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The data obtained through measurement often need to be sorted, analyzed, and calculated to obtain the measurement results. In order to reasonably express the measurement results, it is necessary to correctly use digital expression values and reasonably handle the numbers in the calculation. 1. Significant figures Due to the limited resolution of the measuring instrument, measurement errors are inevitable, and it is impossible to take infinite digits when applying irrational numbers (e, π, etc.) in the calculation process, so the measurement data and measurement results usually obtained are approximate numbers. In order to make the data representation accurate and unified, it is generally stipulated that the absolute value of the absolute error of the approximate number obtained by interception shall not exceed half of its last unit digit, and all the digits of this approximate number from the first non-zero digit to the last digit are called significant digits. Table 2.1 shows the significant digits of several data. ![]() It should be noted that the zeros on the right side of the number cannot be added or subtracted at will, because this is related to the accuracy of the measurement. 37.100 cannot be arbitrarily rewritten as 37.1000, nor can it be arbitrarily rewritten as 37.10, because it does not conform to the principle that the number of significant digits is consistent with the size of the error. The absolute value of the absolute error of 37.1000 is less than 0.00005; the absolute value of the absolute error of 37.10 is less than (0.005). For some data with more digits but fewer significant digits, it is necessary to use the form of multiplying the significant digits by the power of 10 to express it. For example, for the data 43200, if the absolute value of the error is within 50 (i.e. the error does not exceed half of the hundredth place), the number of significant digits should be three (4, 3, 2). In this case, the zeros in the ones and tens places are not significant digits, but in order to express the number of digits of the data, they cannot be removed at will, so it should be expressed in the form of 432×102, 1.32×103 or 1.32×104. In general, in order to clearly express the significant digits of the measured number, the commonly used expression is k×10ｍ In the formula, k- is any number from 1 to less than 10; m- is any integer with any sign. In this expression, the number of digits of k is the significant digit. Therefore, the above data 43200 is generally expressed as 4.320x104. 2. Rules for Rounding Off When it is necessary to reduce the number of digits in a data due to calculation or other reasons, the number rounding off rules should be followed. The rules for rounding off provide that the value of the digit to be discarded shall be measured by the last unit of the digit to be retained: (1) If the value of the digit to be discarded is greater than 0.5 units, the last digit of the digit to be retained shall be increased by 1; (2) If the value of the digit to be discarded is less than 0.5 units, the last digit of the digit to be retained shall remain unchanged; (3) If the value of the digit to be discarded is exactly equal to 0.5 units, the last digit of the digit to be retained shall be made into an even number (i.e., if the last digit of the digit to be retained is an even number, the last digit shall remain unchanged; if it is an odd number, 1 shall be added). Example 2.3 Round the following 7 data to 4 significant figures: 5.142 69, 6.378 501, 2.717 29, 7.691 499, 4.510 500, 3.216 50. 8.343 5. Solution: Use arrows to indicate the rounded results, as shown below: It should be noted that rounding can only be done once, not one by one. For example, when rounding 2.54546 to 2 significant figures, the wrong way is 2.54546→2.5455→2.546→2.55→2.6 The correct way is 2.54546→2.5 In addition, an additional explanation is that the number rounding rule is based on the macroscopic probability of rounding and in, and when the number of rounding is large enough, the rounding errors can offset each other. According to the rounding rule, the number of the last digit to be retained is the unit, and the value that will cause rounding error by discarding the part is within the open interval (0,1). Within this range, with 0.5 as the center, the probability of the corresponding points on both sides appearing on the macroscopic level is equal, and the rounding errors caused are equal in size and opposite in sign. For example, the rounding errors caused by the equal probability of 0.01 and 0.99 are -0.01 and +0.01 respectively; the rounding errors caused by the equal probability of 0.234 and 0.766 are -0.234 and +0.234 respectively. So from a macro point of view, the errors caused by rounding can offset each other when the number of roundings is large enough. The remaining question is how to deal with the rounded part when it is exactly 5. Neither rounding up nor rounding down can offset the rounding errors caused. From a macro point of view, the probability of the last digit of the significant digit to be retained being odd or even is the same. If one case is rounded down and the other case is rounded up, the rounding errors will also offset when the number of roundings is large enough. If the last digit of the significant digit to be retained is odd, adding 1 can increase the chance that the last digit of the significant digit is even, and when an even number is used as a dividend, the chance of being divided out is greater, which is conducive to reducing calculation errors. 