# How to Find Square Root

If N is an approximation to  , a better approximation can be found by using the Taylor series of the square root function:

1. If AC = a and BC = b. OC = AM of a and b, and radius r = QO = OG.
Using Pythagoras’ theorem, QC² = QO² + OC² ∴ QC = √QO² + OC² = QM.
Using Pythagoras’ theorem, OC² = OG² + GC² ∴ GC = √OC² − OG² = GM.
Using similar triangles, = ∴ HC = = HM.

According to historian of mathematics D.E. Smith, Aryabhata’s method for finding the square root was first introduced in Europe by Cataneo—in 1546.

An unknown Babylonian mathematician somehow correctly calculated the square root of 2 to three sexagesimal «digits» after the 1, but it is not known exactly how. The Babylonians knew how to approximate a hypotenuse using

The denominator in the fraction corresponds to the nth root. In the case above the denominator is 2, hence the equation specifies that the square root is to be found. The same identity is used when computing square roots with logarithm tables or slide rules.

## Lucas sequence method

the Lucas sequence of the first kind Un(P,Q) is defined by the recurrence relations:

and the characteristic equation of it is:

it has the discriminant   and the roots:

so when we want  , we can choose   and  , and then calculate   using   and  for large value of  .
The most effective way to calculate   and  is:

then when  :

## A two-variable iterative method

The initialization step of this method is

Then,   (while  ).

The convergence of  , and therefore also of  , is quadratic.

The proof of the method is rather easy. First, rewrite the iterative definition of   as

.

Then it is straightforward to prove by induction that

This can be used to construct a rational approximation to the square root by beginning with an integer. If   is an integer chosen so   is close to  , and   is the difference whose absolute value is minimized, then the first iteration can be written as:

Using the same example as given with the Babylonian method, let   Then, the first iteration gives

Likewise the second iteration gives

Notation for the (principal) square root of .

For example, = 5, since 25 = 5 ⋅ 5, or (5 squared).

Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of the «square» of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.

Hey, students are you looking for Squares and Square Roots Class 8 Maths Formulas? If yes. Then you are at the right place.
In this post, I have listed all the formulas of Squares and Square Roots class 8 that you can use to learn and understand the concepts easily.

If you want to improve your class 8 Math, Squares and Square Roots concepts, then it is super important for you to learn and understand all the formulas.

By using these formulas you will learn about Squares and Square Roots.

With the help of these formulas, you can revise the entire chapter easily.

Suppose you are asked to find the square root of a number. But for that, you need to know what are squares and square roots. The value of the number being multiplied by itself gives the original given number back, known as its square root. For example, let’s assume y is the square root of x, then y = x. This equation can also be written as y = √x. The square root of a number is represented by the symbol ‘√ ‘ Whenever we multiply a number by itself, the positive value we get is a square of the number. We get the original number back by finding out the square root of this positive value.

The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In physics, the RMS current value can also be defined as the «value of the direct current that dissipates the same power in a resistor.»

In the case of a set of n values  , the RMS is

The corresponding formula for a continuous function (or waveform) f(t) defined over the interval   is

and the RMS for a function over all time is

In the case of the RMS statistic of a random process, the expected value is used instead of the mean.

1. In addition to the principal square root, there is a negative square root equal in magnitude but opposite in sign to the principal square root, except for zero, which has double square roots of zero.
2. The factors two and six are used because they approximate the geometric means of the lowest and highest possible values with the given number of digits:   and  .
3. The unrounded estimate has maximum absolute error of 2.65 at 100 and maximum relative error of 26.5% at y=1, 10 and 100
4. If the number is exactly half way between two squares, like 30.5, guess the higher number which is 6 in this case
5. This is incidentally the equation of the tangent line to y=x2 at y=1.
• Dauben, Joseph W. (2007). «Chinese Mathematics I». In Katz, Victor J. (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: Princeton University Press. ISBN 978-0-691-11485-9.
• Gel’fand, Izrael M.; Shen, Alexander (1993). Algebra (3rd ed.). Birkhäuser. p. 120. ISBN 0-8176-3677-3.
• Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0-691-00659-8.
• Smith, David (1958). History of Mathematics. Vol. 2. New York: Dover Publications. ISBN 978-0-486-20430-7.
• Selin, Helaine (2008), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Bibcode:2008ehst.book…..S, ISBN 978-1-4020-4559-2.
1. «Root-mean-square value». A Dictionary of Physics (6 ed.). Oxford University Press. 2009. ISBN 9780199233991.
2. Thompson, Sylvanus P. (1965). Calculus Made Easy. Macmillan International Higher Education. p. 185. ISBN 9781349004874. Retrieved 2020.
3. Jones, Alan R. (2018). Probability, Statistics and Other Frightening Stuff. Routledge. p. 48. ISBN 9781351661386. Retrieved 2020.
4. Cartwright, Kenneth V (Fall 2007). «Determining the Effective or RMS Voltage of Various Waveforms without Calculus» . Technology Interface. 8 (1): 20 pages.

5. Nastase, Adrian S. «How to Derive the RMS Value of Pulse and Square Waveforms». MasteringElectronicsDesign.com. Retrieved 2015.
6. «Make Better AC RMS Measurements with your Digital Multimeter» . Keysight. Keysight. Retrieved 2019.

7. Chris C. Bissell; David A. Chapman (1992). Digital signal transmission (2nd ed.). Cambridge University Press. p. 64. ISBN 978-0-521-42557-5.

8. Weisstein, Eric W. «Root-Mean-Square». MathWorld.
9. «ROOT, TH1:GetRMS». Archived from the original on 2017-06-30. Retrieved .

### Need of Square and Square Roots

Not only in everyday math but also in advanced stages of life square and square roots are applicable. square roots are not only important topics for students but this is applied by engineers at their work. square and square roots are applicable for getting quadratic equations and solving Pythagoras theorem. is based on square roots of a number. There is a need for square and square roots because it is not only applicable to math but it is very useful for deriving standard deviation in statistics; it is useful in Algebra as well as in geometry.

The contents of the course book are:

### The Chapter 6 Explains Regarding:

squares of a number, square of a rational number, square of a large number, finding the product of two consecutive odd-even integers, finding a Pythagorean triplet, it also explains regarding finding square roots by various methods, finding square root through repeated subtraction, finding square root through prime factorization, finding square root by division method, properties of square root and approximate square root by division method.

