Numbers are rounded to other digits - tenths, hundredths, tens, hundreds, etc.
If a number is rounded to any digit, then all digits following this digit are replaced with zeros, and if they are after the decimal point, they are discarded.
Rule #1. If the first of the discarded digits is greater than or equal to 5, then the last of the retained digits is amplified, i.e., increased by one.
Example 1. Given the number 45.769, it needs to be rounded to the nearest tenth. The first digit to be discarded is 6 ˃ 5. Consequently, the last of the retained digits (7) is amplified, i.e., increased by one. And thus the rounded number will be 45.8.
Example 2. Given the number 5.165, it needs to be rounded to the nearest hundredth. The first digit to be discarded is 5 = 5. Consequently, the last of the retained digits (6) is amplified, i.e., increased by one. And thus the rounded number will be 5.17.
Rule #2. If the first of the discarded digits is less than 5, then no amplification is done.
Example: Given the number 45.749, it needs to be rounded to the nearest tenth. The first digit to be discarded is 4
Rule #3. If the discarded digit is 5 and there are no significant digits behind it, then rounding is done to the nearest even number. That is, the last digit remains unchanged if it is even and is enhanced if it is odd.
Example 1: Rounding the number 0.0465 to the third decimal place, we write - 0.046. We do not make amplification, because the last digit stored (6) is even.
Example 2. Rounding the number 0.0415 to the third decimal place, we write - 0.042. We make gains, because the last stored digit (1) is odd.
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Introduction........................................................ ........................................................ .......... | |
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TASK No. 1. Series of preferred numbers.................................................... .... | |
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TASK No. 2. Rounding measurement results.................................................. | |
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TASK No. 3. Processing of measurement results.................................................. | |
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TASK No. 4. Tolerances and fits of smooth cylindrical joints... | |
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TASK No. 5. Tolerances of shape and location.................................................... . | |
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TASK No. 6. Surface roughness.................................................. ..... | |
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TASK No. 7. Dimensional chains.................................................... ............................ | |
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Bibliography................................................ ............................................ |
Task No. 1. Rounding measurement results
When performing measurements, it is important to follow certain rules for rounding and recording their results in technical documentation, since if these rules are not followed, significant errors in the interpretation of measurement results are possible.
Rules for writing numbers
1. The significant digits of a given number are all digits from the first on the left, which is not equal to zero, to the last on the right. In this case, the zeros resulting from the multiplier of 10 are not taken into account.
Examples.
a) Number 12,0has three significant figures.
b) Number 30has two significant figures.
c) Number 12010 8 has three significant figures.
G) 0,51410 -3 has three significant figures.
d) 0,0056has two significant figures.
2. If it is necessary to indicate that a number is exact, the word “exactly” is indicated after the number or the last significant digit is printed in bold. For example: 1 kW/h = 3600 J (exactly) or 1 kW/h = 360 0 J .
3. Records of approximate numbers are distinguished by the number of significant digits. For example, there are numbers 2.4 and 2.40. Writing 2.4 means that only whole and tenths are correct; the true value of the number could be, for example, 2.43 and 2.38. Writing 2.40 means that hundredths are also true: the true value of the number can be 2.403 and 2.398, but not 2.41 and not 2.382. Entry 382 means that all numbers are correct: if for last digit It is impossible to guarantee, then the number should be written 3.810 2. If only the first two digits of the number 4720 are correct, it should be written as: 4710 2 or 4.710 3.
4. The number for which the permissible deviation is indicated must have the last significant figure the same digit as the last significant digit of the deviation.
Examples.
a) Correct: 17,0 + 0,2. Wrong: 17 + 0,2or 17,00 + 0,2.
b) Correct: 12,13+ 0,17. Wrong: 12,13+ 0,2.
c) Correct: 46,40+ 0,15. Wrong: 46,4+ 0,15or 46,402+ 0,15.
5. It is advisable to write down the numerical values of a quantity and its error (deviation) indicating the same unit of quantity. For example: (80.555 + 0.002) kg.
6. It is sometimes advisable to write the intervals between numerical values of quantities in text form, then the preposition “from” means “”, the preposition “to” – “”, the preposition “over” – “>”, the preposition “less” – “<":
"d takes values from 60 to 100" means "60 d100",
"d takes values greater than 120 less than 150" means "120<d< 150",
"d takes values over 30 to 50" means "30<d50".
Rules for rounding numbers
1. Rounding a number is the removal of significant digits to the right to a certain digit with a possible change in the digit of this digit.
2. If the first of the discarded digits (counting from left to right) is less than 5, then the last saved digit is not changed.
Example: Rounding a number 12,23gives up to three significant figures 12,2.
3. If the first of the discarded digits (counting from left to right) is equal to 5, then the last saved digit is increased by one.
