# 1 Answer

Many school books seem to think that since calculators can find square roots, that kids don't need to learn how to find square roots using any pencil-and-paper method. But learning at least the "guess and check" method for finding the square root will actually help the student UNDERSTAND and remember the square root concept itself!

So even though your math book may totally dismiss the topic of finding square roots without a calculator, you can consider to let them practice at least the first method presented here. This method, "guess and check", actually works around what the square root is all about, so I would consider exercises with it as essential to help children understand the concept of square root.

Depending on the child, it might be good to concentrate on teaching the concept of square root without taking the time for paper-pencil calculations. In this case, you can study the guess and check method with the help of a simple calculator that doesn't calculate square roots but can quickly do the multiplications.

Finding square roots by guess & check method

One simple way to find a decimal approximation to, say √2 is to make an initial guess, square the guess, and depending how close you got, improve your guess. Since this method involves squaring the guess (multiplying the number times itself), it actually uses the definition of square root, and so can be very helpful in teaching the concept of square root.

Example: what is √20 ?

Children first learn to find the easy square roots that are whole numbers, but quickly the question arises as to what are the square roots of all these other numbers. You can start out by noting that (dealing here only with the positive roots) since √16 = 4 and √25 = 5, then √20 should be between 4 and 5 somewhere.

Then is the time to make a guess, for example 4.5. Square that, and see if the result is over or under 20, and improve your guess based on that. Repeat the process until you have the desired accuracy (amount of decimals). It's that simple and can be a nice experiment for children.

Example: Find √6 to 4 decimal places

Since 22 = 4 and 32 = 9, we know that √6 is between 2 and 3. Let's just make a guess of it being 2.5. Squaring that we get 2.52 = 6.25. That's too high, so make the guess a little less. Let's try 2.4 next. To find approximation to four decimal places we need to do this till we have five decimal places, and then round the result.

Guess Square of guess High/low

2.4 5.76 Too low

2.45 6.0025 Too high but real close

2.449 5.997601 Too low

2.4495 6.00005025 Too high, so between 2.449 and 2.4495

2.4493 5.99907049 Too low

2.4494 5.99956036 Too low, so between 2.4494 and 2.4495

2.44945 5.9998053025 Too low, so between 2.44945 and 2.4495.

This is enough since we now know it would be rounded to 2.4495 (and not to 2.4494).

Finding square roots using an algorithm

There is also an algorithm that resembles the long division algorithm, and was taught in schools in days before calculators. See the example below to learn it. While learning this algorithm may not be necessary in today's world with calculators, working out some examples can be used as an exercise in basic operations for middle school students, and studying the logic behind it can be a good thinking exercise for high school students.

Example: Find √645 to one decimal place.

First group the numbers under the root in pairs from right to left, leaving either one or two digits on the left (6 in this case). For each pair of numbers you will get one digit in the square root.

To start, find a number whose square is less than or equal to the first pair or first number, and write it above the square root line (2).

2

√6 .45

2

√6 .45

- 4

2 45

2

√6 .45

- 4

(4 _) 2 45

2

√6 .45

- 4

(45) 2 45

Square the 2, giving 4, write that underneath the 6, and subtract. Bring down the next pair of digits. Then double the number above the square root symbol line (highlighted), and write it down in parenthesis with an empty line next to it as shown. Next think what single digit number something could go on the empty line so that forty-something times something would be less than or equal to 245.

45 x 5 = 225

46 x 6 = 276, so 5 works.

2

5

√6 .45 .00

- 4

(45) 2 45

- 2 25

20 00

2

5

√6 .45 .00

- 4

(45) 2 45

- 2 25

(50_) 20 00

2

5

. 3

√6 .45 .00

- 4

(45) 2 45

- 2 25

(503) 20 00

Write 5 on top of line.

Calculate 5 x 45, write that

below 245, subtract, bring down the next pair of digits (in this case the decimal digits 00). Then double the number

above the line (25), and write the doubled number (50) in parenthesis with an empty line next to it as indicated: Think what single digit number something could go on the empty line so that five hundred-something

times something would be less than or equal to 2000.

503 x 3 = 1509

504 x 4 = 2016, so 3 works.

2

5

. 3

√6 .45 .00 .00

- 4

(45) 2 45

- 2 25

(503) 20 00

- 15 09

4

91 00

2

5

. 3

√6 .45 .00 .00

- 4

(45) 2 45

- 2 25

(503) 20 00

- 15 09

(506_)

4

91 00

2

5

. 3 9

√6 .45 .00 .00

- 4

(45) 2 45

- 2 25

(503) 20 00

- 15 09

(506_) 4 91 00

Calculate 3 x 503, write that

below 2000, subtract, bring down the next digits. Then double the 'number' 253 which is above the line (ignoring the decimal point), and write the doubled number 506 in parenthesis with an empty line next to it as indicated: 5068 x 8 = 40544

5069 x 9 = 45621, which is less

than 49100, so 9 works.

