-down numeral are separately stored in the integer part storage portion  23 . In step  32 , therefore, the digit place restoration portion  32  converts these values in different digit places into one numeral. For example, digits 3, 2, 0, 0 which are in different digit places and stored in the memory are converted into one numeral 3200. These steps  31  and  32  perform a round-down operation. Various conventional processes other than the illustrated processing method can be used to perform rounding down. The same principle applies to rounding down performed in the third embodiment described above. 
     In steps  22 - 24 , processing similar to the processing of steps  22 - 24  illustrated in FIG. 7 is performed. The index M stored in the index part storage portion  24  is multiplied by the rounded down integer L, thus producing a rounded down numeral. 
     In accordance with the flowchart illustrated in FIG. 9, 0.205 is rounded down with two significant digits. In step  11 , 0.205 is entered as numerical value x, and 2 is entered as the number of significant digits n. In step  12 , 0.205 is separated into an integer part and an index part, resulting in 205×10 −3 . In step  13 , an index M of −3 is stored in the index part storage portion  24 . In step  14 ,  205  that is the index M is divided into 2, 0, and 5 and stored in turn in the integer part storage portion  23 . In step  31 , digits in the (n+1)th digit place and the following digit places are changed to 0. In this case, the least significant digit 5 is changed to 0. In step  32 , a row of digits 2, 0, 0 in the integer part storage portion  23  is converted into one numeral  200 . In step  22 , numeral −3 stored in the index part storage portion  24  is fetched. In step  23 , calculation 200×10 −3  is performed, resulting in a rounded down numeral of 0.20. 
     A rounding processor in accordance with a fifth embodiment of the present invention is next described. FIG. 10 is a block diagram showing the structure of this rounding processor in accordance with the fifth embodiment. This processor performs rounding processing by making use of rounding up using significant digits. This rounding processor shown in FIG. 10 is similar to the rounding processor shown in FIG. 6 except that the five-judging portion  41 , the odd/even discrimination portion  42 , the round-off judging portion  43 , and the round-down portion  44  are omitted and that only a round-up portion  45  performs a round-up operation. Note that like components are denoted by like reference numerals in various figures and that those components which have been already described will not be described in detail below. The hardware of the rounding processor shown in FIG. 10 which is realized using a computer is similar to that shown in FIG.  2  and so the hardware is not illustrated. 
     Rounding processing executed by the rounding processor described above is described. FIG. 11 is a flowchart illustrating the processing performed by the rounding processor shown in FIG.  10 . The flowchart of FIG. 11 is similar to the flowchart of FIG. 7 except that steps  16 - 21  of the flowchart of FIG. 10 are replaced by steps  41 - 46 . Therefore, only the differences are described below. 
     Referring to FIG. 11, in steps  11 - 15 , the same processing as the processing of steps  11 - 15  illustrated in FIG. 7 is performed. In step  41 , the round-up portion  45  makes a decision as to whether one numeral formed by digits in the (n+1)th and the following digit places is equal to or greater than 1. No round-up operation is performed unless it is equal to or greater than 1, and control goes to step  46 . If this one numeral is equal to or greater than 1, the round-up portion  45  performs rewriting processing of steps  42 - 45 . 
     In step  42 , the original digits from the leftmost digit place to the nth digit place are regained and converted into one numeral. In the next step  43 , 1 is added to this one numeral. In step  44 , the digits in the successive digit places are again stored in turn in the integer part storage portion  23 . In the next step  45 , the digits in the (n+1)th and the following digit places of the integer part K stored are changed to 0. 
     In the next step  46 , an integer/index calculation portion  3  fetches the digits at the successive digit places from the integer part storage portion  23 . The digits are returned to one numeral. These steps  41 - 46  are used to perform a round-up operation. Various conventional processes other than the illustrated processing method can be used to perform rounding up. The same principle applies to the rounding up performed in the third embodiment described above. 
     In steps  22 - 24 , the same processing as the processing of steps  22 - 24  illustrated in FIG. 7 is performed. The index M stored in the index part storage portion  24  is multiplied by the rounded up integer L, and a rounded up numeral is output. 
     In the processing procedure described above, 1270400 is now rounded up with two significant digits. First, in step  11 , 1270400 is entered as a numerical value x, and 2 is entered as the number of significant digits n. Since 1270400 is given by 12704×10 2 , the integer K is 12704 and the index M is 2. Therefore, “2” is stored in the index part storage portion  24  (step  13 ). Digits 1, 2, 7, 0, and 4 are stored in turn in the integer part storage portion  23  (step  14 ). 
