POTW #374: Finding the 1000th Digit of the Square Root of a Large Number

  • MHB
  • Thread starter anemone
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In summary, to find the 1000th digit of the square root of a large number, one can calculate the square root using techniques such as long division or Newton's method, and then count the digits after the decimal point. While this may not have practical use, it can be a challenging mathematical problem that can improve problem-solving skills. This method can be used for any square root, but as the number gets larger, the calculation becomes more complex and time-consuming. Other methods, such as using logarithms or specific algorithms, can also be used. There is no limit to the number of digits that can be found, but as the number of digits increases, the calculation becomes more complex and may require more advanced techniques and computing power.
  • #1
anemone
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MHB
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Here is this week's POTW:

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Let $A=1+10+10^2+\cdots+10^{1997}$. Determine the 1000th digit after the decimal point of the square root of $A$ in base 10.

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
Congratulations to the following members for their correct solution!(Cool)

1. MegaMoh
2. kaliprasad

Sample solution from other:
The answer is the same as the unit digit of $10^{1000}\sqrt{A}$. We have

$10^{1000}\sqrt{A}=10^{1000}\sqrt{\dfrac{10^{1998}-1}{9}}=\dfrac{\sqrt{10^{3998}-10^{2000}}}{3}$

Since $(10^{1997}-1)^2<10^{3998}-10^{2000}<(10^{1999}-4)^2$, so it follows that

$10^{1000}\sqrt{A}$ is between $\dfrac{10^{1999}-7}{3}=33\cdots33$ and $\dfrac{10^{1999}-4}{3}=33\cdots32$

Therefore the answer is 1.
 

Related to POTW #374: Finding the 1000th Digit of the Square Root of a Large Number

1. How do you find the square root of a large number?

To find the square root of a large number, you can use a calculator or a computer program that has the capability to handle large numbers. Alternatively, you can use a method called the long division method, which involves breaking down the number into smaller parts and finding the square root of each part.

2. What is the significance of finding the 1000th digit of the square root of a large number?

The 1000th digit of the square root of a large number is important because it can provide valuable information about the number itself. It can help in determining the number's properties, such as whether it is a perfect square or a prime number, and it can also be used in various mathematical calculations.

3. Is it possible to find the 1000th digit of the square root of any large number?

Yes, it is possible to find the 1000th digit of the square root of any large number using mathematical methods and computer programs. However, as the number gets larger, it becomes more difficult and time-consuming to calculate the 1000th digit accurately.

4. Can finding the 1000th digit of the square root of a large number have practical applications?

Yes, finding the 1000th digit of the square root of a large number can have practical applications in fields such as cryptography, where large numbers are used for encryption and decryption processes. It can also be used in data compression and error correction algorithms.

5. Are there any other methods for finding the 1000th digit of the square root of a large number?

Yes, there are various other methods for finding the 1000th digit of the square root of a large number, such as using the Newton-Raphson method or the binary search method. Each method has its own advantages and limitations, and the choice of method depends on the specific requirements and limitations of the problem at hand.

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