Thus to one decimal place, √645 = 25.4

Visitor comments

I was just writing another comment and somehow the computer submitted it before I was done.I must have tapped the wrong key. So let me just finish by saying that the children are new to the world and are exploring it. Calculating square roots longhand would I believe be fascinating for them and a great way to learn about other topics in math. Oh and by the way I didn't have any lessons at all on square roots until high school and then we didn't learn any way of calculating them.We were taught to factor the number under the radical and extract perfect squares leaving non-perfect squares under the radical. BECAUSE EVEN THE TEACHER DIDN"T KNOW HOW TO DO IT THE RIGHT WAY. Bye and God Bless

Robert Monroe

this is one of the very best sites I have visited for the correct process to solve a problem. You may call me arcaic but when I went to school, they taught the long division to find a square root of a number.

MOSTLY, IT TEACHES ONE TO THINK. Using a calculator is a form of pure laziness. I feel that our children think that getting the basics in school(EARLY) is arcaic. That is why when you go into the store and the bill is 16.75 and you hand the teller a twenty dollar bill, a single dollar bill and 75 cents they haven't a clue what the change should be unless the cash register tells them how much to give you. This leads to LAZY THINKING OR, NOT THINKING AT ALL.

Thank you for your time.

Rush Kerlin

I was looking at the web for the long forgotten routine for finding square roots by hand and I run into your webpage. and wanted to say that many (or all) of the criticism on the standard algorithm calling it ‘archaic’, ‘dead end’ method, etc. in favor of the Babylonian method cannot be justified. The fact of the matter is using paper and pencil to do long division or finding square roots is archaic and is a dead-end process in the 21 st Century, irrespective what routine we use, since we don’t do that anymore for any practical calculations. So the issue is what should we teach to expose students to the fundamental techniques? Babylonian method is a numerical method unlike the other method, and it makes perfect sense to teach the standard routine that works for any numbers first and then other approximate numerical methods, rather than using a predictor-corrector type numerical methods saying they have applications elsewhere. If we go with the predictor-corrector type methods, one has to do an error analysis also, which is not needed with standard method since with the standard routine the correct digits are added one by one with each step (unlike the Babylonian method where the content of the digits may change through each averaging).

Best Wishes,

Karl I. Jacob

Professor, School of Polymer, Textile and Fiber Engineering

Professor, G. W. Woodruff School of Mechanical Engineering

Georgia Institute of Technology

You provided an answer to address, Finding square roots using an algorithm. I noticed that the answer provided was challenged by several people for several reasons. I would like to point out that the solution provided is THE oldest method of solving for square roots in the western world. I was described by Leonardo Picano, otherwise known as Fibonacci, in his book Liber Abaci, Chapter 14. The first edition was "written" in 1202, and the second edition was "written" in 1228. I say "written" because it was literally written by hand, as were all the copies. Johannes Gutenberg's work on the printing press didn't begin until 1436.

Leonardo learned the method from his Arabic travels around the Mediterranean sea, and the Arabs learned it from the Hindu nation around todays India. The method in the example you show, includes some modern interpretation that makes it easier to read. Leonardo also showed a geometric relationship that is related to what we understand as 'chords' today. This is a very simple, non-calculator solution to the question.

David T. Carrott, PhD

I read your suggestion for calculating square root without a calculator. I teach Math for Elementary Teachers and developmental math courses (algebra) to adults. I feel that the focus should be on understanding the number rather than an exercise in following a memorized algorithm. I suggest you have the student determine the pair of perfect squares the number falls between. For example, if finding the sqrt of 645, it falls between the sqrt of 625 which equals 25 and the sqrt of 676 which equals 26. So the sqrt of 645 has to be between 25 and 26. Where does it fall between? There are 50 numbers between 676 and 625. 645 is 20 numbers beyond 625, so 20/50 = 0.4 So the sqrt of 645 is very close to 25.4

This method provides the student with a process that improves their understanding of numbers without expecting them to memorize an algorithm, and it provides an answer to the nearest tenth.

Andrea S. Levy, Ed.D.

