In the days before the calculator, students and teachers had to calculate square roots by hand. Many different methods have emerged to tackle this daunting task, some providing a rough estimate while others calculating the exact value. If you want to know how to compute square roots using simple arithmetic operations, start at step 1 and find out.
Method 1 of 2: use prime number factorization
Step 1. Divide your number into smaller square number factors
This method uses the factors of a number to determine its square root (depending on the starting number, the result can be an exact numerical solution or an approximation). The factors of a number are the numbers that, when multiplied together, make the original number. For example, you can say that the number 8 has the factors 2 and 4. Since 2 * 4 = 8 results. On the other hand, squares are integers that are the square of another integer. For example, 25, 36 and 49 are square numbers because they are also used as 52, 62, and 72 can be written. Square number factors, as you have probably already guessed, are factors that are also square numbers themselves. To find a square root using prime number factorization, first try to break your number down into smaller square number factors.
- Let's use the following example: We want to calculate the square root of 400 by hand. So let's try to divide the number into smaller square numbers first. Since 400 is a multiple of 100, it must also be divisible by 25 - a square number. We divide 400 in our heads by 25 and get 16. 16 happens to be a square number too. So we can put 400 in the square numbers 25 and 16 split, since 25 × 16 = 400.
- We write this: Sqrt (400) = Sqrt (25 × 16)
Step 2. Get the root of your factors
The product rule for square roots says that for all given numbers a and b, Sqrt (a × b) = Sqrt (a) × Sqrt (b). Because of this property, we can take the square root of the two factors and multiply them together to get the final result.
In our example we would take the square root of 25 and 16:
- Sqrt (25 × 16)
- Sqrt (25) × Sqrt (16)
5 × 4 =
Step 3. If your starting number cannot be factored perfectly, reduce your answer to the simplest possible term
In reality, the number you are trying to get the root of cannot always be perfectly broken down into square-number factors. In these cases, you cannot give the exact answer as an integer. But by using the square-number factors you can find, you can turn the root into a smaller, simpler, easier-to-handle problem. To do this, you reduce your number to a combination of factors of square numbers and "non-square numbers" and then simplify.
Take the square root of 147 as an example. 147 cannot be divided into a product of two square numbers, so we do not get a whole number as a result, as in the first example. However, 147 is the product of a square number and another number: 49 and 3. We can use this information to get the simplest possible term as a solution:
- Sqrt (147)
- = Sqrt (49 × 3)
- = Sqrt (49) × Sqrt (3)
- = 7 × Sqrt (3)
Step 4. If necessary, assess the result
Once you've simplified your root like this, it's usually relatively easy to give a rough estimate of the final numerical result by estimating and multiplying the value of the remaining root. One way to do this is to find the square numbers just above and below your number. This will let you know that your result has to be somewhere between these numbers, so you can estimate it.
Let's go back to our example: Since 22 = 4 and 12 = 1, we know that Sqrt (3) must be between 1 and 2 - probably closer to 2 than to 1. We estimate 1, 7. 7 × 1.7 = 11, 9. If we check this answer with the calculator, we find that we are pretty close to the real value, 12, 13, lie.
- This also works with larger numbers. Sqrt (35) must be somewhere between 5 and 6, probably close to 6. 52 = 25 and 62 = 36. 35 is between 25 and 36, so the square root must be between 5 and 6. Since 35 is only one way out of 36, we can safely say that its square root is only slightly lower than 6. The calculator check gives us a value of 5.92 - so we were right.
Step 5. Alternatively, as a first step, reduce the number to its prime number factors
Finding square number factors is not absolutely necessary if you can determine the prime number factors (factors that are also prime numbers). Write the number out in its smallest possible factor. Then, check the factors for equal pairs. If you can find two equal factors, extract both from the root and put one of the two in front of the root.
- For example, let's try to find the square root of 45 using this method. We know that 45 = 9 × 5. In addition, 9 = 3 × 3. So we can write our square root with the following factors: Sqrt (3 × 3 × 5). Now just remove the 3's and drag one of them in front of the root, so you get the simplified square root: '(3) Sqrt (5).' Now it is easy to estimate.
As a final example, let's try to find the square root of 88:
- Sqrt (88)
- = Sqrt (2 × 44)
- = Sqrt (2 × 4 × 11)
- = Sqrt (2 × 2 × 2 × 11). We have several 2's in our root. Since 2 is a prime number, we can remove a pair and put a 2 in front of the root.
