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Extracting Square Roots in your head

  1. Apr 11, 2005 #1
    What are some good quick ways to Extract Square Roots in your head.
    I'm looking for a method that is very prescice and easy
    thanks, hopefully somebody helps me out
  2. jcsd
  3. Jan 29, 2010 #2
    All right. I myself am searching for another method, but this is far by the easiest...and the best.

    Let me tell you the limitations well in advance:
    • This method only works for perfect squares.
    • You can calculate square roots of numbers upto 4 digits.
    Let us take the number 9604.

    It is obvious that the square root will have two digits. Let me make it easy to remember: the root of 3 and 4-digit numbers is always in 2 digits, while the root 2-digit numbers is always a single digit number.

    The first two digits of 9604 are 96. Which number's square, from 0 to 9, makes 96, or anything atleast close to it? 9, of course (as 9 times 9 equals 81). So the first digit of the root is 9.

    Now, what is the last digit of our number? 4. Which numbers square from 0 to 9 ends in 4? It is 8 and 2 (as 8*8=64 and 2*2=4). So these are our second digits of our root. The results we get are 98 and 92 for the root.

    Now we square any one number to see if it is the right root. If the number you square, as in this case, does not produce 9604, the other number is the correct root, i.e. 98.

    As a final note, remember, if the number has three digits, add a zero behind it before you compute it. For example, 961 becomes 0961. This is done so that you take the first two digits of 3-digit numbers, as in the case with the example I just provided, as 09, instead of 96.
    Last edited: Jan 29, 2010
  4. Jan 29, 2010 #3
    alright say i have 879,683,891,711. That is a 12 digit number. Think of this number as 87 with 10 digits after it. Or if you want to be more precise think of it at 88 with 10 digits after it becuase you can say that 87.9 is approx. 88. The integer square root of 88 is 9x9. But you want to get it correct to some more decimals.

    So lets go through another guess which is 9.5. the .5 multiplies by 9 twice, as well as .5. so 9.5 x 9.5 = 9x9 +0.5x9 + 0.5x9 + 0.5x0.5. .5 x 9 twice is 9. so we are already at 81+9 which is 90. thats no good it too high. So now my guess would be that it is 9.38 or something ilke that. But since we have a 12 digit number, we have to shift 9.38 (12-2)/2 digits over. Which is 5. So my guess would be that it is 938000 is the root.

    And i havent checked but i can tell you for sure that my answer is friggen close.

  5. Jan 29, 2010 #4
    Dacruick's method is pretty straightforward and widely applicable. Here's a noteworthy comment:

    When Dac' put the long radicand into scientific notation (when he said "Think of this number as 87 with 10 digits after it") he made sure to place the decimal somewhere that would provide an even number of "digits after it". IE 88X10^10 rather than 8.8x10^11.

    The trick wouldn't have worked otherwise.
  6. Jan 29, 2010 #5
    The method I use is actually quite similar

    While dacruick's method is good, it still relies too much on guessing. Instead of guessing, you could try to actually approximate it better

    First of all, as einsteinoid said, it depends on whether the number has an even number of digits or not.

    Example: 328: taking 32 would yield 5 as the first digit, and thats not even close to the real answer. So it depends on whether the number of digits is even or odd.

    In fact, what you're supposed to do is to write the number in a form p x 100^k, with k an integer. By decreasing k, u gain accuracy.

    So you want to calculate the square root of 879,683,891,711. This number can be written as

    87.9683891711 x 100^5

    The first perfect square below 87 is 81, so our first digit must be 9. Hence the root of this number will be at least:

    9 x 10^5

    For the second digit, we rewrite the number to:

    8796.83891711 x 100^4

    Now we know that the square root of this number is supposed to lie between 90x10^4 and 100x10^4
    Furthermore, we also know the difference between 90² and 91² equals to 90+91. Same goes for 91² and 92², the difference between them is 91+92. So for numbers between 90² and 100², we can say that the difference between them and 90² is about 180*(x-90). The difference between 8796 and 8100 is about 700, so we could fit 180 3 times in it. (Care has to be taken when it seems to fit almost exactly, in that case we cannot make the approximation that the difference between squares remains constant in the interval). So the next digit must be 3.

    93^2 yields (100x86+49) = 8649. Therefore, the root must be at least

    93 x 10^4

    For the next digit, again we rewrite the number, now looking at the first 6 digits:

    879683.891711 x 100^3

    This lies between 930² and 940². The difference between squares between 930² and 940² is approximately 1860. Thus we have:

    879683 - 864900 = about 14800. 8x1860 is just a little higher then 14800, so the next digit must be a 7.

    937² gives (1000x874+63^2) = 874000 + 7600/2 + 169 = 877969

    So our solution will lie between 937x10^3 (note it's 10^3, not 100^3) and 398x10^3. If your mind can still handle it, you could try and do the next digit.

    As you can see, the longer you repeat this, the closer you will get to the real root.

    Edit: When looking for the third digit, 8x1860 seemed quite close to 14800, so what happens if u do make the wrong assumption? What if I assumed it was about 15000 and
    took 8 as the next digit?

    938² would give me 1000x876+62^2 = 876000+7400/2 + 144 = 879844 > 879683, so we would have to lower it to 7.
    Last edited: Jan 29, 2010
  7. Jan 29, 2010 #6
  8. Jan 29, 2010 #7
    Ahh, Newton's method is pretty cool!
  9. Jan 30, 2010 #8
    Depends how accurate you want to be and what is the purpose.

    Usually crude estimates are good enough.
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