I'm currently a student at MCC I'm taking a course that is for Elementary Math Teachers. We are supposed to do a lesson plan so that we can teach elementary children how to use the Pythagorean theorem. I need to learn how to break down Pythagorean theorm for an elementary child. I got stuck at the square rooting part.

Read my answer to this question.

The method you show in the article is archaic. There is a MUCH more efficient algorithm. (This is the algorithm actually used behind the scenes inside a calculator when you hit the square root button.)

1. Estimate the square root to at least 1 digit.

2. Divide this estimate into the number whose square root you want to find.

3. Find the average of the quotient and the divisor. The result becomes the new estimate.

The beauty of this method is that the accuracy of the estimate grows extremely rapidly. Each cycle will essentially double the number of correct digits. From a 1-digit starting point you can get a 4-digit result in two cycles. If you know a square root already to a few digits, such as sqrt(2)=1.414, a single cycle of divide and average will give you double the digits (eight, in this case).

In addition to giving a way to find square roots by hand, this method can be used if all you have is a cheap 4-function calculator. If students can get square roots manually, they will not find square roots to be so mysterious. Also, this method is a good first example of an itterative solution of a problem.

David Chandler

This other way is called Babylonian method of guess and divide, and it truly is faster. It is also the same as you would get applying Newton's method. See for example finding the square root of 20 using 10 as the initial guess:

Guess Divide Find average

10 20/10 = 2 average 10 and 2 to give new guess of 6

6 20/6 = 3.333 average 3.333 and 6 gives 4.6666

4.666 20/4.666= 4.1414 average 4.666,4.1414= 4.4048

4.4048 20/4.4048=4.5454 average = 4.4700

4.4700 20/4.4700=4.4742 average = 4.4721

4.4721 20/4.4721=4.47217 average = 4.47214

This is already to 4 decimal places

4.47214 20/4.47214=4.472132 average =4.472135

4.472135 20/4.472135=4.472137 average = 4.42136

The poster asserts that the article's method is "archaic" and that the "Babylonian Method" is more efficient. At first glance, this would appear to be so, because the poster's example finds the square root of the two digit whole number 20 instead of the article's example of 645.

However, I actually worked out the article's example (square root of 645) using both methods and found that the Babylonian Method required 9 "cycles of divide and average" to arrive at the answer. Also, the Babylonian Method requires the student to perform 5 digit long division - no small feat for an elementary or middle school student. The article's method, on the other hand, only requires the student to perform one 4 step, long division problem by working out at the most a half a dozen or so 4 digit x 1 digit multiplication problems.

It is therefore reasonable to conclude that the Babylonian Method is more suitable to solve by calculator or solve by computer, while the article's method is more suitable to solve by pencil-and-paper.

Since the subject of the article was how to teach an elementary or middle school student to easily find square roots with a paper-and-pencil method, the article's "archaic" method seems to be the most fitting.

Alex

In response to Alex's post, How did it take you 9 cycles to produce 25.4 using the Babylonian Method on 645? It takes 1.5 steps if you use your guess as 25

1)645/25=25.8

(25+25.8)/2=25.4

2)645/25.4=~25.39

The Babylonian method is very effective if one already knows many perfect squares to approximate the original value. I find that students cannot follow the reasons behind the algorithm in this post, while the divide and average method seems to be more intuitive if they have worked with averages before.

Daniel

I am doubtful about teaching the long division method for extracting square roots. The Babylonian method is easier to remember and understand, and it affords just as much practice in basic arithmetic. More importantly, it has clear connections to topics such as Newton's method and recursive sequences that will be encountered in calculus and beyond. The long division method is somewhat faster for manual calculation, but it leads to no other important topics -- it is a dead end.

David

I was trained on old computer circuitry and binary hardware algorithms. The method used to calculate the root of 645 is the method used in high performance binary calculations since it only requires shift, subtract, and compare which are all single cycle/stage instructions or are diverted to a co-processor. Convert a number to binary, split it into 2 bit groups, and use the above routine. Multiply and divide require 10's to hundreds of cycles/stages and kill preformance and pipelines. It is faster to perform a square root than a divide since divide works through 1 bit per cycle/stage and square root steps through 2 bits per cycle.

Brad

Can we find the nth root by division method. if yes then please tell me ?

or tell how we can 3rd, 4th, root by division method.

Amar Deep

Yes, we can. It looks quite tedious to do by hand, but the algorithm exists for any root and is similar to the square root one. See these links: an example of using division method for finding cube root, and info about the nth root algorithm (or paper-pencil method).

what is the square root of -1?

Tamara Yardley

-1 cannot have a square root (at least, not a real one) because any two numbers with the same "sign" (+/- positive or negative), when multiplied, will equal a positive number. Try it: +2 × +2 = 4 and -2 × -2 = 4.

