Square root

Square root Техника
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Square root

Notation for the (principal) square root of .

Square root

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.

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.

Содержание
  1. Properties and uses
  2. Square roots of positive integers
  3. As decimal expansions
  4. As expansions in other numeral systems
  5. As periodic continued fractions
  6. Square roots of negative and complex numbers
  7. Principal square root of a complex number
  8. Nth roots and polynomial roots
  9. Square roots of matrices and operators
  10. In integral domains, including fields
  11. In rings in general
  12. Geometric construction of the square root
  13. Square root algorithm to find the square root of 2685
  14. Recent Articles
  15. The SAT Math Test: How To Be Prepared To Face It And Survive
  16. Fundamental Counting Principle
  17. Square Root Function
  18. What is Square Root Function?
  19. Domain and Range of Square Root Function
  20. Square Root Graph
  21. Graphing Any Square Root Function
  22. Properties of Square Root Function
  23. FAQs on Square Root Function
  24. What is Parent Square Root Function?
  25. What is the Formula of Square Root Function?
  26. What is the Derivative of Square Root Function?
  27. How to Graph Square Root of x?
  28. What is Square Root Function Domain?
  29. What is the Difference Between Cube Root Graph and Square Root Graph?
  30. What are the Asymptotes of Square Root Function?
  31. What is the Integral of Square Root Function?
  32. Square Root of a Number
  33. Square Root of a Number Properties
  34. Facts of Square Root of a Number
  35. How do you Calculate the Square Root of a Number?
  36. Square Root of a Number Using Prime Factorization Method
  37. Square Root of a Number Using Long Division Method
  38. Square Root of a Decimal Number
  39. Estimating the Square Root of a Number
  40. Solved Examples
  41. FAQs
  42. Latest Updates
  43. NCERT Books for Class 3 Maths
  44. CBSE Class 3 EVS Syllabus
  45. CBSE Class 3 Hindi Syllabus
  46. CBSE Class 3 Maths Syllabus
  47. Other examples showing how to find the square root of a number.
  48. Can the square root of a number be negative?
  49. Can the square root of a number be a real number?
  50. Recent Articles
  51. The SAT Math Test: How To Be Prepared To Face It And Survive
  52. Fundamental Counting Principle
  53. What is Square Root?
  54. Repeated Subtraction Method of Square Root
  55. Square Root by Prime Factorization Method
  56. Examples
  57. Square Root by Estimation Method
  58. Examples
  59. Square Root Formula
  60. Square Root of a Negative Number
  61. Examples
  62. Conclusion
  63. Practice Problems
  64. Recommended Reading
  65. FAQs
  66. What is the square root of a number?
  67. How to find the square root of a number?
  68. Can square root be negative?
  69. What is the formula for calculating the square root of a number?
  70. What are the applications of the square root formula?
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  75. More examples showing how to quickly estimate the square root
  76. Another estimation method
  77. Estimate the square root FAQs
  78. Recent Articles
  79. The SAT Math Test: How To Be Prepared To Face It And Survive
  80. Fundamental Counting Principle
  81. Square Root of Decimals
  82. What is Square Root of Decimals
  83. How to Find the Square Root of Decimals?
  84. Square Root by Estimation Method
  85. Square Root by Long Division Method
  86. FAQs on Square Root of Decimals
  87. What Is the Square Root of Decimals in Math?
  88. Which Method is Used to Find the Square Root of Non-Perfect Square Numbers?
  89. What are the Applications of the Square Root of Decimals?
  90. How To Find Square Root of Decimals by Estimation?
  91. What Are the Steps to Find Square Root of Decimals by Long Division Method?

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.

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 decimal expansions

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.

 
 

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 .

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 √.

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.

Square roots of matrices and operators

In integral domains, including fields

Each element of an integral domain has no more than 2 square roots. The difference of two squares identity u2 − v2 = (u − v)(u + v) is proved using the commutativity of multiplication. If and are square roots of the same element, then u2 − v2 = 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.

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.

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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.

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.

  • 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.