Example: Rounding a number 0,145gives up to two digits 0,15.
Note . In cases where the results of previous rounding should be taken into account, proceed as follows.
4. If the discarded digit is obtained as a result of rounding down, then the last remaining digit is increased by one (with a transition to the next digits, if necessary), otherwise - vice versa. This applies to both fractions and integers.
Example: Rounding a number 0,25(obtained as a result of the previous rounding of the number 0,252) gives 0,3.
4. If the first of the discarded digits (counting from left to right) is more than 5, then the last saved digit is increased by one.
Example: Rounding a number 0,156gives to two significant figures 0,16.
5. Rounding is performed immediately to the desired number of significant figures, and not in stages.
Example: Rounding a number 565,46gives up to three significant figures 565.
6. Whole numbers are rounded according to the same rules as fractions.
Example: Rounding a number 23456gives to two significant figures 2310 3
The numerical value of the measurement result must end with a digit of the same digit as the error value.
Example:Number 235,732 + 0,15should be rounded to 235,73 + 0,15, but not until 235,7 + 0,15.
7. If the first of the discarded digits (counting from left to right) is less than five, then the remaining digits do not change.
Example: 442,749+ 0,4rounded up to 442,7+ 0,4.
8. If the first digit to be discarded is greater than or equal to five, then the last digit to be retained is increased by one.
Example: 37,268 + 0,5rounded up to 37,3 + 0,5; 37,253 + 0,5 must be roundedbefore 37,3 + 0,5.
9. Rounding should be done immediately to the desired number of significant figures; rounding incrementally may lead to errors.
Example: Step by step rounding of a measurement result 220,46+ 4gives at the first stage 220,5+ 4and on the second 221+ 4, while the correct rounding result is 220+ 4.
10. If the error of a measuring instrument is indicated with only one or two significant digits, and the calculated error value is obtained with a large number of digits, only the first one or two significant digits should be left in the final value of the calculated error, respectively. Moreover, if the resulting number begins with the digits 1 or 2, then discarding the second character leads to a very large error (up to 3050%), which is unacceptable. If the resulting number begins with the number 3 or more, for example, with the number 9, then preserving the second character, i.e. indicating an error, for example, 0.94 instead of 0.9, is misinformation, since the original data does not provide such accuracy.
Based on this, the following rule has been established in practice: if the resulting number begins with a significant digit equal to or greater than 3, then only one is retained in it; if it begins with significant figures less than 3, i.e. from numbers 1 and 2, then two significant figures are stored in it. In accordance with this rule, the standardized values of errors of measuring instruments are established: two significant figures are indicated in the numbers 1.5 and 2.5%, but in numbers 0.5; 4; 6% only one significant figure is indicated.
Example:On an accuracy class voltmeter 2,5with measurement limit x TO = 300 In a reading of the measured voltage x = 267,5Q. In what form should the measurement result be recorded in the report?
It is more convenient to calculate the error in the following order: first you need to find the absolute error, and then the relative one. Absolute error X = 0 X TO/100, for the reduced voltmeter error 0 = 2.5% and the measurement limits (measurement range) of the device X TO= 300 V: X= 2.5300/100 = 7.5 V ~ 8 V; relative error = X100/X = 7,5100/267,5 = 2,81 % ~ 2,8 % .
Since the first significant digit of the absolute error value (7.5 V) is greater than three, this value should be rounded according to the usual rounding rules to 8 V, but in the relative error value (2.81%) the first significant digit is less than 3, so here two decimal places must be retained in the answer and = 2.8% must be indicated. Received value X= 267.5 V must be rounded to the same decimal place as the rounded absolute error value, i.e. up to whole units of volts.
Thus, the final answer should state: “The measurement was made with a relative error of = 2.8%. The measured voltage X= (268+ 8) B".
In this case, it is more clear to indicate the limits of the uncertainty interval of the measured value in the form X= (260276) V or 260 VX276 V.
Many people are interested in how to round numbers. This need often arises among people who connect their lives with accounting or other activities that require calculations. Rounding can be done to whole numbers, tenths, and so on. And you need to know how to do it correctly so that the calculations are more or less accurate.
What is a round number anyway? This is the one that ends in 0 (for the most part). In everyday life, the ability to round numbers makes shopping trips much easier. Standing at the checkout, you can roughly estimate the total cost of purchases and compare how much a kilogram of the same product costs in bags of different weights. With numbers reduced to a convenient form, it is easier to make mental calculations without resorting to a calculator.
Why are numbers rounded?
People tend to round any numbers in cases where it is necessary to perform more simplified operations. For example, a melon weighs 3,150 kilograms. When a person tells his friends about how many grams the southern fruit has, he may be considered a not very interesting interlocutor. Phrases like “So I bought a three-kilogram melon” sound much more concise without delving into all sorts of unnecessary details.
