How to calculate a square root without a calculator
and should your child learn how to do it
Most people in today's world feel that since calculators can find square roots, that children don't need to learn how to find square roots using any pencil-and-paper method. However, learning at least the "guess and check" method for finding the square root will actually help the students 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, consider letting students learn and practice at least the "guess and check" method. Since it actually deals with the CONCEPT of square root, I would consider it as essential for students to learn.
Depending on the situation and the students, the "guess and check" method can either be performed with a simple calculator that doesn't have a square root button or with paper & pencil calculations.
Finding square roots by guess & check method
To find a decimal approximation to, say √2, first make an initial guess, then 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 uses the actual definition of square root, and so can be very helpful in teaching the concept of square root.
Example: what is square root of 20?
You can start out by noting that since √16 = 4 and √25 = 5, then √20 must be between 4 and 5.
Then make a guess for √20; let's say for example that it is 4.5. Square that, see if the result is over or under 20, and improve your guess based on that. Repeat this process until you have the desired accuracy (amount of decimals). It's that simple and can be a nice experiment for students!
Example: Find √6 to 4 decimal places
Since 22 = 4 and 32 = 9, we know that √6 is between 2 and 3. Let's guess (or estimate) that it is 2.5. Squaring that we get 2.52 = 6.25. That's too high, so we reduce our estimate a little. Let's try 2.4 next. To find the square root of 6 to four decimal places we need to repeat this process until we have five decimals, and then we will round the result.
|Estimate||Square of estimate||High/low|
|2.45||6.0025||Too high but real close|
|2.4495||6.00005025||Too high so the square root of 6 must be between 2.449 and 2.4495.|
|2.4494||5.99956036||Too low, so the square root of 6 must be between 2.4494 and 2.4495|
|2.44945||5.9998053025||Too low, so the square root of 6 must be between 2.44945 and 2.4495.|
This is enough iterations since we know now that √6 would be rounded to 2.4495 (and not to 2.4494).
Finding square roots using an algorithm
There is also an algorithm for square roots that resembles the long division algorithm, and it 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):
Then continue this way:
|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.
|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.
|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
I was happy to see that you recommended the "estimate and check" method. This is what I also recommended to my daughter, who is now studying square roots in her home school curriculum. The "estimate and check" method is a good exercise in estimating, multiplying, and also memorizing perfect squares.
Another method, more suitable for students in an algebra class, would be to simplify the radical using the accepted method. Then find the remaining square root with an estimation method. For example, To find SQRT(1400), simplify to SQRT(100)*SQRT(14), which is equal to 10*SQRT(14). Then find SQRT(14) by an estimation method. For square roots of perfect squares, no estimation would even be needed.
Instructor of Mathematics
Noticed several of the comments related to using an algorithm to find the square root of a number. Some comments appeared to say that finding the result with a paper and pen vs calculator is archaic. That may be so. However, when I was in my freshman year at high school (early 70's) Herr Quinnell mentioned - as class was getting out - some of the things one can do with math - including finding square roots. So, I asked him how this was done. He showed me the algorithm method on the board.
I cannot speak to the value of generally knowing how this is used in other professions. In electronics engineering, finding square root is an integral part of design. We have parts called resistors. They aid in limiting current in circuits. These parts have wattage ratings. The value of a resistor is measured in "ohms". In a math sense this can be found by dividing volts by amperes. 10 volts divided by 0.001 ampere is a resistance of 10,000 ohms. As a square root example if I know the 10,000 ohm resistor has a rating of 0.25 watts I can calculate the maximum worst case voltage that could appear across it, before damage could occur. This is found by taking the resistance value - multiplying the wattage rating - and finding the square root. Square root of 2500 is 50. This part could withstand 50 volts.
My point - I could have calculated the result using 'artificial means'. Because somebody took the time to show me how to do square root on a chalkboard, I did not need to hunt down a calculator. By the time I would have found the calculator I've already figured out an answer. Taking the time to show students how things like square root are done has value. They may not actually put this to use later in life - but some just might.
Garth Price, CET
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.
Karl I. Jacob
Professor, School of Polymer, Textile and Fiber Engineering
Professor, G. W. Woodruff School of Mechanical Engineering
Georgia Institute of Technology
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.
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.
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:
|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|
|4.472135||20/4.472135=4.472137||average = 4.472136|
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.
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.
or tell how we can 3rd, 4th, root by division method.
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 information about the nth root algorithm (or paper-pencil method).
-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.
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!
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
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.
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.
Newbury Park, Ca.
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!")
Another example of using the square root algorithm
An explanation of why this square root algorithm works.
Free worksheets for square roots, including a worksheet generator
A geometric view of the square root algorithm
Square roots by Divide-and-Average
Explanation and example of the ancient algorithm for approximating square roots.
Square Root Algorithms
Formulas for a recurrence relation and Newton's iteration that can be used to approximate square roots. For the mathematically minded.
A new method of getting the square root of a special group of numbers in an easier way.