## Iterative methods for reciprocal square roots

• Applying Newton’s method to the equation   produces a method that converges quadratically using three multiplications per step:

• Another iteration is obtained by Halley’s method, which is the Householder’s method of order two. This converges cubically, but involves five multiplications per iteration:[]
, and
.
• If doing fixed-point arithmetic, the multiplication by 3 and division by 8 can implemented using shifts and adds. If using floating-point, Halley’s method can be reduced to four multiplications per iteration by precomputing   and adjusting all the other constants to compensate:
, and
.

The first way of writing Goldschmidt’s algorithm begins

(typically using a table lookup)

until   is sufficiently close to 1, or a fixed number of iterations. The iterations converge to

, and
.

Note that it is possible to omit either   and   from the computation, and if both are desired then   may be used at the end rather than computing it through in each iteration.

A second form, using fused multiply-add operations, begins

(typically using a table lookup)

until   is sufficiently close to 0, or a fixed number of iterations. This converges to

, and
.

Square roots of positive numbers are not in general rational numbers, and so cannot be written as a terminating or recurring decimal expression. Therefore in general any attempt to compute a square root expressed in decimal form can only yield an approximation, though a sequence of increasingly accurate approximations can be obtained.

where and 10 are the natural and base-10 logarithms.

as it allows one to adjust the estimate by some amount and measure the square of the adjustment in terms of the original estimate and its square. Furthermore,   when is close to 0, because the tangent line to the graph of   at  , as a function of alone, is  . Thus, small adjustments to can be planned out by setting   to , or  .

1. Start with an arbitrary positive start value x. The closer to the square root of a, the fewer the iterations that will be needed to achieve the desired precision.
2. Replace x by the average (x + a/x) / 2 between x and a/x.
3. Repeat from step 2, using this average as the new value of x.

That is, if an arbitrary guess for   is x0, and xn + 1 = (xn + a/xn) / 2, then each xn is an approximation of   which is better for large n than for small n. If a is positive, the convergence is quadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If , the convergence is only linear.

Using the identity

The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.

Another useful method for calculating the square root is the shifting nth root algorithm, applied for .

«Heron’s method» redirects here. For the formula used to find the area of a triangle, see Heron’s formula.

More precisely, if is our initial guess of   and is the error in our estimate such that S = (x+ ε)2, then we can expand the binomial

and solve for the error term

since  .

Therefore, we can compensate for the error and update our old estimate as

1. Begin with an arbitrary positive starting value (the closer to the actual square root of , the better).
2. Let xn + 1 be the average of and (using the arithmetic mean to approximate the geometric mean).
3. Repeat step 2 until the desired accuracy is achieved.

It can also be represented as:

This algorithm works equally well in the -adic numbers, but cannot be used to identify real square roots with -adic square roots; one can, for example, construct a sequence of rational numbers by this method that converges to +3 in the reals, but to −3 in the 2-adics.

To calculate , where = 125348, to six significant figures, use the rough estimation method above to get

Therefore, ≈ 354.045.

Semilog graphs comparing the speed of convergence of Heron’s method to find the square root of 100 for different initial guesses. Negative guesses converge to the negative root, positive guesses to the positive root. Note that values closer to the root converge faster, and all approximations are overestimates. In the SVG file, hover over a graph to display its points.

Suppose that x0 > 0 and S > 0. Then for any natural number n, xn > 0. Let the relative error in xn be defined by

-1}» data-class=»mwe-math-fallback-image-inline»>

Then it can be shown that

And thus that

and consequently that convergence is assured, and quadratic.

#### Worst case for convergence

<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b965d9c243f4fa62fe825c21a5c171a8e5d8c2" data-alt="{\begin{aligned}S&=1;&x_{0}&=2;&x_{1}&=1.250;&\varepsilon _{1}&=0.250.\\S&=10;&x_{0}&=2;&x_{1}&=3.500;&\varepsilon _{1}&<0.107.\\S&=10;&x_{0}&=6;&x_{1}&=3.833;&\varepsilon _{1}&<0.213.\\S&=100;&x_{0}&=6;&x_{1}&=11.333;&\varepsilon _{1}&

Thus in any case,

<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5421d029f479921e0c154a095df9a6314eab837a" data-alt="\varepsilon _{2}<2^{-5}
<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edfb1c400b6a50ee05f9788500fdb114d6201ec8" data-alt="\varepsilon _{3}<2^{-11}
<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc199c55d6ae1f04351b7b7aae217d0fb9a5b21" data-alt="\varepsilon _{4}<2^{-23}
<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3425039f9411fb9109b4174cb04d837dc38fe81" data-alt="\varepsilon _{5}<2^{-47}
<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73f882414b129be7dc41538e4ce0d60c51891d7" data-alt="\varepsilon _{6}<2^{-95}
<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb09dcdf99c0c344075f8f6097c36fa976be0469" data-alt="\varepsilon _{7}<2^{-191}
<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca951a28fe668eb7abc2f3e5af2b968c1689f5c5" data-alt="\varepsilon _{8}<2^{-383}

Rounding errors will slow the convergence. It is recommended to keep at least one extra digit beyond the desired accuracy of the being calculated to minimize round off error.

Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted
,
, or
) of a real number. Arithmetically, it means given
, a procedure for finding a number which when multiplied by itself, yields
; algebraically, it means a procedure for finding the non-negative root of the equation
; geometrically, it means given two line segments, a procedure for constructing their geometric mean.

The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number (which is not already a perfect square) has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal notation system.

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The method employed depends on what the result is to be used for (i.e. how accurate it has to be), how much effort one is willing to put into the procedure, and what tools are at hand. The methods may be roughly classified as those suitable for mental calculation, those usually requiring at least paper and pencil, and those which are implemented as programs to be executed on a digital electronic computer or other computing device. Algorithms may take into account convergence (how many iterations are required to achieve a specified precision), computational complexity of individual operations (i.e. division) or iterations, and error propagation (the accuracy of the final result).