Interestingly, even in science there is no need to always deal with the most accurate numbers possible. But if we are talking about periodic infinite fractions, which have the form 3.33333333...3, then this becomes impossible. Therefore, the most logical option would be to simply round them. As a rule, the result is then slightly distorted. So how do you round numbers?
Some important rules when rounding numbers
So, if you wanted to round a number, is it important to understand the basic principles of rounding? This is a modification operation aimed at reducing the number of decimal places. To carry out this action, you need to know several important rules:
- If the number of the required digit is in the range of 5-9, rounding is carried out upward.
- If the number of the required digit is in the range 1-4, rounding is done downwards.
For example, we have the number 59. We need to round it. To do this, you need to take the number 9 and add one to it to get 60. This is the answer to the question of how to round numbers. Now let's look at special cases. Actually, we figured out how to round a number to tens using this example. Now all that remains is to use this knowledge in practice.
How to round a number to whole numbers
It often happens that there is a need to round, for example, the number 5.9. This procedure is not difficult. First we need to omit the comma, and when we round, the already familiar number 60 appears before our eyes. Now we put the comma in place, and we get 6.0. And since zeros in decimal fractions are usually omitted, we end up with the number 6.
A similar operation can be performed with more complex numbers. For example, how do you round numbers like 5.49 to integers? It all depends on what goals you set for yourself. In general, according to the rules of mathematics, 5.49 is still not 5.5. Therefore, it cannot be rounded up. But you can round it up to 5.5, after which it becomes legal to round up to 6. But this trick doesn’t always work, so you need to be extremely careful.
In principle, an example of correct rounding of a number to tenths has already been discussed above, so now it is important to display only the main principle. Essentially, everything happens in approximately the same way. If the digit that is in the second position after the decimal point is in the range 5-9, then it is removed altogether, and the digit in front of it is increased by one. If it is less than 5, then this figure is removed, and the previous one remains in its place.
For example, at 4.59 to 4.6, the number “9” disappears, and one is added to the five. But when rounding 4.41, the unit is omitted, and the four remains unchanged.
How do marketers take advantage of the mass consumer's inability to round numbers?
It turns out that most people in the world do not have the habit of assessing the real cost of a product, which is actively exploited by marketers. Everyone knows promotion slogans like “Buy for only 9.99.” Yes, we consciously understand that this is essentially ten dollars. Nevertheless, our brain is designed in such a way that it perceives only the first digit. So the simple operation of bringing a number into a convenient form should become a habit.
Very often, rounding allows you to better evaluate intermediate successes expressed in numerical form. For example, a person began to earn $550 a month. An optimist will say that it is almost 600, a pessimist will say that it is a little more than 500. It seems that there is a difference, but it is more pleasant for the brain to “see” that the object has achieved something more (or vice versa).
There are a huge number of examples where the ability to round turns out to be incredibly useful. It is important to be creative and avoid loading yourself with unnecessary information whenever possible. Then success will be immediate.
Let's look at examples of how to round numbers to tenths using rounding rules.
Rule for rounding numbers to tenths.
To round a decimal fraction to tenths, you must leave only one digit after the decimal point and discard all other digits that follow it.
If the first of the discarded digits is 0, 1, 2, 3 or 4, then the previous digit is not changed.
If the first of the discarded digits is 5, 6, 7, 8 or 9, then we increase the previous digit by one.
Examples.
Round to the nearest tenth:
To round a number to tenths, leave the first digit after the decimal point and discard the rest. Since the first digit discarded is 5, we increase the previous digit by one. They read: “Twenty-three point seven five hundredths is approximately equal to twenty three point eight tenths.”
To round this number to tenths, we leave only the first digit after the decimal point and discard the rest. The first digit discarded is 1, so we do not change the previous digit. They read: “Three hundred forty-eight point thirty-one hundredths is approximately equal to three hundred forty-one point three tenths.”
When rounding to tenths, we leave one digit after the decimal point and discard the rest. The first of the discarded digits is 6, which means we increase the previous one by one. They read: “Forty-nine point nine, nine hundred sixty-two thousandths is approximately equal to fifty point zero, zero tenths.”
We round to the nearest tenth, so after the decimal point we leave only the first of the digits, and discard the rest. The first of the discarded digits is 4, which means we leave the previous digit unchanged. They read: “Seven point twenty-eight thousandths is approximately equal to seven point zero tenths.”
To round a given number to tenths, leave one digit after the decimal point, and discard all those following it. Since the first digit discarded is 7, therefore, we add one to the previous one. They read: “Fifty-six point eight thousand seven hundred six ten thousandths is approximately equal to fifty six point nine tenths.”
And a couple more examples for rounding to tenths:
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