     In step  15 , the digits 7, 0, and 4 in the (n+1)th and the following digit places are taken and converted into one numeral  704 . Since this numeral is greater than 1, control goes to step  42  (step  41 ). In step  42 , the digits 1 and 2 in the digit places up to the nth digit place of the integer K, i.e., up to the second digit place, are extracted and converted into one numeral 12. In step  43 , “1” is added to this numeral, resulting in  13 . The digits of the numeral are stored again in turn in the integer part storage portion  23  (step  44 ). As a result of this processing, digits 1, 3, 7, 0, and 4 are stored in turn in the integer part storage portion  23 . The digits 7, 0, and 4 in the (n+1)th and the following digit places, i.e., the third and the following digit places, are changed to 0 (step  45 ). The digits stored in the integer part storage portion  23  are 1, 3, 0, 0, and 0. Converting these digits back to one numeral gives rise to 13000 (step  46 ). Finally, “2” is read from the index part storage portion  24  (step  22 ). A multiplication 13000×10 2  is performed (step  23 ). A product of 1300000 is output (step  24 ). 
     A rounding processor in accordance with a sixth embodiment of the present invention is next described. FIG. 12 is a block diagram showing the configuration of the rounding processor in accordance with the sixth embodiment. This rounding processor serves to perform rounding processing by rounding off using significant digits. This rounding processor shown in FIG. 12 is similar to the rounding processor shown in FIG. 6 except that the five-judging portion  41  and the odd/even discrimination portion  42  of the rounding processor shown in FIG. 6 are omitted and that a round-down portion  44  performs rounding down or a round-up portion  45  performs rounding up, depending on the result of the decision made by the round-off judging portion  43 . Like component are indicated by like reference numerals in various figures and that those components which have been already described will not be described in detail. The hardware of the rounding processor shown in FIG. 12 realized using a computer is similar to that shown in FIG.  2  and thus will not be illustrated. 
     The rounding processor described above performs rounding processing as described below. FIG. 13 is a flowchart illustrating the rounding processing performed by the rounding processor shown in FIG.  12 . The flowchart of FIG. 13 is similar to the flowchart of FIG. 7 except that steps  15 - 21  of FIG. 7 are replaced by steps  51 - 53 . 
     As illustrated in FIG. 13, in steps  11 - 14 , the same processing is performed as the processing done by steps  11 - 14  illustrated in FIG.  7 . In step  51 , a round-off judging portion  4  reads the digit in the (n+1)th digit place from the integer part storage portion  23  and makes a decision as to whether the digit is equal to or greater than 5. If the result of the decision is YES, control goes to step  53 , where rounding up is performed. If the result of the decision is NO, control proceeds to step  52 , where rounding down is carried out. These rounding up and rounding down can be the operations illustrated in FIG. 9 (steps  31  and  32 ) or the operations illustrated in FIG. 11 (from step  15  to step  46 ). Furthermore, various conventional algorithms can be employed. 
     In steps  22 - 24 , the same processing as the processing executed by the steps  22 - 24  illustrated in FIG. 7 is performed. The index M stored in the index part storage portion  24  is multiplied by the rounded down integer L, producing a rounded off numeral. 
     In the third through sixth embodiments described above, the numeral to be processed is separated into an integer part and an index part. Therefore, the digit place to be rounded can be specified by the number of significant digits. Where rounding specified by JIS is performed as in the third embodiment, the numeral is divided into an integer part and an index part. Then, the digit in the (n+1)th and digits following it are taken as one numeral and divided by 5. Consequently, a decision as to whether the digit in the (n+1)th digit place is 5 and the rightmost nonzero digit can be made with a quite few conditional branches. Hence, the processing can be performed quickly. Furthermore, the program can be created with greater ease. 
     As described above, with the rounding processor in accordance with the present invention, it is easy to create a program. Also, the processing time can be shortened. This means that the processor can be realized by software without almost affected by the computer&#39;s processing capability. The processing procedure in accordance with the present invention can be applied to various applications using rounding. Examples of application of the present invention include: 
     (1) The rounding processing is incorporated as a function in a spreadsheet program. Examples of spreadsheet software programs include Excel, Lotus 1-2-3, and Sanshiro (distributed in Japan). 
     (2) The rounding processing is incorporated in software programs for finding nitrogen oxide concentrations and software programs for finding the concentrations of constituents of chemicals. 
     (3) The rounding processing is incorporated in analytical instruments and measuring instruments including electrochemical analysis instruments, optochemical analysis instruments, electromagnetic analysis instruments, isolation analysis instruments, and thermoanalysis instruments. 
     (4) The rounding processing is incorporated in instruments for measuring physical quantities and physical properties including thermometers, hygrometers, dewpoint sensors, photometers, and colorimeters. 
     (5) The rounding processing is incorporated in special instruments such as development and research-related instruments, biological instruments, agricultural instruments, forest-associated instruments, marine product-associated instruments, stockbreeding-associated instruments, meteorological instruments, and ocean observational instruments. 
     (6) The rounding processing is used for operations for examinations and registrations in ISO 9000 and ISO 14000. For example, it can be incorporated in testing, measuring, and monitoring hardware or can be used with software programs for tests, measurements, and monitoring.