Procedures for finding square roots (particularly the square root of 2) have been known since at least the period of ancient Babylon in the 17th century BCE. Heron’s method from first century Egypt was the first ascertainable algorithm for computing square root. Modern analytic methods began to be developed after introduction of the Arabic numeral system to western Europe in the early Renaissance. Today, nearly all computing devices have a fast and accurate square root function, either as a programming language construct, a compiler intrinsic or library function, or as a hardware operator, based on one of the described procedures.

Abramowitz, Miltonn; Stegun, Irene A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Courier Dover Publications. p. 17. ISBN 978-0-486-61272-0.

Bailey, David; Borwein, Jonathan (2012). «Ancient Indian Square Roots: An Exercise in Forensic Paleo-Mathematics» . American Mathematical Monthly. Vol. 119, no. 8. pp. 646–657. Retrieved .

Campbell-Kelly, Martin (September 2009). «Origin of Computing». Scientific American. 301 (3): 62–69. Bibcode:2009SciAm.301c..62C. doi:10.1038/scientificamerican0909-62. JSTOR 26001527. PMID 19708529.

Cooke, Roger (2008). Classical algebra: its nature, origins, and uses. John Wiley and Sons. p. 59. ISBN 978-0-470-25952-8.

Fowler, David; Robson, Eleanor (1998). «Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context» . Historia Mathematica. 25 (4): 376. doi:.

Gower, John C. (1958). «A Note on an Iterative Method for Root Extraction». The Computer Journal. 1 (3): 142–143. doi:.

Guy, Martin; UKC (1985). «Fast integer square root by Mr. Woo’s abacus algorithm (archived)». Archived from the original on 2012-03-06.

Heath, Thomas (1921). A History of Greek Mathematics, Vol. 2. Oxford: Clarendon Press. pp. 323–324.

Lomont, Chris (2003). «Fast Inverse Square Root» .

Markstein, Peter (November 2004). Software Division and Square Root Using Goldschmidt’s Algorithms . 6th Conference on Real Numbers and Computers. Dagstuhl, Germany. CiteSeerX .

Piñeiro, José-Alejandro; Díaz Bruguera, Javier (December 2002). «High-Speed Double-Precision Computationof Reciprocal, Division, Square Root, and Inverse Square Root». IEEE Transactions on Computers. 51 (12): 1377–1388. doi:10.1109/TC.2002.1146704.

Sardina, Manny (2007). «General Method for Extracting Roots using (Folded) Continued Fractions». Surrey (UK).

Simply Curious (5 June 2018). «Bucking down to the Bakhshali manuscript». Simply Curious blog. Retrieved .

Steinarson, Arne; Corbit, Dann; Hendry, Mathew (2003). «Integer Square Root function».

Wilkes, M.V.; Wheeler, D.J.; Gill, S. (1951). The Preparation of Programs for an Electronic Digital Computer. Oxford: Addison-Wesley. pp. 323–324. OCLC 475783493.

Square root is one of the most important functions in Mathematics which has a wide range of applications in day to day life and also scientific calculations. Square root of any number in Mathematics is that number which when multiplied by itself gives the product equal to the number whose square root is to be determined. Square root of a number is represented as the number written within the symbol ‘√’. Square root of a number ‘x’ is written as √x. Square root of a number can be represented in exponential form as the number to the power ½. Square root of a number ‘x’ can be written in exponential form as (x)

### What is a Perfect Square Number and Square Root Formula?

It is very important to understand what is a square root of a perfect square number before making yourself clear about what is a root in math. What is a perfect square number in Mathematics can be that number which is obtained as a product by multiplying any integer by itself. The square root formula when used for perfect square numbers will yield a number which is an integer as the answer. i.e. Square root of a perfect square number is always an integer.

### What is a Root in Math?

There are several methods to find the square root of a number among which a few familiar ones are:

1. Prime factorization method

2. Repeated Subtraction Method

3. Guess and check method

4. Number line method

5. Long division method

### Finding Square Root Formula by Prime Factorization Method

Prime factorization method is a method in which the numbers are expressed as a product of their prime factors. The identical prime factors are paired and the product of one element from each pair gives the square root of the number. This method can also be used to find whether a number is a perfect square or not. However, this method cannot be used to find the square root of decimal numbers which are not perfect squares.

Evaluate the root of 576.

So, 576 can be written as a product of prime numbers as:

576 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3

Square root of 576 = 2 x 2 x 2 x 3 = 24

### Square Root Formula Using Repeated Subtraction Method

This is a method in which the number whose square root is to be determined is repeatedly subtracted by the consecutive odd number till the difference becomes zero. The number of subtractions gives the root of the number.This method can only be used to find the square root of perfect square numbers.

Estimate the Square root of 16

The number is subtracted from odd numbers starting from 1.

16 — 1 = 15

15 — 3 = 12

12 — 5 = 7

7 — 7 = 0

Number of subtractions here is 4. So, the square root of 16 is 4.

### Average Method of Square Root Formula:

In this method, the concept of average is used to find the square root of a given decimal number. This method can be conveniently used to find the square root of whole numbers upto a few decimal places.

Evaluate the square root of 3 using the average method.

The two square numbers in between which ;3’ lies are 1 and 4. So, the square root of 3 lies between 1 and 2. Find the average of these two numbers to get the square root of 3.

Square root of 3 = (1 + 2)/ 2 = 3/ 2 = 1.5 which is not accurate. So, finding the average is further continued as

Square root of 3 = (1.5 + 2)/2 = 1.75 which is approximately equal to square root of 3.

### Fun Facts About Square Root Formula:

• Square and square root operations are inverse mathematical operations with respect to each other.

• Square root of a square of a number is the number itself.

• The square of square root of a number is the number itself.

The article has presented complete insight about the Square Root formula that will help students to practice and learn.

### In electrical engineering

where   refers to the direct current (or average) component of the signal, and   is the alternating current component of the signal.

#### Average electrical power

Electrical engineers often need to know the power, P, dissipated by an electrical resistance, R. It is easy to do the calculation when there is a constant current, I, through the resistance. For a load of R ohms, power is defined simply as:

However, if the current is a time-varying function, I(t), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function is periodic (such as household AC power), it is still meaningful to discuss the average power dissipated over time, which is calculated by taking the average power dissipation:

So, the RMS value, IRMS, of the function I(t) is the constant current that yields the same power dissipation as the time-averaged power dissipation of the current I(t).