The square root algorithm, which helps to get the square root without using a calculator, is not taught a lot in school today.

Teachers usually teach the process once or never and usually have students use a calculator afterward.

Sorry, there are no shortcuts for this square root algorithm, so you will need patience before you master this algorithm!

Square root algorithm

Square root algorithm to find the square root of 2685

Example:  Square-root of 2685

Square-root-of-2685-image

First, always group the numbers in pairs starting from right to left and it is OK if there is only one number left in the leftmost
position.

I will use a space to show that they are being grouped in pairs. In this case, we have two pairs of 2 numbers.

Square-root-of-2685-image

Then, using the first pair, (26) find a number whose square is less than 26.

5 × 5 = 25 and 6 × 6 = 36, so your number is 5. Put 5 on top of the square root sign.

Square-root-of-2685-image

Subtract the square of the number on top that is 5 from the first pair.

: When you first start, you always take the square root of the pair or single number in the leftmost position. After that, you always double the number on top.

Since 52 = 5 × 5 = 25, subtract 25 from 26.

Square-root-of-2685-image

Bring down the next pair of numbers (85).

Square-root-of-2685-image

Then, double 5 to get 10 and put 10 next to 185 on the left side.

Square-root-of-2685-image

For the number 10, you will try to find a number you can put next to 0, which will also go next to 5.

Let’s call this number z. Then, find z such that 10z × z is less or equal to 185, but never bigger than 185.

If I put 2 next to 10, I get 102 × 2 = 204 and 204 is bigger than 185, so z = 2 is not the number I need.

However, if I put 1 next to 10, I get 101 × 1 = 101 and 101 is smaller than 185.

Thus, z = 1 is the number I will put next to 10 and 5.

101 is the number I will subtract from 185.

Put the 1 on top of the square root sign next to 5

Square-root-of-2685-image

Subtract 101 from 185

Square-root-of-2685-image

Bring down a pair of zeros next to 84 to continue the process and put a decimal point next to 51.

Square-root-of-2685-image

Double 51 to get 102 and put 102 next to 8400 on the left side.

Square-root-of-2685-image

For the number 102, you will try to find a number you can put next to 2, which will also go next to the decimal point.

Let’s call this number z again. Then, find z such that 102z × z is less or equal to 8400, but never bigger than 8400.

If I put 9 next to 102, I get 1029 × 9 = 9261 and 9261 is bigger than 8400.

However, if I put 8 next to 1028, I get 1028 × 8 = 8224 and 8224 is smaller than 8400.
Thus, z = 8 is the number I will put next to 102 and the decimal point.

8224 is the number I will subtract from 8400.

Put 8 on top of the square root sign next to 102 and the decimal point.

Square-root-of-2685-image

Subtract 8224 from 8400

Square-root-of-2685-image

Bring down a pair of zeros next to 176. Double 518 ignoring the decimal point to get 1036 and put 1036 next to 176000.

Square-root-of-2685-image

For the number 1036, you will try to find a number you can put next to 6, which will also go next to 8.

Let’s call this number z again. Then, find z such that 1036z × z is less or equal to 17600, but never bigger than 17600.

If I put 2 next to 1036, I get 10362 × 2 = 20724.

Since when z = 2, I get 20724 and 20724 is bigger than 17600, z = 2 is not the number I need.

However, if I put 1 next to 1036, I get 10361 × 1 = 10361 and 10361 is less than 17600.
Thus, 10361 is the number you will subtract from 17600.

1 is the number you will put next to 1036 and 8.

Put 1 on top of the square root sign next to the decimal point.

Subtract 10361 from 17600

Square-root-of-2685-image

I will stop here because it is a lot of work to put this square root algorithm here for you. Appreciate my effort? Look for the donation button below and please make a donation to keep this website running.

Feel free to continue the square root algorithm to get more decimal numbers, but I think I am done.

If you were looking for the square root of 785 for instance, the square root algorithm does not change at all.

Notice that when you group the number in pairs starting from right to left, there will be only 1 number in the leftmost
position. That number is 7 and that is fine.