3. Effective safety digits In practical applications, the method of taking only effective digits of measured data and discarding the rest may also discard some information for reference. In particular, when the measurement result is used as an intermediate result and a large number of calculations are required, rounding errors may accumulate rapidly. Therefore, in addition to the effective digits, 1 to 2 more digits can be taken to the right as safety digits. The data represented in this way is called effective safety digits. The specific method for processing the measured data is: first determine the position of the least significant digit by the error (generally expressed as 1 to 2 effective digits); then take 1 to 2 more safety digits to the right from the least significant digit; finally, discard the rest of the digits according to the digital rounding rules. Example 2.4 The original data obtained by measuring a certain voltage is 1.83549±0.014V. Try to determine the effective safety digits. Solution: Determine the position of the least significant digit by the error, i.e. 1.83549 0.014. It can be seen that the least significant digit is the percentile (digit 8); the number of significant digits is 2 (1, 8). Take 2 more safety digits to the right, i.e. the tenth digit and the hundredth digit (digits 3, 5). Process the remaining digits according to the rounding rule, and get the effective safety number of 1.835. IV. Rules for the operation of significant digits 1. When the number of numbers involved in the addition and subtraction operation does not exceed 10: (1) Take the number with the least decimal places as the base number, and round the remaining numbers to one more decimal place than the base number; (2) Perform the addition and subtraction operation; (3) Round the result to the same number of decimal places as the base number; (4) When only two numbers are added (or subtracted), round them to the same number of decimal places. Example 2.5 13.65+0.00823+1.633=? Solution (1) 13.65 has the fewest digits after the decimal point (2 digits), so it is taken as the base number, and the other two numbers are rounded off according to the base number: 0.00823 → 0.008 (one more decimal place than the base number is retained) 1.633 → 1.633 (one more decimal place than the base number is retained) (2) Calculation: 13.65 + 0.008 + 1.633 = 15.291 (3) The result of rounding off according to the number of digits after the decimal point of the base number is: 15.291 → 15.29 Multiplication and division operations (1) Take the number with the least number of significant digits as the base number, and round off the other numbers to one more significant digit than the base number (regardless of the position of the decimal point); (2) Perform multiplication and division operations; (3) Round off the calculation result to make the number of significant digits the same as the base number; (4) When the first significant digit of the base number is "8" or "9", the calculation result can have one more significant digit than the base number. Example 2.7 0.0121×25.64×1.05782=? Solution (1) The base number 0.0121 is a 3-digit significant figure. Round the other two numbers to the following: 25.64 → 25.64 (one more significant figure than the base number) 1.05782 → 1.058 (one more significant figure than the base number) (2) Perform multiplication and division operations: 0.0121 × 25.64 × 1.058 = 0.3282 (3) Round the result to the same number of significant figures as the base number: 0.3282 → 0.328 3. The number of significant figures of the result of exponentiation and square root is the same as the original number or one more significant figure is retained. 4. The logarithm of the logarithm operation should have the same number of significant figures as the real number. For example, take lg32.8 = 1.52. When looking up the table, the number should be the same as the number of significant figures. 5. Trigonometric function The number of digits used in the function value usually increases as the angle error decreases. The corresponding relationship is shown in Table 2.2 6. The number of digits that should be retained in the calculation results of the intermediate steps of multi-step operations is 1 more than the number of digits required to be retained in a single operation (only one addition, subtraction, multiplication or division, square root or exponentiation operation is required). This information belongs to the purchase line network. If you need to reprint it, please indicate the source. For more information, please go to the purchase line network!
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