Average power can also be found using the same method that in the case of a time-varying voltage, V(t), with RMS value VRMS,

This equation can be used for any periodic waveform, such as a sinusoidal or sawtooth waveform, allowing us to calculate the mean power delivered into a specified load.

By taking the square root of both these equations and multiplying them together, the power is found to be:

Both derivations depend on voltage and current being proportional (that is, the load, R, is purely resistive). Reactive loads (that is, loads capable of not just dissipating energy but also storing it) are discussed under the topic of AC power.

In the common case of alternating current when I(t) is a sinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If Ip is defined to be the peak current, then:

where t is time and ω is the angular frequency (ω = 2/T, where T is the period of the wave).

Since Ip is a positive constant:

Using a trigonometric identity to eliminate squaring of trig function:

but since the interval is a whole number of complete cycles (per definition of RMS), the sine terms will cancel out, leaving:

A similar analysis leads to the analogous equation for sinusoidal voltage:

where IP represents the peak current and VP represents the peak voltage.

Because of their usefulness in carrying out power calculations, listed voltages for power outlets (for example, 120 V in the US, or 230 V in Europe) are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from the above formula, which implies VP = VRMS × , assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 × , or about 170 volts. The peak-to-peak voltage, being double this, is about 340 volts. A similar calculation indicates that the peak mains voltage in Europe is about 325 volts, and the peak-to-peak mains voltage, about 650 volts.

RMS quantities such as electric current are usually calculated over one cycle. However, for some purposes the RMS current over a longer period is required when calculating transmission power losses. The same principle applies, and (for example) a current of 10 amps used for 12 hours each 24-hour day represents an average current of 5 amps, but an RMS current of 7.07 amps, in the long term.

The term RMS power is sometimes erroneously used in the audio industry as a synonym for mean power or average power (it is proportional to the square of the RMS voltage or RMS current in a resistive load). For a discussion of audio power measurements and their shortcomings, see Audio power.

where R represents the gas constant, 8.314 J/(mol·K), T is the temperature of the gas in kelvins, and M is the molar mass of the gas in kilograms per mole. In physics, speed is defined as the scalar magnitude of velocity. For a stationary gas, the average speed of its molecules can be in the order of thousands of km/hr, even though the average velocity of its molecules is zero.

When two data sets — one set from theoretical prediction and the other from actual measurement of some physical variable, for instance — are compared, the RMS of the pairwise differences of the two data sets can serve as a measure how far on average the error is from 0. The mean of the absolute values of the pairwise differences could be a useful measure of the variability of the differences. However, the RMS of the differences is usually the preferred measure, probably due to mathematical convention and compatibility with other formulae.

In the days before calculators, students and professors alike had to calculate square roots by hand. Several different methods have evolved for tackling this daunting process, some giving a rough approximation, others giving an exact value. To learn how to find a number’s square root using only simple operations, please see Step 1 below to get started.

#### Understanding the Process

1. Consider the number you are calculating the square root of as the area S of a square. Because a square’s area is L2 where L is the length of one of its sides, therefore, by trying to find the square root of your number, you are trying to calculate the length L of the side of that square.

2. Specify letter variables for each digit of your answer. Assign the variable A as the first digit of L (the square root we are trying to calculate). B will be its second digit, C its third, and so on.

3. Specify letter variables for each «chunk» of your starting number. Assign the variable Sato the first pair of digits in S (your starting value), Sb the second pair of digits, etc.

4. Understand this method’s connection to long division. This method of finding a square root is essentially a long division problem that divides your starting number by its square root, thus giving its square root as an answer. Just like in a long division problem, in which you are only interested by the next one digit at a time, here, you are interested by the next two digits at a time (which correspond to the next digit at a time for the square root).

5. Find the biggest number whose square is less than or equal to Sa. The first digit A in our answer is then the biggest integer where the square does not exceed Sa (meaning A so that A² ≤ Sa < (A+1)²). In our example, Sa = 7, and 2² ≤ 7 < 3², so A = 2.

• Note that, for instance, if you wanted to divide 88962 by 7 via long division, the first step would be similar: you would be looking at the first digit of 88962 (8) and you would want the biggest digit that, when multiplied by 7, is lower than or equal to 8. Essentially, you’re finding d so that 7×d ≤ 8 < 7×(d+1). In this case, d would be equal to 1.
6. Visualize the square whose area you are beginning to solve. Your answer, the square root of your starting number, is L, which describes the length of a square with area S (your starting number). Your values for A,B,C, represent the digits in the value L. Another way of saying this is that, for a two-digit answer, 10A + B = L, while for a three-digit answer, 100A +10B + C = L, and so on.

• In our example, (10A+B)² = L2 = S = 100A² + 2×10A×B + B². Remember that 10A+B represents our answer L with B in the units position and A in the tens position. For instance, 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 the area of the biggest square inside, is the area of the smallest square, and 10A×B is the area of each of the two remaining rectangles. By performing this long, convoluted process, we find the area of the entire square by adding up the areas of the squares and rectangles inside it.
7. Subtract A² from Sa. Drop one pair (Sb) of digits from S. Sa Sb is nearly the total area of the square, which you just subtracted the area of the bigger internal square from. The remainder is can be though of as the number N1, which we obtained in step 4 (N1 =380 in our example). N1 is equal to 2×10A×B + B² (area of the two rectangles plus area of the small square).

8. Look for N1 = 2×10A×B + B², also written as N1 = (2×10A + B) × B. In our example, you already know N1 (380) and A (2), so you need to find B. B is most likely not going to be an integer, so you must actually find the biggest integer B so that (2×10A + B) × B ≤ N1. So, you have: N1 < (2×10A + (B+1)) × (B+1).)

9. Solve. To solve this equation, multiply A by 2, shift it in the position of the tens (which is equivalent to multiplying by 10), place B in the position of the units, and multiply the resulting 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 the bottom right quadrant in step 4. In step 5, you find the biggest integer B that fits on the underscore so that (2×10A + B) × B ≤ N1.

10. Subtract the area (2×10A + B) × B from the total area. This gives you the area S-(10A+B)² not yet accounted for (and which will be used to calculate the next digits in a similar fashion).

• Is 28 a perfect number?