Just start with 7 and say to yourself, » The square of what number is less than 7 and proceed an already outlined above.

I thought I said I was done. Why am I still talking about this square root algorithm?

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Square Root Function

The square root function involves the square root symbol √ (which is read as «square root of»). The square root of a number ‘x’ is a number ‘y’ such that y2 = x. i.e., if y2 = x ⇒ y = √x. i.e., if ‘x’ is the square of ‘y’ then ‘y’ is the square root of ‘x’. Some examples are:

  • 22 = 4 ⇒ √4 = 2
  • 42 = 16 ⇒ √16 = 4

We know that the square root of a number can be either positive or negative. i.e.,√4 = ±2. But while defining the square root function, we restrict its range to be the set of all positive real numbers (otherwise it won’t become a function at all), and hence in the case of the square root function, the result is always positive. Let us use all these facts to understand the square root function.

What is Square Root Function?

square root function

f(x) = √x is the parent square root function but when the transformations are applied to it, it may look like f(x) = a√(b(x — h)) + k, where a, b, h, and k are numbers such that

  • ‘a’ is the vertical dilation
  • ‘b’ is the horizontal dilation
  • ‘h’ is the horizontal translation
  • ‘k’ is the vertical translation

Domain and Range of Square Root Function

The square root of a negative number is NOT a real number. i.e., the square root function cannot accept negative numbers as inputs. i.e.,

  • The domain of the square root function f(x) = √x is the set of all non-negative real numbers. i.e., the square root function domain is [0, ∞). Note that it includes 0 as well in the domain.

In general, the square root of a number can be either positive or negative. i.e., √25 = 5 or -5 as 52 = 25 and (-5)2 = 25. But the range of the square root function (i.e., its y-values) is restricted to only positive numbers, because otherwise, it fails the vertical line test and it won’t be a function if an input has two outputs. Thus,

  • The range of the square root function f(x) = √x is also the same as its domain [0, ∞). Note that it includes 0 as well in the range.

Square Root Graph

Square Root Graph

Note that when some transformations are applied to the graph, the graph may not lie in the first quadrant itself.

Graphing Any Square Root Function

We have seen how to graph the parent square root function f(x) = √x. Here are the steps that are useful in graphing any square root function that is of the form f(x) = a√(b(x — h)) + k in general.

  • Step 1: Identify the domain of the function by setting «the expression inside the square root» to greater than or equal to 0 and solving for x.
  • Step 2: The range of any square root function is always y ≥ k where ‘k’ is the vertical translation of the function f(x) = a√(b(x — h)) + k.
  • Step 3: Construct a table of values with two columns x and y, take some random numbers for x (from the domain only) starting from the first value of the domain, substitute them in the given function and find the corresponding values of y.
  • Step 4: Plot all the points on the plane and connect them by a curve and also extend the curve following the same trend.

Note: Computing the x-intercept and y-intercept would also help in graphing the square root function.

Example: Graph the square root function f(x) = √(x — 2) + 3.

To find its domain, x — 2 ≥ 0 ⇒ x ≥ 2.

Its vertical shift is 3 and hence its range is y ≥ 3.

Now, we will construct a table with some values greater than 2 (as the domain is x ≥ 2). Choose some values for x such that √(x — 2) is a perfect square so that the calculation becomes easier.

Now, plot these points and join them by a curve.

Graphing Square Root Function

We can also graph the square root function by applying the transformations on the parent square root graph f(x) = √x.

Properties of Square Root Function

Here are the important points/properties that are to be noted about the square root function f(x) = √x.

  • Its domain is [0, ∞).
  • Its range is [0, ∞).
  • It has no relative maxima but it has a minimum at (0, 0).
  • A square root function has no asymptotes.
  • It is an increasing function throughout its domain [0, ∞).
  • The square root function f(x) = √x has critical point at (0, 0) and it has no inflection points.

FAQs on Square Root Function

What is Parent Square Root Function?

The parent square root function is f(x) = √x. This function may be translated/dilated/reflected and can transform to the form f(x) = a√(b(x — h)) + k.