Since a square root of a number must equal that number when multiplied by itself. When you multiply this number by itself, and set it up as a full equation ( n * n = x ), the two factors (n and n) are either both positive or both negative since they are the same number. Therefore, their product will be positive. No real number multiplied by itself will equal a negative number, so -1 cannot have a real square root.

Blake

Square root of -1 is not a real number. It is denoted by i and called the imaginary unit. From i and its multiples we get pure imaginary numbers, such as 2i, 5.6i, -12i an so on. It leads to a whole new number system of complex numbers where numbers have a real part and an imaginary part (for example 5 + 3i or -20 - 40i). And there is a lot of fascinating mathematics done with this number system!

I was trying to find on the net, the old way of doing square roots by long division. YEAH I found it. Read the responses and would disagree with many of the posters.

Finding the square of 645 is easy if you know 252 and 262 but I never memorized the squares of numbers from 1 to 30 or so, I only memorized up to 12X12 (old imperial system)

Guessing the square of 645 is around 25 is great but if you guess it's 2 then you have a larger problem ahead of you.

I see the 'other' posters are finding easier quicker ways...that is the trouble today. Let's look for an easy way with no understanding. With your method anyone with long division and simple multiplication skills can do it. The simplest solution is buy a calculator and avoid all mental skills. LOL

square root of 645 hmmmm 20

645/20 = 32.25 average of 52.25 = 26.25

645/26.25 = 24.57 average of 50.82 = 25.41

Averaging method seems to work, but it isn't teaching much division...sorta like the higher/lower on The Price is Right.

My guess of the square of 645 is 25.41....wow it works the first time, what did I learn, nothing.

Using the Averaging method, what is the square root of 9331671....my first guesstimate is 10, have fun!

9331671/10 = 933167.1 + 10 = 9331681.1/2 = 466588.55

9331671/466588.55 = 19.999785 + 466588.55 = 466607.57/2 = 233303.285

9331671/233303.285 = 39.99802 + 233303.285 = 233343.27/2 = 116671.235

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Oh yeah, these are kids in grade 3 or 4 doing long math with 8 digit numbers...so much for averaging. And what is the degree of significance since we are working with one decimal place or 3....don't want to 'average' too soon or we could lose significant digits. If we are working with billions dropping digits too soon can make a HUGE difference.

adrian

I'm a layman who came to the site via a Google search on "how to calculate a square root." I read the presentation, then looked at the responses. I must say that I was dismayed at the comment offered up by Andrea S. Levy, Ed.D., where she suggested that memorizing an algorithm is less desirable than understanding a number.

I presently work as a technical writer for a firm that writes credit union banking software. Understanding all the algorithms used in the financial world is utterly essential for us to do what we do. In fact, one of the calculations we use to determine the amortization of a consumer loan with fees in a given time period is strikingly similar to your square root presentation. The calculation must be written by the software engineer for the machine, so it does ultimately reside in the mind of a human being. If the engineer doesn't know the algorithm, thousands of consumers will bear the consequences. I suggest that memorization is simply another tool in the box. Use it when its appropriate.

Best regards,

Michael Kelly

Newbury Park, Ca.

The last commenter on the page (Adrian) said that she never learned the squares from 1 to 30. This brings to mind a trick I recently learned for finding squares close to 50. Start with the square of 50, 2500, add 100 times the distance between 50 and the number, and then add the square of the distance of 50 and the number. For instance, 432 = 2500 - 700 + 49 = 1849. This comes from the simple FOIL identity (50 + x)2 = 2500 - 100x + x2. In this identity, x is the distance between 50 and the number. If the number is 43 (as in my example), x is -7. If the number is 54, x is 4. So if you memorize your squares from 1 to 25, you get the squares of 26 through 75 "for free".

If the idea of memorizing the squares of 1 to 25 seems daunting, it's not. A few weeks ago, before knowing this trick, I knew just up to 13 offhand, with a few others scattered here and there. I drew up a table in Excel listing numbers 1 to 25 side by side with their squares, printed it out and put it on the wall of my cubicle. The squares I don't have memorized in those first 25 I can now get in a few seconds (for instance, for the square of 23 I am still counting up from 20 squared: 400, 441, 484, *529*). Even with not quite knowing them all I can find squares from 1 to 75 in under 10 seconds (thought process for finding 73 squared offhand: "73 is 23 greater than 50. What's 23 squared again? 400, 441, 484, 529! 2500 + 2300 + 529 = 5329. Done!")

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