Yes. A «perfect» number is a positive integer which is the sum of all of its positive divisors (except itself). Thus, 28 = 1 + 2 + 4 + 7 + 14.

• What is the square root of 0.000121?

.011. If you are calculating it by hand, here’s what to do: 121’s sqrt is 11. To get .000121, you simply find an amount of 0’s after the decimal point and the number 11 afterwards. Then multiply by itself. .011*.011= .000121.

• Can I write the cube root of x as (1/x)^1/3?

No, it’s (x)^1/3.

200 characters left

• This method works for any base, not just in base 10 (decimal).

• In the example, 1.73 can be considered to be a «remainder» : 780.14 = 27.9² + 1.73.

• Moving the decimal point by an increment of two digits in a number (factor of 100), moves the decimal point by increments of one digit in its square root (factor of 10).

• Be sure to separate the digits into pairs from the decimal point. Separating 79,520,789,182.47897 as «79 52 07 89 18 2.4 78 97″ will yield a useless number.

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Let’s understand what is a square root of a number, how to find the square root of a number, and what are the properties of the square root of a number.

## What is Square Root?

The square root of a number is that factor of a number which when multiplied by itself gives the original number. Squares and square roots are special exponents.

The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.

If square of a number $a$ is $b$, then square root of the number $b$ is $a$.

It is very easy to find the square root of a number that is a perfect square. Perfect squares are those positive numbers that

• Repeated Subtraction Method of Square Root
• Square Root by Prime Factorization Method
• Square Root by Estimation Method

### Repeated Subtraction Method of Square Root

We subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach 0. The number of times we subtract is the square root of the given number. This method works only for perfect square numbers. Let us find the square root of $36$ using this method.

$36 – 1 = 35$

$35 – 3 = 32$

$32 – 5 = 27$

$27 – 7 = 20$

$20 – 9 = 11$

$11 – 11 = 0$

### Square Root by Prime Factorization Method

Step 1: Divide the given number into its prime factors.

Step 2: Form pairs of similar factors such that both factors in each pair are equal.

Step 3: Take one factor from the pair.

Step 4: Find the product of the factors obtained by taking one factor from each pair.

Step 5: That product is the square root of the given number.

Let’s consider some examples to understand the process.

### Examples

Let’s consider some examples to understand the process.

Ex 1: Find the square root of $144$.

Ex 2: Find the square root of $324$.

### Square Root by Estimation Method

Estimation and approximation refer to a reasonable guess of the actual value to make calculations easier and more realistic. This method helps in estimating and approximating the square root of a given number. This method is used for numbers that are not perfect squares.

Let’s consider some examples to understand the process.

### Examples

First, find the nearest perfect square numbers to which are less than and greater than $15$.

These numbers are

• $9$ on the lower side since, $3 \times 3 = 9$
• $16$ on the upper side since, $4 \times 4 = 16$

Let us find the squares of numbers $3.8$ and $3.9$ (numbers closer to $4$).

Now, repeat the process and check between $3.85$ and $3.9$.

• The number of times the process is repeated depends on the number of decimal places required.
• This is a very long process and time-consuming.

## Square Root Formula

When $n= 2$, we call it square root. We can use any of the above methods for finding the square root, such as prime factorization, long division, and so on.

## Square Root of a Negative Number

The square root of a negative number cannot be a real number, since a square is either a positive number or zero. But complex numbers have the solutions to the square root of a negative number.

Let’s consider some examples to understand the process.

### Examples

Ex 1: Square root of $-16$.

$-16 = 16 \times \left(-1 \right)$

## Conclusion

The square root of a number is that factor of a number which when multiplied by itself gives the original number. There are $3$ methods of finding the square root of a number – Repeated Subtraction Method of Square Root, Square Root by Prime Factorization Method, and Square Root by Estimation Method.

## Practice Problems

1. Find the square root of the following numbers using the repeated subtraction method
• $25$
• $121$
• $196$
2. Find the square root of the following numbers using the prime factorization method
• $196$
• $1296$
• $784$
3. Find the square root of the following numbers using the approximation method
• $31$
• $53$
• $90$

## FAQs

### Can square root be negative?

Yes, the square root of a number can be negative. In fact, all the perfect squares like $4$, $9$, $25$, etc. have two square roots, one is a positive value and one is a negative value.

For example, the square roots of $4$ are $-2$ and $2$. Since, $\left(-2 \right) \times \left(-2 \right)$ is also equal to $4$. Similarly, the square roots of $9$ are $3$ and $-3$, and so on.

### What are the applications of the square root formula?

There are various applications of the square root formula:
a) The square root formula is mainly used in algebra and geometry.
b) It helps in finding the roots of a quadratic equation.
c) It is widely used by engineers.

## Nth roots and polynomial roots

A cube root of   is a number   such that  ; it is denoted

If is an integer greater than two, a th root of   is a number   such that  ; it is denoted

Given any polynomial , a root of is a number such that p(y) = 0. For example, the th roots of are the roots of the polynomial (in )

Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of th roots.

## Properties and uses

The graph of the function f(x) = √x, made up of half a parabola with a vertical directrix

The principal square root function   (usually just referred to as the «square root function») is a function that maps the set of nonnegative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.

The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers, the latter being a superset of the rational numbers).

For all real numbers x,

<span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0640b9c1359d9e56c1047817a6abaeb9d12f0eb9" data-alt="
\sqrt{x^2} = \left|x\right| =
\begin{cases}
x, & \mbox{if }x \ge 0 \\
-x, & \mbox{if }x
(see absolute value)

For all nonnegative real numbers x and y,

The square root function is continuous for all nonnegative x, and differentiable for all positive x. If f denotes the square root function, whose derivative is given by:

The square root of a nonnegative number is used in the definition of Euclidean norm (and distance), as well as in generalizations such as Hilbert spaces. It defines an important concept of standard deviation used in probability theory and statistics. It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry. Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws.

## In rings in general

Unlike in an integral domain, a square root in an arbitrary (unital) ring need not be unique up to sign. For example, in the ring   of integers modulo 8 (which is commutative, but has zero divisors), the element 1 has four distinct square roots: ±1 and ±3.

Another example is provided by the ring of quaternions   which has no zero divisors, but is not commutative. Here, the element −1 has infinitely many square roots, including , , and . In fact, the set of square roots of −1 is exactly

A square root of 0 is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0. For example, in   any multiple of is a square root of 0.