= So our simplified square root is (2) Sqrt (2 × 11) or (2) Sqrt (2) Sqrt (11).
Now we can guess Sqrt (2) and Sqrt (11) and find an estimated answer if that's what we want.
Method 2 of 2: calculate square roots manually
Use written division
Step 1. Separate the digits of your number into pairs
This method works in a similar way to the written division and gives the exact result of the square root digit-to-digit. While not mandatory, the process is easier to oversee if you divide your workspace into workable quadrants. First, draw a vertical line that divides your work area into two quadrants. Then draw a shorter horizontal line at the top of your right quadrant to divide the right area into a smaller and a larger quadrant. Now, divide the digits of your number into pairs of two, starting with your decimal point. For example, 79.520.789.182, 47897, "7 95 20 78 91 82, 47 89 70". Write your number in your left area at the top.
As an example, let's try to find the square root of 780, 14. Draw the two lines to divide your work area and write 7 80.14 at the top of the left quadrant
Step 2. Find the largest integer n whose square is less than or equal to the number (or pair of numbers) on the far left
Start with your starting number with the digits on the far left. Regardless of whether it is a digit or a pair. Find a square number that is less than or equal to. Then take the square root of that square number. That number is n. Write n in the upper right area and write the square of n in the lower right area.
- In our example, the leftmost digit is the number 7. We know: 22 = 4 ≤ 7 < 32 = 9. So n = 2, since 2 is the smallest integer whose square is less than or equal to 7. Write 2 in the upper right area. That is the first digit of our solution. Write 4 (the square of 2) in the lower right area. This number is relevant for the next step.
Step 3. Subtract the number you just got from the pair of numbers on the far left
As in the written division, in the next step we need to subtract the square number we just found from the area we just worked on. Write the number under the area and subtract, write the result below.
In our example, write 4 under 7, then subtract. We get the answer
Step 4. Drag the next pair of numbers down on the left
Drag the next range of the number whose square root we want to find next to the subtracted value we just calculated. Now multiply the number in the upper right quadrant by 2 and write the result in the lower right quadrant. Save space for the multiplication in the next step by writing '"_ × _ ="' next to the number.
In our example the next pair is "80": Write "80" next to the 3. Double the number in the upper right gives 4: So write "4_ × _ =" in the lower right quadrant
Step 5. Fill in the placeholders
You need to fill in all the placeholders on the right with the same whole number. To do this, you use the largest possible whole number, which means that the multiplication on the right is less than or equal to the current number on the left.
In our example, if we fill the spaces with 8, we get 4 (8) × 8 = 48 × 8 = 384. So a number greater than 380. 8 is too big. So we try with 7. Write 7 in the free fields and solve: 4 (7) × 7 = 329. With 7 we get 329, which is less than 380. Write 7 in the upper right area. That is the second digit of our solution
Step 6. Subtract the number you just calculated from the current number on the left
Continue to follow the written division principle and subtract accordingly. Take the result of the multiplication problem in the right square and subtract it from the current number on the left, write the answer below.
- In our example: Subtract 329 from 380 and you get 51.
Step 7. Repeat step 4
Drag the next section of the number down. When you reach the decimal point, write a decimal point in your answer on the right. Then, multiply the number on the right by 2 and write it next to the open-ended multiplication calculation ("_ × _"), just like above.
In our example: Since we now come to the decimal point, we write a decimal point in our current solution in the upper right corner. Then we pull the next pair (14) down. Twice the number on the right-hand side (27) results in 54, so write "54 _ × _ =" in the lower right quadrant
Step 8. Repeat steps 5 and 6
Find the largest digit to replace the placeholders and do the multiplication.
In our example: 549 times 9 results in 4941 and this result is less than the number on the left (5114). 549 × 10 = 5490, which is too much. So write a 9 in the top right and subtract the result of the multiplication from the number on the left: 5114 minus 4941 gives 173
Step 9. To calculate more digits, drag a pair of zeros down and repeat steps 4, 5, and 6
For added accuracy, repeat the process and find the hundredths, thousandths, etc. of your answer. Repeat the process until you are satisfied with the accuracy of your solution.
Understand the process
Step 1. Consider the number you are calculating the square root for as the area S of a square
Since the area of a square L2 with L as the length of one side, use the square root of S to find the length L of one side of the square.
Step 2. Assign a letter to each part of your answer
Denote the first digit of L (the square root we want to compute) as A. B is the second digit, C is the third, and so on.