What is the Formula of Square Root Function?

The formula for the square root function is f(x) = √x. It means the output of each input value is equal to the square root of the input value. For example, f(25) = √25 = 5. Note that all inputs and outputs of a square root function are always non-negative.

What is the Derivative of Square Root Function?

The derivative of the square root function f(x) = √x is calculated by the power rule of differentiation, d(xn)/dx = nxn-1. By this rule, d(√x)/dx = d(x1/2)/dx = (1/2) x(1/2) — 1 = (1/2) x-1/2 = 1/(2√x). Thus, the derivative of √x is 1/(2√x).

How to Graph Square Root of x?

To graph the square root of x, just note that its inputs and outputs are all non-negative and hence its graph lies in the first quadrant. Further, to get the clear shape of the graph, calculate some points on it, by taking some random numbers for x and computing corresponding y-values for them.

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What is Square Root Function Domain?

What is the Difference Between Cube Root Graph and Square Root Graph?

The cube root graph can take in any real number as input and produces any real number as output. But the square root function takes in and produces only the non-negative real numbers.

What are the Asymptotes of Square Root Function?

There is no horizontal line or vertical line that can break the graph of square root function and hence it has no vertical/horizontal asymptotes.

What is the Integral of Square Root Function?

The integral of the square root function √x can be found using the power rule of integration ∫xn dx = xn+1/(n + 1) + C.. Using this, ∫√x dx = ∫x1/2 dx = x(1/2 + 1)/(1/2 + 1) + C = x3/2/(3/2) + C = (2/3) x3/2 + C.

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Just as the division is the inverse operation of multiplication, the square root of a number is the inverse squaring procedure. The square root of any number is multiplied by itself, giving the product equal to the original number.

For example, if \(y\) is the integer and the square root is \(x\), it can be said that \(y×y = x\). The square root is represented by the symbol \(√\). In this article, we will be studying more about the square root of integers, with different examples.

Square Root of a Number

Therefore, the square root of \(y\) is denoted as \(\sqrt y \).

Square Root of a Number Properties

1. A perfect square number has a perfect square root.

2. If the last digit of a number is even, then its square root will also be even.

3. The odd perfect number has a square root that is odd.

4. The square root of a negative number is not defined or undefined because the perfect square can never be negative.

Facts of Square Root of a Number

How do you Calculate the Square Root of a Number?

Let us discuss each way of finding the square root one by one with the help of examples.

Square Root of a Number Using Prime Factorization Method

Suppose we need to find the square root of a number. The prime factorization method means that we break the number into all its prime factors and then use those prime factors to find the square root.

This is explained below with the help of an example.

Example: Find the square root of \(576\) by the prime factorization method.

Thus, prime factors of \(576\) are

\(576 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\)

Hence, the square root of \(576\) is \(24\).

Square root

Square Root of a Number Using Long Division Method

Finding the square root of a large number using the prime factorization method is quite time-consuming. Thus, we use another method known as the long division method to find the square root of large numbers.

This process involves several steps and is explained below with the help of an example.

Example: Find the square root of \(7569\) using the division method.

Step 1: Firstly, put a bar over every pair of digits in the given number, starting from the unit’s place. Thus, for \(7569\), we will refer to \(75\) as the first group and \(69\) as the second group.

Square root

Step 2: Start with the first group, i.e., \(75\). Think of the nearest perfect square number equal to or less than \(75\)

Therefore, \(8\) becomes the divisor as well as the first digit of the quotient. Carry out the long division to get \(11\) as the remainder.

Square root
Square root

Given that \(1169\) ends with \(9\), the last digit of the new divisor will be \(7\) as \(7 \times 7 = 49\). Let us verify this:

\(167 \times 7 = 1169\), which gives us the required units place digit as \(7\). Therefore, we put \(7\) in the units place of the final quotient and get \(167\) as the new divisor.

Square root

Step 5: Carry out the long division, and thus the remainder will be \(0\).Therefore, the square root of \(7569\) is \(87\).

