The Gifted Girl is a 2017 American film directed by Mark Webb. The following plot introduction is from toutiao encyclopedia.
In a small town in Central Florida, a seven-year-old girl, Mary (McKenna Grace), lives with her uncle, Frank (Chris Evans), a shipwright who takes on odd jobs from time to time. Mary is surrounded by many positive adult role models who help her a lot, including Frank's neighbor Roberta (Octavia Spencer) and her teacher Bonnie (Jenny Slater). Mary is also a mathematical genius with a brilliant brain.
Frank's mother Evelyn (Lindsay Duncan) believes that Mary belongs to a special school that is more suitable for adults, but Frank wants Mary to have a more regular childhood—he believes that doing so is to honor his sister—that is, Mary's mother's last wishes. The controversy sparked a legal battle over Mary's future custody.
Today we will explore the two oral arithmetic problems that Mary demonstrated her mathematical talent in the class on the first day of enrollment, can ordinary people mentally calculate like 7-year-old Mary?
Mary grew up loving mathematics, she used an Apple notebook to learn mathematics, and she also read a lot of advanced mathematics books and practiced calculus on her own. Her uncle (Captain America) sent her to ordinary elementary school, hoping that she would be able to live a happy ordinary life and avoid the path of loneliness and pain of genius.
On the first day of school, Mary was dismissive of the simple arithmetic questions posed by the teacher, extremely impatient, and bored. Teacher Bonnie wanted to dampen her sharpness and told Mary to stand up and answer the question. The difficulty of the questions gradually increases. When Ms. Bonnie asked 57× 135 = how much? The teacher didn't know the answer either, so he quickly picked up the calculator. The answers that came out were no different from Mary's.
To Bonnie's surprise, Mary also figured out the square root of the answer.
From easy to difficult, let's first explore, how to mentally calculate 57×135 =? Finally, let's talk about any mental arithmetic squares.
Because the movie only has the answer and no problem solving process, we have to reason ourselves, restore the problem solving steps of the genius girl's brain, and see if ordinary people can complete the task.
<h1 class="pgc-h-decimal" data-index="01" > is mentally arithmetic multiplication using crossover</h1>
It is likely that Mary mentally calculated this multiplication problem using cross-cutting. The mental arithmetic process is shown in the following figure.
Why is it calculated this way?
Please take a look at the following figure, the calculation principle is clear at a glance:
Decompose 135 and 57 into 130+5 and 50+7, and you can use the cross method to calculate.
<h1 class="pgc-h-decimal" data-index="02" > lattice multiplication</h1>
Considering that Mary was born into a family of Western mathematicians, it is possible that she will use the lattice multiplication that western mathematical circles are familiar with.
The so-called lattice multiplication is to use several meridians to represent a multiplier, several parallel lines to represent another multiplier, and count how many intersections of the lattice formed by the longitude and latitude lines to get the answer. Using Chinese characters as an example, calculating 3×3 can write a "tian" character, counting a number with 9 intersections, you get 3× 3 = 9.
Now we use lattice multiplication to calculate 135×57.
Draw 140 meridians and 60 parallel lines on paper to get 8400 intersections. Subtract the extra intersections and you get the answer.
The extra 5 meridians produce 5×60 = 300 intersections.
The extra 3 parallels produce 3× 140 = 420 crosspoints.
There are 3 × 5 = 15 intersections that are repeated.
Therefore, the columnar calculation results are as follows:
8400—300—420+15
=8400—705
=7695
<h1 class="pgc-h-decimal" data-index="03" > how do ancient chinese mathematicians open the square? </h1>
Now let's look at the next question:
We set the answer precision to a decimal point to retain a significant digit.
Let's first consider how ancient mathematicians squared.
Ancient mathematicians in China used the complete square formula to square, which is suitable for mental arithmetic.
First it can be determined that the square root integer part of 7695 is two digits and the ten digits are 8.
So how are single digits determined?
(a+b)²=a²+2ab+b²
=a²+b(2a+b)
a has been determined to be 80, and the idea for determining b is as follows:
The square area is 7695, minus the large square area of 6400, there are still 1295 left.
b(2a+b)<1295
b(160+b)<1295
It can be seen that b=7 is more appropriate, and b=8 is greater than 1295.
7(160+7)
=1120+49
=1169
Now that it is determined that the square root integer part of 7695 is 87, the next step is to determine the decile after the decimal point.
∵1295—1169=126
∴b(2a+b)<126
Substitute a = 87 into the above equation
b(174+b)<126
So, what about b should be a few tenths?
It can be seen that b= 0.8 is greater than 126, so
b=7
Calculate:
0.7(174+0.7)
=70+49+2.8+0.49
=121.8+0.49
=122.29<126
The answer retains one decimal place, which needs to be calculated to two decimal places and rounded to retain one significant digit.
Don't want to continue to open the party, what to do?
Mentally calculate 87.75² = 87.75× 87.75.
Use the cross-cutting method to mentally calculate.
