How to Calculate the Square Root by Hand (with Pictures)

2024 Author: Samantha Chapman | [email protected]. Last modified: 2024-01-15 13:47

Before the advent of computers, students and professors had to calculate square roots by hand. Several methods have been developed to deal with this cumbersome process: some give approximate results, others give exact values. To learn how to find the square root of a number using just simple operations, read on.

Steps

Method 1 of 2: Using Prime Factorization

Step 1. Factor your number into perfect squares

This method uses the factors of a number to find its square root (depending on the type of number, you can find an exact numerical answer or a simple approximation). The factors of a number are any set of other numbers that when multiplied together give the number itself as a result. For example, you could say that the factors of 8 are 2 and 4, because 2 x 4 = 8. Perfect squares, on the other hand, are whole numbers, the product of other whole numbers. For example, 25, 36, and 49 are perfect squares, because they are 5 respectively², 6² and 7². The perfect square factors are, as you can guess, factors which are themselves perfect squares. To start finding the square root through prime factorization, you can initially try reducing your number to its prime factors which are squares.

• Let's take an example. We want to find the square root of 400 by hand. To start, let's try dividing the number into factors that are perfect squares. Since 400 is a multiple of 100, we know that it is divisible by 25 - a perfect square. A quick split in mind lets us know that 25 goes into 400 16 times. Coincidentally, 16 is also a perfect square. Thus, the perfect square factors of 400 are

  Step 25

  Step 16., because 25 x 16 = 400.

• We could write it as: Sqrt (400) = Sqrt (25 x 16)

Step 2. Take the square root of your factors which are perfect squares

The property of the product of square roots states that for any number to And b, Sqrt (a x b) = Sqrt (a) x Sqrt (b). Based on this property, we can take the square roots of our factors which are perfect squares and multiply them together to get our answer.

• In our example, we will have to take the square roots of 25 and 16. Read below:

  • Sqrt (25 x 16)
  • Sqrt (25) x Sqrt (16)

  • 5 x 4 =

    Step 20.

  

  Step 3. If your number isn't a perfect factor, reduce it to a minimum

  In real life, for the most part, the numbers you have to find the square roots of will not be nice "round" numbers with perfectly quadratic factors, such as 400. In these cases, it may be impossible to find the correct answer as an integer.. Instead, by finding all the possible factors that are perfect squares, you can find the answer in terms of a smaller, simpler, and easier to manage square root. To do this, you need to reduce your number to a combination of factors of perfect and non-perfect squares, and then simplify.

  • Let's take the square root of 147 as an example. 147 is not the product of two perfect squares, so we can't find an exact integer, as we tried earlier. However, it is the product of a perfect square and another number - 49 and 3. We can use this information to write your answer as follows in simpler terms:

    • Sqrt (147)
    • = Sqrt (49 x 3)
    • = Sqrt (49) x Sqrt (3)
    • = 7 x Sqrt (3)

    

    Step 4. If needed, make a rough estimate

    With your square root in the form of smaller factors, it's usually easy to find a rough estimate of a numerical value by guessing the remaining square root values and multiplying them. One way to help you make this estimate is to find perfect squares on both sides of your square root number. You will know that the decimal value of your square root will be between these two numbers: in this way you will be able to approximate a value between them.

    • Let's go back to our example. Since 2² = 4 and 1² = 1, we know that Sqrt (3) is between 1 and 2 - probably closer to 2 than to 1. Suppose we have 1.7 x 1.7 = 11, 9. If we do the test with our calculator, we can see that we are close enough to the correct answer 12, 13.

      This also works with larger numbers. For example, Sqrt (35) can be estimated between 5 and 6 (probably very close to 6). 5² = 25 and 6² = 36. 35 is between 25 and 36, so its square root must be between 5 and 6. Since 35 is one digit less than 36, we can say with certainty that its square root is just less than 6. Doing the test with the calculator, we find about 5, 92 - we were right.

      

      Step 5. Alternatively, as a first step reduce your number to its minimum terms

      It is not necessary to find perfectly quadratic factors if you can determine the prime factors of a number (those factors that are also prime numbers). Write your number in the form of its prime factors. Then look for possible combinations of prime numbers among your factors. When you find two identical prime factors, remove both of these numbers from within the square root and put only one of these numbers outside the square root.

      • For example, we find the square root of 45 using this method. We know that 45 = 9 x 5 and that 9 = 3 x 3. We can then write our square root in the form of factors: Sqrt (3 x 3 x 5). Simply remove the 3 and put just one off the square root: (3) Sqrt (5). At this point it is easy to make an estimate.