## In frequency domain

The RMS can be computed in the frequency domain, using Parseval’s theorem. For a sampled signal  , where   is the sampling period,

where   and N is the sample size, that is, the number of observations in the sample and FFT coefficients.

In this case, the RMS computed in the time domain is the same as in the frequency domain:

## What is Square of a Number?

In mathematics, square of a number is the result of multiplying the number by itself. The word square is usually equivalent to raising a number to the power of 2 and denoted by the superscript 2.

, the square of 4 is written as 42 which gives 16 as the answer. In this case, 16 is the square of number 4.

Below is a list of squares of the first twelve numbers:

1 x 1 = 1                           7 x 7 = 49
2 x 2 = 4                           8 x 8 = 64
3 x 3 = 9                           9 x 9 = 81
4 x 4 = 16                        10 x 10 = 100
5 x 5 = 25                        11 x 11 = 121
6 x 6 = 36                        12 x 12 = 144

### Squaring Negative Numbers

The square of a negative number is a positive number. For instance, -3 x -3 would become 9, however – 3 x 3 = -9, this is because -3 is a different number to 3.

## Square roots of negative and complex numbers

First leaf of the complex square root

Second leaf of the complex square root

Using the Riemann surface of the square root, it is shown how the two leaves fit together

The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes by j, especially in the context of electricity where «i» traditionally represents electric current) and called the imaginary unit, which is defined such that i2 = −1. Using this notation, we can think of i as the square root of −1, but we also have (−i)2 = i2 = −1 and so −i is also a square root of −1. By convention, the principal square root of −1 is i, or more generally, if x is any nonnegative number, then the principal square root of −x is

The right side (as well as its negative) is indeed a square root of −x, since

For every non-zero complex number z there exist precisely two numbers w such that w2 = z: the principal square root of z (defined below), and its negative.

### Principal square root of a complex number

To find a definition for the square root that allows us to consistently choose a single value, called the principal value, we start by observing that any complex number   can be viewed as a point in the plane,   expressed using Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair   where   is the distance of the point from the origin, and   is the angle that the line from the origin to the point makes with the positive real ( ) axis. In complex analysis, the location of this point is conventionally written   If

then the principal square root of   is defined to be the following:

The principal square root function is thus defined using the nonpositive real axis as a branch cut.
If   is a non-negative real number (which happens if and only if  ) then the principal square root of   is   in other words, the principal square root of a non-negative real number is just the usual non-negative square root.
It is important that <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27eb4930922e649015e37b8d782b6b1ad7d55b9" data-alt="{\displaystyle -\pi   because if, for example,   (so  ) then the principal square root is but using   would instead produce the other square root

The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even continuous). The above Taylor series for   remains valid for complex numbers   with <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e657241d23e0514c31745c2d302fffa61a77ed" data-alt="{\displaystyle |x|

The above can also be expressed in terms of trigonometric functions:

where is the sign of (except that, here, sgn(0) = 1). In particular, the imaginary parts of the original number and the principal value of its square root have the same sign. The real part of the principal value of the square root is always nonnegative.

For example, the principal square roots of are given by:

where <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aeabfa69185eedc1363619bb534927d46eb8aff" data-alt="{\displaystyle -\pi   and <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679df4a1684cbd40b9ff7ac9bbed56bffe6a2903" data-alt="{\displaystyle -\pi  .

•
Counterexample for the principal square root: z = −1 and w = −1
This equality is valid only when <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616dd651f429490e09f3398a52a279be4c00a279" data-alt="{\displaystyle -\pi
•
Counterexample for the principal square root: and z = −1
This equality is valid only when <span data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd8739a5027cfcf839f6d2f8105a1e94a115622" data-alt="{\displaystyle -\pi
•
Counterexample for the principal square root: z = −1)
This equality is valid only when

A similar problem appears with other complex functions with branch cuts, e.g., the complex logarithm and the relations logz + logw = log(zw) or log(z*) = log(z)* which are not true in general.

if the branch includes +i or

if the branch includes −i, while the right-hand side becomes

where the last equality,   is a consequence of the choice of branch in the redefinition of √.

## Negative or complex square

If S < 0, then its principal square root is

If S = a+bi where a and b are real and b ≠ 0, then its principal square root is

is the modulus of S. The principal square root of a complex number is defined to be the root with the non-negative real part.

## What is Square Root of a Number?

The square root is an inverse operation of squaring a number. In other words, the square root is an operation that undoes an exponent of 2. A square root of a number x is such that a number y is the square of x, simplify written as y2 = x.

, 5 and – 5 are both square roots of 25 because:

5 x 5 = 25 and -5 x -5 =25.

The square root of a number x is denoted with a radical sign √x or x 1/2. For instance, the square root of 16 is represented as √16 = 4. A number whose square root is calculated is referred to as radicand. In this expression, √16 = 4, number 16 is the radicand.

• A perfect square number has a perfect square root.
• An even perfect number has the square root that is even.
• The odd perfect number has the square root that is odd.
• The square root of a negative number is undefined.
• Only numbers ending with an even number of zeros have square roots.

### Finding the Square Root of Numbers

• :
This method involves successful and repeated subtraction of odd numbers such as 1, 3, 5, and 7 from the number until zero is reached. The square of the number is equal to the number or frequency of subtraction performed on the number. Suppose we need to calculate the square of a perfect number like 16, the number of subtractions performed is 4, so the square root of 16 is 4.
• :
In this method, a perfect square number is factorized by successive divisions. The prime factors are grouped into pairs, and the product of each number is calculated. The product is therefore, the square root of the number. To find the square of a perfect number such as: 144 is performed as:
1. 144 = 2 × 2 × 2 × 2 × 3 × 3.
2. Pair the prime factors.
3. Selecting one number from each pair.
4. 2 × 2 × 3 = 12.
5. Thus, the √144 = 12.
• Division method is a suitable technique of calculating the square of a large number.

