Square root refers to a mathematical operation method, which is used to find thesquare rootThe square root is the inverse of square root.Square root is a square root operation.
The operation of finding the square root of a number is called extraction of square root, where a is called the number to be extracted.In the real number range, a must be greater than or equal to zero, that is, a isNonnegative number；In the range of complex numbers, the square of i is defined as - 1, that is, the square root of - 1 is ± i, which is recorded as itwo=-1。
Theoretical basis
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Square root is the inverse operation of square root. As long as you know the calculation method of square root, square root can be easily solved.
If the ten digit value is set as A and the one digit value is set as B, it is A × 10+B. According to the square of the sum of two digits, there are: (A × 10+B)two=(A×10)two+2(A×10)×B+Btwo=(Atwo)×100+(20A+B)×B。
Example 359twocomputing method
1、3two=9，
2、(20×3+5)×5=325，
3、(20×35+9)×9=6381，
4. These numbers are combined by two digits: 90000+32500+6381=128881.359two=128881。
Reverse these calculation steps to square root.In the same way, we can get the methods of cube and n-th power.
Prescription history
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The achievements of ancient mathematics in China are brilliant. The classic mathematical works of our country came out as early as the first century BC《Chapter Nine Arithmetic》There is a calculation basedKaipingmethod.LaternorthSong DynastymathematicianJia XianThe prescription technique was further improved and a mature procedural prescription method was formed:Multiplication flattening method。
In ancient calculation books, "square root" was sometimes used to directly refer to square root.For example, in the "Zhoubi Suanjing", "Pythagorean strands are multiplied by each other, and they are string solid, and the square root is divided into them, that is, the chord." M is equivalent to a right triangle, where we have known to hook a, strand b, and find the chord c,
, that is, square root operation.In the Xiechawei Suanjing of the Song Dynasty and the Suanfa Tongzong of the Ming Dynasty, the "common examples of words" explained the "prescription" as "self multiplication reduction", that is, the method of leveling.[2]
According to historical records, the Kaiping method was not introduced abroad until the fifth century AD[1]。
computational procedure
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1. Number of squares to be extractedintegerThe part is divided into segments every two digits from one digit to the left, separated by apostrophes and divided into several segments, indicating that the square root is several digits;
Written square root method
2. According to the number in the first paragraph on the left, get the number on the highest order of the square root (3 in the vertical form);
3. Subtract the square of the highest digit from the number of the first paragraph, and write the number of the second paragraph to the right of their difference to form the first remainder (256 in the vertical);
4. Multiply the highest digit obtained by 20 to divide the first remainder, and the maximum integer obtained is taken as the trial quotient (20 × 3 divided by 256, the maximum integer obtained is 4, that is, the trial quotient is 4);
5. Add 20 times of the highest digit of the square root and multiply by the test quotient. If the product obtained is less than or equal to the remainder, the test quotient is the second digit of the square root;If the product obtained is greater than the remainder, reduce the trial quotient and try again (in the vertical formula, (20 × 3+4) × 4=256, indicating that trial quotient 4 is the second digit of the square root);
6. Use the same method to continue to find the number on other squares of the square root
In case of inexhaustible range, its approximate value can be calculated according to the required accuracy
For example, the approximate value (accurate to 0.01) can be obtained by listing the vertical form on the right above.
Written calculationSquare root operationIt is complicated and has less direct application in practice, but this method can be used to obtain the approximate value of the square root of a number with arbitrary accuracy
Instance 1Square root formula
For example, A=5:
5 is between the square of 2 and the square of 3;between.The initial value of 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 is acceptable, and the intermediate value of 2.5 is preferred.
Step 1: 2.5+(5/2.5-2.5) 1/2=2.2;Negative feedback when the input value is greater than the output value;
That is, 5/2.5=2, 2-2.5=- 0.5, - 0.5 × 1/2=- 0.25, 2.5+(- 0.25)=2.25, taking two digits 2.2.
Step 2: 2.2+(5/2.2-2.2) 1/2=2.23;The input value is less than the output value, positive feedback;
That is, 5/2.2=2.27272, 2.27272-2.2=0.07272, 0.07272 × 1/2=0.03636, 2.2+0.03636=2.23636.Take 3 digits 2.23.
Step 3: 2.23+(5/2.23-2.23) 1/2=2.236.
That is, 5/2.23=2.2421525, 2.2421525-2.23=0.0121525, 0.0121525 × 1/2=0.00607, 2.23+0.006=2.236, four digits are taken.
Take one more digit in each step.This method is also called feedback square root. Even if you input an incorrect value, it doesn't matter. The output value will automatically adjust to close to the accurate value.
For example, A=200:
200 is between the square of 10 and the square of 20.The initial value can be 11, 12, 13, 14, 15, 16, 17, 18, 19.Take 15:15+(200/15-15) 1/2=14.Taking 19 also yields 14:19+(200/19-19) 1/2=14.
14+（200/14-14）1/2=14.1。
14.1+（200/14.1-14.1)1/2=14.14。
Example 2Exact square root formula
For a number C to be squared, first try to estimate a number a as close as possible to the square root, so that C=a ² ± b, and b ≤ a, then
For example, √ 3000:
Because 3000=3025-25=55 ² - 25
So √ 3000 ≈ 55-25/(2 × 55) - 25 ²/(8 × 55 ³)=54.7722577......, the accuracy of 7 significant figures is obtained at one time.
