Find the law of approximate numbers, and the formula for the value range of approximate numbers

For example, it is impossible to calculate the exact number of the population of our country, but it is possible to give an approximate number. For example, China's population is 1.3 billion, and 1.3 billion is an approximate number.

Approximation 1 Four calculations of approximate numbers.

1 Addition and subtraction.

Under normal circumstances, approximate numbers are added and subtracted, and the least accurate known number is accurate to which digit is exact, and the sum or difference can only be accurate to this digit at most. For example, a classmate weighed kilograms last year, and this year's weight has increased by kilograms compared to last year. When finding this year's weight, add up these two approximate numbers.

Because it is accurate to the tenth place, the accuracy of the ratio (to the percentile) is lower, so the sum added can only be accurate to the tenth place at most.

To make it easy to see how reliable the results are, we add a "? ? " is used to denote a truncated number.

As you can see, since the first addition cannot be determined from the percentile, the sum of the added from the percentile cannot be determined.

The addition and subtraction of approximate numbers can generally be carried out according to the following rules: (1) Determine which digit the calculation result can be accurate to. (2) "Round" the mantissa of the known number that exceeds this digit to the next digit of this digit.

3) Do the math and round the last digit of the calculated number.

Example 1 Find the sum of approximate numbers and .

Round the "thousandth" to the thousandth first, get, and then do the addition.

Round the "percentile" to the percentile and get.

Example 2 Find the difference between the approximate number and the other.

Round the "10,000th percentile" first.

Rounding up to the thousandth.

Example 3 Finding the Approximate Number726 and.

Round up to the tenths.

2 Multiplication and division.

In general, approximate numbers are multiplied and divided, and the known number with the fewest significant digits has as many significant digits, and the product or quotient can only have the same number of significant digits at most.

For example, in the case of approximate and multiplication, it can be seen from the vertical formula that only the first two digits of the product are determined, that is, there can only be two significant digits. This is the same as the number of significant figures for the second factor.

3 8.？？The multiplication and division of approximate numbers in this paragraph can generally be carried out according to the following rules: (1) Determine how many significant figures there are in the result. (2) Round up the number of significant figures in the known number to only one more than the number needed in the result.

3) Perform the calculations and "round" the calculated numbers to the number of significant figures.

Example 4 The product of the sum.

Round up to the nearest unit.

Rounding "to the single digit" gives you 79.

Example 5 Find the quotient of the approximate number divided by it.

3 Mixed operations.

The mixed operation of approximate numbers can be calculated step by step according to the order of operation and the calculation rules of approximate numbers, but the result of the intermediate operation is one more digit than the final result.

Example 6 Calculation.

According to the known data, the result of the final operation is a two-digit number, so the result of the intermediate operation is a three-digit number!

Approximate numberThere is no formula for the value range, and the formula is rounding.

An approximate number is a number that is close to an exact number. Among them, the exact number is the original data of the number, which has not been expressed before any operation such as reduction, simplification, or rounding.

An approximate number is a number that is not much different from the original data obtained by rounding, advancing or removing the tailing. For example, the population of our country cannot be calculated accurately, but we can say an approximate number, for example, our population is 1.3 billion, and 1.3 billion is an approximate number.

Significant figures: Numbers that are close to the actual number, but do not match exactly, are called approximate numbers.

For approximate numbers, it is often necessary to know how accurate they are.

The accuracy of an approximate number is usually expressed in one of two ways:

Use rounding.

Expression. An approximate number is rounded to which digit it is, and the approximate number is said to be exact.

In addition, there are two methods: one and one tail. It is expressed in terms of the number of significant figures. Rounded approximate numbers, starting with the first non-zero number on the left and ending with the last digit, all digits are called significant digits of the number.

1. Rounding method: When taking the decimal approximation, if the highest digit of the mantissa is four or smaller than four, remove the mantissa number. If the highest digit of the mantissa is five or more than five, the mantissa is rounded off and the previous digit is rounded, and this method of taking an approximate number is called rounding.

2. Advance method: The advance method is to remove the number of the excess part and add 1 to the last number of the reserved part. The resulting approximation is an excess approximation (larger than the exact value).

In real life, rounding is not necessarily possible, and the advance method will be used (as long as the omitted bit is greater than zero, the limb must be added to one place).

3. Tail removal method: The tail removal method is to remove the decimal part of the number, and take the hand defeat of the integer part of the potato world, which is a commonly used mathematical value method, and the value is an approximate value (smaller than the accurate value), which is also called the tail principle.

A number that is close to an exact number (slightly more or less than an exact number) is called an approximate number.

The rounding method is the most commonly used method to find the approximate number, mainly to see that the mantissa it omits is 4 or more than 4 hours, and the mantissa of the code is rounded; If the number in the highest digit of the omitted mantissa is 5 or greater than 5, the mantissa number should be omitted and the previous digit should be advanced by one. For example, 3096401 3.1 million, 1 3 = From the above two examples, it can be seen that the approximate number is smaller than the exact value when "the four rounds of the model song", and the approximate number is larger than the exact value when "five into".

Now in the second grade, there is no specific requirement for you to approximate the number, and there is no one standard for the same. The following requirements can be made: if it is a four-digit number, it is approximated to a hundred; If it's a three-digit number, it's approximately ten digits.

For the same question, the reserved digits should be the same. That is to say, if the fruit is reserved to the tenth place, everyone is reserved to the tenth place; If it is reserved to the hundreds, it is also reserved to the hundreds.

For example: 416-251 70 (both numbers are reserved to the tenth place for calculation), 416-251 100 (both numbers are reserved to the hundred place for calculation), 416-251 is lost to the sun stove 120 (416 is reserved to the tenth place is about 420, 251 is reserved to the hundred place is about equal to 300). 416-251 50 (416 to the hundred place is about equal to 400, 251 to the tenth place is about equal to 250).

1 Rounding, this is the most commonly used.

2 into one method, 3 to the tail method.

The latter two are only used in special cases.

Among them, the exact number is the original data of the number, and it has not been expressed before any operations such as reduction, simplification, or rounding. The near-imitation is a number that is not much different from the original data obtained by rounding, advancing or removing the tailing. Such as:

It is impossible to calculate the exact number of China's population, but we can say an approximate number, for example, if we talk about our population of 1.3 billion, 1.3 billion is an approximate number.

One method: transporting goods, loading oil, taking a ship, taking a car, ......Tail method: make cakes, divide the east and west, and make clothes.

Advance a method. The method of further one[1] is to remove the number of the excess part and add 1 to the last number of the reserved part. The resulting approximation is an excess approximation (i.e., greater than the exact value).

In our real life, rounding is not always possible, and sometimes it is used to advance (i.e., as long as the omitted digit is greater than zero, it must be rounded by one place).

In order to make the results more in line with objective reality or to make the results meaningful.

Tail removal method. The tail removal method is a common mathematical method of removing the decimal part of a number and taking its integer part, and the value is an approximate value (the verification hand is smaller than the exact value), and this method is often used in life. It is also called the tail removal principle.