Step 3. Give each pair of digits a variable name
Assign the variable S to the first parta, the second part pb, etc.
Step 4. Understand the relationship to the written division
As with a written division, in which you are only ever interested in the next digit, you are only interested in the next two digits at a time (which ultimately each result in a digit of the square root)
Step 5. Find the largest number whose square is less than or equal to S.a is.
The first digit A of this square root is then the largest integer, the Sa does not exceed (this means that A is chosen so that A² = Sa ≤ (A + 1) ²). In our example, S isa = 7 and 2² ≤ 7 <3², so A = 2.
For example, if you want to divide 88962 using written division by 7, then the first step is similar. You would look at the first digit of 88962 (8) and you would find the largest number that, when multiplied by 7, is less than or equal to 8. That is, a number d such that 7 × d ≤ 8 <7 × (d + 1). The number d would then be 1
Step 6. Imagine the square whose area you want to calculate
Your answer, the square root of your starting number, is L, which is the length of one side of S (your starting number). The values A, B, C, stand for the individual digits of L. In other words, for a two-digit solution 10A + B = L, while for a three-digit solution 100A + 10B + C = L, etc.
- In our example is (10A + B) 2 = L2 = S = 100A² + 2 x 10A x B + B². 10A + B stands for our solution L, with B in the ones place and A in the tens. With A = 1 and B = 2, 10A + B would simply be the number 12. (10A + B) 2 is the area of the entire square while 100A² the area of the largest inner square, B² the area of the smallest square and 10A × B is the area of the two remaining rectangles. With the help of this long, nested process, we find the area of the entire square by adding up the areas of the squares and rectangles it contains.
Step 7. Subtract A² from Sa.
Draw a pair (pb) from places from S downwards. S.a S.b is almost the entire area of the square from which you just subtracted the larger inner square. The remainder can be seen as the number N1 that we got in step 4 (N1 = 380 in our example). N1 corresponds to 2 × 10A × B + B² (area of the two rectangles, plus area of the small square).
Step 8. Search for N1 = 2 × 10A × B + B², also written N1 = (2 × 10A + B) × B
In our example we already know N1 (380) and A (2), so all you have to do is find B. B is unlikely to be an integer, so you really need to find the largest integer such that (2 × 10A + B) × B ≤ N1. So you have: N1 <(2 × 10A + (B + 1)) × (B + 1).)
Step 9. Solve
To solve this equation, multiply A by 2, move it to the tens place (equivalent to multiplying by 10), place B to the ones place, and multiply the number by B. In other words, solve (2 × 10A + B) × B This is exactly what you do when you write "N_ × _ =" (with N = 2 × A) in step 4 in the lower right quadrant. And in step 5 you are looking for the largest integer B that fits the place of the underscores, so that: (2 × 10A + B) × B ≤ N1.
Step 10. Subtract the area (2 × 10A + B) × B from the total area
This gives you the area S- (10A + B) ², which we have not yet considered (and which we use to calculate the next digit in a similar form).
Step 11. To calculate the next digit C, repeat this process
Delete the next pair of digits (pc) of S to get N2 on the left and look for the greatest integer C such that: (2 × 10 × (10A + B) + C) × C ≤ N2 (corresponds to the representation of "AB", followed by "_ × _ =" and finding the largest integer that fits the bars).
- Moving the decimal point forward by two digits (factor of 100) moves the decimal point of the square root one place forward (factor of 10).
- In our example, you can consider 1.73 to be the "remainder": 780, 14 = 27, 9² + 1.73.
- This method works for any number system, not just base 10 (decimal).
- You can adapt the presentation of your calculation to your own requirements. Some people prefer to write the result over the starting number.
- An alternative method that uses continuous fractions can be as follows: √z = √ (x ^ 2 + y) = x + y / (2x + y / (2x + y / (2x +…))). For example, to find the square root of 780.14, find the integer whose square is closest to 780.14, which is 28. So z = 780.14, x = 28, and y = -3.86. Set one and with the first term alone you get x + y / (2x) = 78207/2800 or about 27.931 (1); with the next term, 4374188/156607 or about 27.930986 (5). Each term brings an additional 3 decimal places of precision.
- Make sure to split the digits into pairs starting from the decimal point. 79,520,789,182, 47897 to be subdivided as follows: "79 52 07 89 18 2, 4 78 97 "results in a useless number.