The following are the steps involved:
1. A bar is placed over every pair of digits starting from the right-hand side.
2. Divide left end number by a number whose square is less or equivalent to the numbers under the left end.
3. Take this number as the divisor and quotient. Similarly, take the leftmost number as the dividend.
4. Divide to get the result.
5. Pull down the next number with a bar to the right-hand side of the remainder.
6. Multiply the divisor by 2.
7. To the right of this new divisor, find a suitable dividend. This process is repeated until we get zero as the remainder. The square of the number therefore is equal to the quotient.

I am thinking of two numbers. Both numbers are square numbers greater than $1$. If the sum of these numbers is $100$, what are the values of the two numbers?

True or False: There are nine perfect squares between $0$ and $100$.

## In common waveforms

Sine, square, triangle, and sawtooth waveforms. In each, the centerline is at 0, the positive peak is at   and the negative peak is at

A rectangular pulse wave of duty cycle D, the ratio between the pulse duration ( ) and the period (T); illustrated here with a = 1.

Graph of a sine wave’s voltage vs. time (in degrees), showing RMS, peak (PK), and peak-to-peak (PP) voltages.

If the waveform is a pure sine wave, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform, which may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is:

Peak-to-peak

For other waveforms, the relationships are not the same as they are for sine waves. For example, for either a triangular or sawtooth wave

Peak-to-peak

### In waveform combinations

Alternatively, for waveforms that are perfectly positively correlated, or «in phase» with each other, their RMS values sum directly.

## Relationship to other statistics

From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the «error» / square deviation as well.

## Square roots of positive integers

A positive number has two square roots, one positive, and one negative, which are opposite to each other. When talking of the square root of a positive integer, it is usually the positive square root that is meant.

The square roots of an integer are algebraic integers—more specifically quadratic integers.

The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since   only roots of those primes having an odd power in the factorization are necessary. More precisely, the square root of a prime factorization is

### As expansions in other numeral systems

As with before, the square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore have non-repeating digits in any standard positional notation system.

The square roots of small integers are used in both the SHA-1 and SHA-2 hash function designs to provide nothing up my sleeve numbers.

### As periodic continued fractions

One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrange c.. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers.

## Formula to Find Square Root of A Number

The square of 4 is 16 or 4 = 16 and √16 = 4. It is easy to find the square root of 16 or such numbers because they are perfect squares. But for the numbers like 10, 7, 20, 24, and many more, it becomes a bit difficult and calculative. There is no such formula to find the square root of a number. But there are various methods and tricks by which calculation of square roots can be easy. In different sections, we will learn about them.

### Representation of Square Roots

The symbol used to represent the square root is known as the radical symbol. ‘√’ is the square root symbol. If x is the original number, then √x is the square root of the number. Here x is also known as the radicand. Radical and square root both represent the same thing. The square root formula can be given by:

X = √Y, where X is the square root of Y. On the other hand, X = Y. Therefore, where Y is the square of X.

### Properties of Squares And Square Roots

In Math, the square root function takes a positive value as input and gives the input value’s square root as the output. Hence, it is said to be a one to one function. For example,

F(a) = √a.

Let us consider if x = 25, then the value returned will be 5. Below we have some of the properties of squares and square roots.

• A number can have a square root if it ends with an even number of zeros.

• A number can be a perfect square. Then, there will be a perfect square root also.

• Multiplication of two square root values is possible. For example, √2 × √3 = √6.

• If you find square root of a negative number, you get a complex number as a result. Perfect squares are never negative.

• We get a radical number by multiplying two same square roots. If you multiply √7 × √7, you get 7 back, which is a nonsquare root number.

• Numbers ending with 1, 4, 5, and 6 will have a square root.

• Perfect square root does not exist for numbers ending with 2, 3, 7, and 8.

### Easy Way to Find Square Root of A Number

We need to find the square root of a number, whether it’s a perfect or imperfect square. You already know if it’s an ideal square, then we can use the prime factorization method to factorize and find the square root easily. For other numbers, we have the division method.

### Prime Factorization Method

It’s one of the easiest ways to find the square root of a number. For example, let’s take the number 256. Now break 256 into its prime factors. Therefore 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Now we need to pair in doubles of similar numbers and consider them once only, (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2). Hence, √256 = (2 × 2 × 2 × 2) = 16.

### Long Division Method

It is the square root formula used for imperfect squares or large numbers. Let’s consider number 3. It is not a perfect square number. Taking this example, we have illustrated the long division method below.

Writing 3 with a decimal point and groups of two zeros for the decimal. Putting all of this under the long square root symbol (which looks like division), and we start dividing.

1√3.00 00 00 00 00

Putting the largest number whose square is less than or equal to 3 and above the 3. Square it and subtract it from 3, which in this case is 1.

Bringing down a pair of zeros and multiplying the answer so far by two. The next digit in the answer will be put at the end of the doubled answer and multiply that by the new digit. This product is subtracted, and two more zeros are brought down. We have to continue this process for as many digits of accuracy as we want.

The square root of 3 results in 1.732.

Now let us consider the number 24.

1. Give Some Applications of the Square Root Formula.

The square root formula is used in geometry, calculus, algebra, quadratic equations, computing, and other mathematical applications.

1. Give the Square Roots of 5, 7, 9, and 11.

The square roots of 5, 7, 9, and 11 are 2.236, 2.646, 3.0, and 3.317, respectively.

## Squares and Square Roots Class 8 Maths Formulas

Square Number: The square of a number is the product of the number with the number itself The, square of x = (x × x), denoted by x 2.

For positive numbers $a$ and $b$, we have

### Properties of the Square Number:

• A number ending in 2, 3, 7 or 8 is never a perfect square.
• A number ending in an odd number of zeros is never a perfect square.
• The square of an even number is even.
• The square of an odd number is odd.
• The square of a proper fraction is smaller than the fraction.
• For every natural number $n$, we have $\left\{(n+1)^{2}-n^{2}\right\}=\{(n+1)+n\}$.
• Sum of first $n$ odd natural numbers $=n^{2}$.
• If $m, n, p$ are natural numbers such that $\left(m^{2}+n^{2}\right)=p^{2}$, then $(m, n, p)$ is called a Pythagorean triplet.
• For every natural number $m>1,\left(2 m, m^{2}-1, m^{2}+1\right)$ is a Pythagorean triplet.
• There are $2 n$ non-perfect square numbers between the squares of the number $n$ and $(n+1)$
• The numbers which can be expressed as the product of the number with itself are called square numbers or perfect squares.
For example, $1,4,9,16,25, \ldots .$
• If a natural number $m$ can be expressed as $n^{2}$, where $n$ is also a natural number, then, $m$ is called a square number.