      • As a final example problem, let's try to find the square root of 88:

        • Sqrt (88)
        • = Sqrt (2 x 44)
        • = Sqrt (2 x 4 x 11)
        • = Sqrt (2 x 2 x 2 x 11). We have several 2's in our square root. Since 2 is a prime number, we can remove a couple of them and put one out of the square root.
        • = our least terms square root is (2) Sqrt (2 x 11) o (2) Sqrt (2) Sqrt (11). At this point, we can estimate Sqrt (2) and Sqrt (11) to find an approximate answer.

        Method 2 of 2: Finding the Square Root Manually

        Use the Column Split Method

        

        Step 1. Separate the digits of your number into pairs

        This method uses a similar process to column division to find an exact square root, digit by digit. While it's not essential, you can make this process easier if you organize your workspace visually and work on your piece number. First of all, draw a vertical line that separates your workspace into two sections, then draw a shorter horizontal line at the top, at the top of the right-hand section, to divide it into a small upper part into a larger lower part. Then, starting with the decimal point, divide the digits into pairs: for example, 79.520.789.182, 47897 becomes "7 95 20 78 91 82, 47 89 70". Write it on the top left.

        For example, let's try to calculate the square root of 780, 14. Draw two segments to divide your workspace as above and write "7 80, 14" at the top in the left space. It may happen that on the far left there is only one number as well as that there are two. You will write your answer (the square root of 780, 14) in the space at the top right

        

        Step 2. Find the largest integer n whose square is less than or equal to the leftmost number or pair of numbers

        Start with the leftmost piece, which will be either a single number or a pair of digits. Find the largest perfect square that is less than equal to that group, then take the square root of this perfect square. This number is n. Write n in the upper left space and write the square of n in the lower right quadrant.

        In our example, the leftmost group is the single number 7. Since we know that 2² = 4 ≤ 7 < 3² = 9, we can say that n = 2, because it is the largest integer whose square is less than or equal to 7. Write 2 in the upper right square. This is the first digit of our answer. Write 4 (the square of 2) in the lower right quadrant. This number will be important in the next step.

        

        Step 3. Subtract the newly calculated number from the leftmost pair

        As with the division by column, the next step is to subtract the square just found from the group we have just analyzed. Write this number under the first group and subtract, writing under your answer.

        • In our example, we will write 4 under 7, then we will do the subtraction. This will give us as a result

          Step 3..

        

        Step 4. Write down the following group of two digits

        Move the next group of two digits to the bottom, next to the subtraction result you just found. Then multiply the number in the upper right quadrant by two and bring it back to the lower right. Next to the number you just transcribed, add '"_x_ ="'.

        In the example, the next pair is "80": write "80" next to 3. The product of the upper right number by 2 is 4: write "4_ × _ =" in the lower right quadrant

        

        Step 5. Fill in the blanks in the right quadrant

        You must enter the same integer. This number must be the largest integer which allows the multiplication result in the right quadrant to be less than or equal to the number on the left.

        In the example, entering 8, you get 48 multiplied by 8 equals 384, which is greater than 380. So 8 is too big. 7 on the other hand is fine. Enter 7 in the multiplication and calculate: 47 times 7 equals 329. Write 7 in the upper right: this is the second digit of the square root of 780, 14

        

        Step 6. Subtract the number you just calculated from the number you have on the left

        Continue with the division by column. Put the result of the multiplication in the right quadrant and subtract it from the number on the left, writing below what it does.

        In our case, subtract 329 from 380, which gives 51

        

        Step 7. Repeat step 4

        Lower the following group of two digits. When you encounter the comma, also write it in your result in the upper right quadrant. Then multiply the number in the upper right by two and write it next to the group ("_ x _"), as done previously.

        In our example, since there is a comma in 780, 14, write the comma in the square root at the top right. Lower the next pair of digits to the left, which is 14. The product of the upper right number (27) by 2 is 54: write "54_ × _ =" in the lower right quadrant

        

        Step 8. Repeat steps 5 and 6

        Find the largest digit to insert in the blanks on the right that gives a lesser result equal to the number on the left. Then solve the problem.

        In the example, 549 times 9 gives 4941, which is less than or equal to the left number (5114). Write 9 at the top right and subtract the multiplication result from the number on the left: 5114 minus 4941 gives 173

        

        Step 9. If you want to find more digits, write a pair of 0's in the bottom left and repeat steps 4, 5 and 6

        You can go ahead with this procedure to find cents, thousandths, etc. Continue until you get to the required decimals.

        Understanding the Process

        

        Step 1. To understand how this method works, consider the number whose square root you want to calculate as the surface S of a square

        It follows that what you are calculating is the length L of the side of that square. You want to find the number L whose square L² = S. Finding the square root of S, find the L side of the square.