See the following figure for the process.
Thus it is known that the arithmetic square root of 7695 retains the decimal place after 87.7.
After these two problems have been solved, I think ordinary people can mentally calculate them according to the above methods.
Opening square is an unavoidable problem, and the Pythagorean theorem, the cosine theorem, quadratic equations, triskele products, etc. all involve open squares. Let's review the history of open squares.
<h1 class="pgc-h-decimal" data-index="04" how did the ancient Greeks square it >? </h1>
The ancient Babylonians, the ancient Greeks (e.g. Archimedes, Helen) and the computer square method were all iterative methods.
What is the iterative method? Take a look at the image below:
The iterative method of opening square contains the general idea of calculating irrational numbers, that is, the successive approximation method of gradual approximating irrational numbers with rational numbers. Successive approximations are a universal method widely used in scientific computing. The essence of the iterative method is convergence and automatic error correction. Even if the initial value is wrong, a closer approximation can be calculated in the end.
How should the initial value of the square root be set? Recommend a good approximation formula, please see the following figure:
An approximate formula for calculating the square root
Example Question (1): Calculate the arithmetic square root of 17
It is easy to know that the integer part of the root number 17 is 4, which is substituted into the formula of the above figure, and the result of the following figure can be obtained by oral calculation:
Calculates the initial value of root number 17
Take the initial value of 4.125, use the iterative method to square, the process is as follows:
X₀=4.125
【4+（17-4×4）÷（2×4）=4.125】
X₁=4.1231060606
(X₀+17/X₀)÷2=X₁=(4.125+4.12121212121)÷2=4.1231060606 （17÷4.125=4.12121212121）
X₂=4.12310562561768
（17÷4.12310606060606=4.12310519062931）
X₃=4.12310562561766
【X₃=(X₂+17/X₂）÷2】（17÷4.12310562561768=4.12310562561764）
The comparison results with the built-in scientific calculator for Windows 10 systems are as follows:
根号17=4.1231056256176605498214098559741
The iterative method is a cycle of division and arithmetic means.
<h1 class="pgc-h-decimal" data-index="05" > How does Newton square? </h1>
The infinite series method based on Newton's generalized binomial theorem opens squares.
Pre-Modern Sculpture in the British Library: Newton in Computation
【Infinite series method】
In the summer of 1661, Newton left his hometown of Ursop to attend Trinity College, Cambridge. In 1663, Newton began to read Euclid's Primitives of Geometry, followed by Descartes' Geometry (René Descartes introduced coordinate systems and founded analytic geometry, which was published in 1637 along with his Methodology, providing the basis for Newton and Leibniz's subsequent proposals for calculus). In early 1665, Newton discovered the generalized binomial theorem, and later proposed the "flow number method" (differential calculus). In 1666, Newton invented the "countercurrent number method" (integrals). Although Newton was only a nameless man at Cambridge University at the time, his contributions and extraordinary achievements to mathematics were great.
How to describe the binomial theorem in one sentence?
Expands the "arbitrary real power" of the Sum of Two numbers into the form of "Sum".
In 1654 , Pascal established the binomial theorem for general positive integer powers. In 1665, Newton continued to explore the situation in which the index was a fraction and a negative number on the basis of previous research, and discovered the generalized binomial theorem, which generalized the binomial theorem from special to general to infinite series, which was a very remarkable achievement. Newton found that the use of infinite series expansion is not only convenient for solving area problems, but also can be used for square operations and logarithmic calculations. This was very helpful for Heavy Astronomical Calculations after Newton. Newton expanded a logarithm into an infinite series, calculating it to the 55th decimal place.
Binomial expansion
The figure above is the mathematical principle of the infinite series method. The unfolding formula of the open square is shown in the following figure:
Open square and open square take the binomial expansion of the reciprocal
Please enjoy an example from Newton, find an approximation of the root number 7.
Screenshot 1
Screenshot 2
Screenshot 3
Let's look at an example problem: using the infinite series method to find the square root of 215.
215 squares
< h1 class="pgc-h-decimal" data-index="06" > stills</h1>
Grandma appeared.
Pretty like not like a powerhouse.
QED on the chalkboard is an abbreviation in Latin, equivalent to Chinese testimony or testimony.
This movie is very warm, I recommend everyone to watch.
In 2014, the strongest brain invited the Chinese rain man Zhou Wei
The performance was performed in 16 figures to the power of 14, and the audience was breathtaking.
Is there anything more powerful?
Yes, the Indian woman Shakhuntara calculates 201 digits to the power of 23, and the answer is 9 digits. Her speed beat the best computer in the world at the time.
Hua Luogeng published an article in the journal dissecting the mystery of speed calculation. Please read the link below:
【"Reprint" about the algorithm of opening the high power", "Genius and Exercise" (text / Hua Luogeng) - today's headlines] https://m.toutiao.com/is/JTc6K1e/
Science has not yet been popularized, and the media still needs to work. Thanks for reading, goodbye.