If you have any Confusion related to Squares and Square Roots Class 8 Maths Formulas, then feel free to ask in the comments section down below.

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## In integral domains, including fields

Each element of an integral domain has no more than 2 square roots. The difference of two squares identity u2v2 = (uv)(u + v) is proved using the commutativity of multiplication. If and are square roots of the same element, then u2v2 = 0. Because there are no zero divisors this implies or u + v = 0, where the latter means that two roots are additive inverses of each other. In other words if an element a square root of an element exists, then the only square roots of are and . The only square root of 0 in an integral domain is 0 itself.

In a field of characteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that u = u. If the field is finite of characteristic 2 then every element has a unique square root. In a field of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any.

Given an odd prime number , let q = pe for some positive integer . A non-zero element of the field with elements is a quadratic residue if it has a square root in . Otherwise, it is a quadratic non-residue. There are (q − 1)/2 quadratic residues and (q − 1)/2 quadratic non-residues; zero is not counted in either class. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.

## Continued fraction expansion

Quadratic irrationals (numbers of the form  , where a, b and c are integers), and in particular, square roots of integers, have periodic continued fractions. Sometimes what is desired is finding not the numerical value of a square root, but rather its continued fraction expansion, and hence its rational approximation. Let S be the positive number for which we are required to find the square root. Then assuming a to be a number that serves as an initial guess and r to be the remainder term, we can write   Since we have  , we can express the square root of S as

By applying this expression for   to the denominator term of the fraction, we have

For , the value of   is 1, so its representation is:

Proceeding this way, we get a generalized continued fraction for the square root as

Step 2 is to reduce the continued fraction from the bottom up, one denominator at a time, to yield a rational fraction whose numerator and denominator are integers. The reduction proceeds thus (taking the first three denominators):

Finally (step 3), divide the numerator by the denominator of the rational fraction to obtain the approximate value of the root:

rounded to three digits of precision.

The actual value of is 1.41 to three significant digits. The relative error is 0.17%, so the rational fraction is good to almost three digits of precision. Taking more denominators gives successively better approximations: four denominators yields the fraction  , good to almost 4 digits of precision, etc.

In general, the larger the denominator of a rational fraction, the better the approximation. It can also be shown that truncating a continued fraction yields a rational fraction that is the best approximation to the root of any fraction with denominator less than or equal to the denominator of that fraction — e.g., no fraction with a denominator less than or equal to 70 is as good an approximation to as 99/70.

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## Approximations that depend on the floating point representation

A number is represented in a floating point format as   which is also called scientific notation. Its square root is   and similar formulae would apply for cube roots and logarithms. On the face of it, this is no improvement in simplicity, but suppose that only an approximation is required: then just   is good to an order of magnitude. Next, recognise that some powers, , will be odd, thus for 3141.59 = 3.1415910 rather than deal with fractional powers of the base, multiply the mantissa by the base and subtract one from the power to make it even. The adjusted representation will become the equivalent of 31.415910 so that the square root will be 10.

A table with only three entries could be enlarged by incorporating additional bits of the mantissa. However, with computers, rather than calculate an interpolation into a table, it is often better to find some simpler calculation giving equivalent results. Everything now depends on the exact details of the format of the representation, plus what operations are available to access and manipulate the parts of the number. For example, Fortran offers an EXPONENT(x) function to obtain the power. Effort expended in devising a good initial approximation is to be recouped by thereby avoiding the additional iterations of the refinement process that would have been needed for a poor approximation. Since these are few (one iteration requires a divide, an add, and a halving) the constraint is severe.

So for a 32-bit single precision floating point number in IEEE format (where notably, the power has a bias of 127 added for the represented form) you can get the approximate logarithm by interpreting its binary representation as a 32-bit integer, scaling it by  , and removing a bias of 127, i.e.

For example, 1.0 is represented by a hexadecimal number 0x3F800000, which would represent   if taken as an integer. Using the formula above you get  , as expected from  . In a similar fashion you get 0.5 from 1.5 (0x3FC00000).

/* Assumes that float is in the IEEE 754 single precision floating point format */

/* Convert type, preserving bit pattern */

* ((((val.i / 2^m) - b) / 2) + b) * 2^m = ((val.i - 2^m) / 2) + ((b + 1) / 2) * 2^m)

* b = exponent bias
* m = number of mantissa bits

/* Subtract 2^m. */
/* Divide by 2. */
/* Add ((b + 1) / 2) * 2^m. */

/* Interpret again as float */



The three mathematical operations forming the core of the above function can be expressed in a single line. An additional adjustment can be added to reduce the maximum relative error. So, the three operations, not including the cast, can be rewritten as




### Reciprocal of the square root



/* The next line can be repeated any number of times to increase accuracy */



## Geometric construction of the square root

The square root of a positive number is usually defined as the side length of a square with the area equal to the given number. But the square shape is not necessary for it: if one of two similar planar Euclidean objects has the area a times greater than another, then the ratio of their linear sizes is  .

A square root can be constructed with a compass and straightedge. In his Elements, Euclid (fl. 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13. Since the geometric mean of a and b is  , one can construct   simply by taking .

The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.

Euclid’s second proof in Book VI depends on the theory of similar triangles. Let AHB be a line segment of length with AH = a and HB = b. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as h. Then, using Thales’ theorem and, as in the proof of Pythagoras’ theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don’t need that, but it is the essence of the proof of Pythagoras’ theorem) so that AH:CH is as HC:HB, i.e. a/h = h/b, from which we conclude by cross-multiplication that h2 = ab, and finally that  . When marking the midpoint O of the line segment AB and drawing the radius OC of length (a + b)/2, then clearly OC > CH, i.e.   (with equality if and only if ), which is the arithmetic–geometric mean inequality for two variables and, as noted above, is the basis of the Ancient Greek understanding of «Heron’s method».

Another method of geometric construction uses right triangles and induction:   can be constructed, and once   has been constructed, the right triangle with legs 1 and   has a hypotenuse of  . Constructing successive square roots in this manner yields the Spiral of Theodorus depicted above.

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