Algorithm to find square root of a number

Square Root Algorithm Derivation

The Algorithm:

Use this algorithm to find the square root of any real number.

Step 1: Group the number in "twos" from the decimal place. (If you have a number with an odd number of digits, the group to the far left will only be a group of 1 digit.)

Step 2: Start with the first group of two (the group on the left). This group may be a group of only one number if your number has an odd number of digits. Find the greatest square less than or equal to that group of digits and its square root will be your first approximation of the entire square root.

Step 3: Subtract the current approximation squared and bring down the next group of numbers behind it. This is your next number to work with.

Step 4: Double the current approximation of the root.

Step 5: Find the "ones" digit of the doubled number that would result in a number which divides into the number you are currently working with- with the smallest possible remainder. This is the next number in your approximation of the square root.

Step 6: Multiply the "ones" digit by the doubled number plus the "ones" digit. Subtract this number from the number you are currently working with and bring down the next group of numbers behind it. This is your next group of numbers to work with.

Step 7: Repeat steps 4 through 6 until you get an approximation with an acceptable number of significant digits.

Diagram of Algorithm:

The Derivation:

The square root of a number, N, is the number, M, so that M2 = N. The square root algorithm is set up so that we take the square root of a number in the form of (X + R)2. The square root of this number is obviously (X + R). X represents the current approximation for the square root, and R represents the remainder of the number left over from the approximation. Our approximation will always be the correct square root of the number truncated (not rounded) to the number of digits in our approximation. If we expand our number (X + R)2 it will equal X2 + 2RX + R2. This gives us the basis for our derivation of the square root algorithm.

Step 1: The square root of a number between 1 and 100 is a number between 1 and 10. Furthermore, the square root of a number between 100 and 10000 is a number between 10 and 100. And so on. Therefore, if a number is divided into groups of two digits each from the decimal point to the left, the number of digits in the integer part of the square root of that number will be equal to the number of groups of figures.

Step 2: Only the first group of two (or one) digits determine the first digit of the square root. If the first two (or one) digits are a perfect square, then there is nothing left over and the process can be repeated on the next two digits of the number. This is usually not the case, which means there is a part of the first two (or one) digits which has not been accounted for and our approximation is not perfect. This leads into Step 3.

Step 3: Take the expanded value of our number: X2 + 2RX + R2. We subtract off the current approximation, X2, which results in 2RX + R2, the part of the original number which is not accounted for in our approximation. This gives us our next value to work with.

Step 4: Rewriting 2RX + R2 gives us R(2X + R). We see that our current approximation, X, must be doubled, resulting in 2X, which are the first digits of the number we will be working with.

Step 5: In order to find the next approximation, we need the R value. This number must divide into the next grouping with the smallest remainder, as shown by R(2X + R). (R obviously divides this number.)

Step 6: Since you've found an approximation based on this number, you must subtract of R(2X + R) so you take into account any remainder from your previous group of numbers.

Step 7: The procedure can be repeated as many times as necessary until either no remainder is found, meaning you have the actual square root, or until an acceptable number of decimal places are estimated.

Diagram of Derivation:

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The square root algorithm, which helps to get the square root without using a calculator, is not taught a lot in school today.

Teachers usually teach the process once or never and usually have students use a calculator afterward.

If you want to understand how to get the square root without using a calculator, study the following example carefully!

Sorry, there are no shortcuts for this square root algorithm, so you will need patience before you master this algorithm!

Square root algorithm to find the square root of 2685

Example:  Square-root of 2685

First, always group the numbers in pairs starting from right to left and it is OK if there is only one number left in the leftmost position.

I will use a space to show that they are being grouped in pairs. In this case, we have two pairs of 2 numbers.

Then, using the first pair, (26) find a number whose square is less than 26.

5 × 5 = 25 and 6 × 6 = 36, so your number is 5. Put 5 on top of the square root sign.

Subtract the square of the number on top that is 5 from the first pair.

Very important! : When you first start, you always take the square root of the pair or single number in the leftmost position. After that, you always double the number on top.

Since 52 = 5 × 5 = 25, subtract 25 from 26.

Bring down the next pair of numbers (85).

Then, double 5 to get 10 and put 10 next to 185 on the left side.

For the number 10, you will try to find a number you can put next to 0, which will also go next to 5.

Let's call this number z. Then, find z such that 10z × z is less or equal to 185, but never bigger than 185.

If I put 2 next to 10, I get 102 × 2 = 204 and 204 is bigger than 185, so z = 2 is not the number I need.

However, if I put 1 next to 10, I get 101 × 1 = 101 and 101 is smaller than 185.

Thus, z = 1 is the number I will put next to 10 and 5.

101 is the number I will subtract from 185.

Put the 1 on top of the square root sign next to 5

Subtract 101 from 185

Bring down a pair of zeros next to 84 to continue the process and put a decimal point next to 51.

Double 51 to get 102 and put 102 next to 8400 on the left side.

For the number 102, you will try to find a number you can put next to 2, which will also go next to the decimal point.

Let's call this number z again. Then, find z such that 102z × z is less or equal to 8400, but never bigger than 8400.

If I put 9 next to 102, I get 1029 × 9 = 9261 and 9261 is bigger than 8400.

However, if I put 8 next to 1028, I get 1028 × 8 = 8224 and 8224 is smaller than 8400. Thus, z = 8 is the number I will put next to 102 and the decimal point.

8224 is the number I will subtract from 8400.

Put 8 on top of the square root sign next to 102 and the decimal point.

Subtract 8224 from 8400

Bring down a pair of zeros next to 176. Double 518 ignoring the decimal point to get 1036 and put 1036 next to 176000.

For the number 1036, you will try to find a number you can put next to 6, which will also go next to 8.

Let's call this number z again. Then, find z such that 1036z × z is less or equal to 17600, but never bigger than 17600.

If I put 2 next to 1036, I get 10362 × 2 = 20724.

Since when z = 2, I get 20724 and 20724 is bigger than 17600, z = 2 is not the number I need.

However, if I put 1 next to 1036, I get 10361 × 1 = 10361 and 10361 is less than 17600. Thus, 10361 is the number you will subtract from 17600.

1 is the number you will put next to 1036 and 8.

Put 1 on top of the square root sign next to the decimal point.

Subtract 10361 from 17600

I will stop here because it is a lot of work to put this square root algorithm here for you. Appreciate my effort? Look for the donation button below and please make a donation to keep this website running.

Feel free to continue the square root algorithm to get more decimal numbers, but I think I am done.

If you were looking for the square root of 785 for instance, the square root algorithm does not change at all.

Notice that when you group the number in pairs starting from right to left, there will be only 1 number in the leftmost position. That number is 7 and that is fine.

Just start with 7 and say to yourself, " The square of what number is less than 7 and proceed an already outlined above.

I thought I said I was done. Why am I still talking about this square root algorithm?

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