        

        Step 2. Specify the variables for each digit of your answer

        Assign variable A as the first digit of L (the square root we are trying to calculate). B will be the second digit, C the third and so on.

        

        Step 3. Specify the variables for each group of your starting number

        Assign the variable S_TO to the first couple of digits in S (your starting value), S_B. to the second couple of digits, and so on.

        

        Step 4. Just as in the calculation of divisions we consider one digit at a time, so in the calculation of the square root we consider one pair of digits at a time (which is one digit at a time of the square root)

        

        Step 5. Consider the largest number whose square is less than S_TO.

        The first digit A in our answer is the largest integer whose square does not exceed S._TO (i.e. such that A² ≤ S_TO<(A + 1) ²). In our example, S_TO = 7 and 2² ≤ 7 <3², so A = 2.

        Note that, dividing 88962 by 7, the first step would be similar: you would consider the first digit of 88962 (8) and look for the largest digit which, multiplied by 7, is equal to or less than 8. Which means d such that 7 × d ≤ 8 <7 × (d + 1). d would therefore be 1

        

        Step 6. Display the square whose area you are calculating

        Your answer, the square root of your starting number, is L, which describes the length of the side of a square of area S (your starting number in parentheses. The values A, B and C represent the digits of the number L. Another way to put it is that, for a two-digit result, 10A + B = L, while, for a three-digit result, 100A + 10B + C = L and so on.

        In our example, (10A + B) ² = L² = S = 100A² + 2x10AxB + B². Remember that 10A + B represents our answer L with B in the units position and A in the tens. For example, with A = 1 and B = 2, 10A + B is simply the number 12. (10A + B) ² is the area of the whole square, while 100A² is the area of the largest square, B² is the area of the smallest square e 10AxB is the area of each of the two remaining rectangles. Continuing with this long and complex procedure, we find the area of the whole square by adding the areas of the squares and rectangles that compose it.

        

        Step 7. Subtract A² from S_TO.

        To consider the factor 100, a pair of digits (S_B.): "S_TOS._B."must be the total area of the square and from this we subtracted 100A² (the area of the largest square). What remains is the number N1 obtained on the left in step 4 (380 in the example). That number is equal to 2 × 10A × B + B² (the area of the two rectangles added to the area of the smaller square).

        

        Step 8. Calculate N1 = 2 × 10A × B + B², also written as N1 = (2 × 10A + B) × B

        You know N1 (= 380) and A (= 2), and you want to find B. In the equation above, B will probably not be an integer, so you will need to find the major integer B so that (2 × 10A + B) × B ≤ N1 - since B + 1 is too large, then you will have: N1 <(2 × 10A + (B + 1)) × (B + 1).

        

        Step 9. To solve, multiply A by 2, move it to the decimals (which would be equal to multiplying by 10), put B in the units position, and multiply that number by B

        That number is (2 × 10A + B) × B, which is exactly the same as writing "N_ × _ =" (with N = 2 × A) in the lower right quadrant in step 4. In step 5, you look for the largest integer which, substituted in multiplication, gives (2 × 10A + B) × B ≤ N1.

        

        Step 10. Subtract the area (2 × 10A + B) × B from the total area (on the left, in step 6), which corresponds to the area S- (10A + B) ², not yet taken into account (and which will be used to calculate the next digit in the same way)

        

        Step 11. To calculate the figure C below, repeat the process:

        lowers the next pair of digits from S (S_C.) to get N2 on the left and look for the largest C number so that (2 × 10 × (10A + B) + C) × C ≤ N2 (which is like writing the product times 2 of the two-digit number "AB" followed by "_ × _ =" and find the largest number that can be inserted into the multiplication).

        Advice

        • Moving the comma by two into a decimal number (factor 100) is the same as moving the comma by one into the square root (factor 10).
        • In the example, 1.73 can be considered as a "remainder": 780, 14 = 27, 9² + 1.73.
        • This method works with any type of base, not just the decimal.
        • You can represent your calculations in the way that is most convenient for you. Some write the result above the starting number.
        • For an alternative method use the formula: √z = √ (x ^ 2 + y) = x + y / (2x + y / (2x + y / (2x +…))). For example, to calculate the square root of 780, 14, the integer whose square is closest to 780, 14 is 28, hence z = 780, 14, x = 28, and y = -3, 86. Entering i values and calculating for x + y / (2x) we obtain (in minimum terms) 78207/2800 or, by approximating, 27, 931 (1); the next term, 4374188/156607 or, approximating, 27, 930986 (5). Each term adds about 3 decimals of precision to the previous one.