Square root

Square Root of a Decimal Number

Step 1: Put a bar over every pair of digits in the given number, starting from the unit’s place and moving left for an integral part. Further, start with the tenth place and move right for the decimal part. Thus, \(38\), forms a group for an integral part, and \(44\) forms the group for the decimal part.

Square root
Square root
Square root

Given that \(244\) ends with \(4\), the last digit of the new divisor can be either \(2\) or \(8\). If it is \(2\),

\(122 \times 2 = 244\), which gives us the required units, place digit as \(2\). Therefore, we put \(2\) in the decimal part of the final quotient and get \(122\) as the new divisor.

Square root
Square root

Estimating the Square Root of a Number

Consider a real-life situation:

Solved Examples

Square root

\( = 3 \times 13 = 39\)

Q.2. Find the square root of \(841\) using the division method.
Ans: Let us do the division as shown below.

Square root

Hence, the square root of \(841\) is \(29\).

Q.4. Find the square root of \(2\) correct up to three decimal places.
Ans: We will find the square root of \(2\) by the division method.

Square root

Q.5. Find the square root of \(144\) by the prime factorization method.
Ans: The prime factors of \(144\) are:

Square root

Hence, the square root of \(144\) is \(12\).

FAQs

Q.1. How do you calculate the square root of a number?
Ans: Square root of a number can be calculated by either the prime factorization method or the long division method.

Q.3. How do you find the square root of an imperfect square?
Ans: The square root of an imperfect square can find out by the long division method.

Q.4. What is the square of \(19\)?
Ans: The square of \(19\) is \(361\).

Q.5. How do you find the square root of a perfect square?
Ans: The square root of perfect squares can find out by using prime factorization.

We hope this detailed article on Square Root of a Number helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest.

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Before understanding what the square root of a number is, it is important to understand the meaning of root of a number.

Square root of a number

The root of a number is an equal factor of the number. For example, here is how to find the root of 16.

First, we need to factor 16. The different ways to factor 16 are shown below.

16 = 1 × 16

16 = 2 × 8

16 = 4 × 4

The root of 16 is 4 because 4 is the equal factor for 16. We call 4 the

square root

of 16 and we write √16 = 4.

4 is called square root because we have to square 4 or raise 4 to a power of 2 to get 16.

Other examples showing how to find the square root of a number.

Find the square root of 4.You can factor 4 in two different ways.

4 = 1 × 4

4 = 2 × 2

The equal factor is 2, so 2 is the square root of 4 and we write √4 = 2.

Find the square root of 64.

64 = 1 × 64

64 = 2 × 32

64 = 4 × 16

68 = 8 × 8

The equal factor is 8, so the square of 64 is 8 and we write √64 = 8.

Can the square root of a number be negative?

Yes, absolutely! If you multiply a positive number by itself, you get a positive product. If you multiply a negative number by itself, you also get a positive product.

For example, since -8 × -8 = 64, -8 is also a square root of 64. However, unless otherwise stated, the square root sign (  refers to the positive root of a number, also called principal square root.

Can the square root of a number be a real number?

For the numbers above, the square root was equal to an integer.

it is not always possible to get the square root as an integer.

Sometimes, you may get a real number when finding the square root.

For example, use the square root calculator below to find the square root of 5.

The result includes lots of numbers after the decimal point.

Ready for big time challenge? Just like long division, learn how to compute the square root without a calculator for any number that is not a perfect square.I promise you will not sweat too much!

Teachers! Do you want a ready made square roots table that students can quickly refer to as they solve their basic math problems? Get the square roots table.

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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.

https://youtube.com/watch?v=nEJYRtgHL-4%3Ffeature%3Doembed

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

What is the square root of a number?

How to find the square root of a number?

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 is the formula for calculating the square root of a number?

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.

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One way to estimate the square root of any number, especially numbers that are not perfect squares, is to find a whole number greater than the square root and another whole number less than the square root of that number that is not a perfect square. 

Take a close look at the figure below so you can learn the process. There are some important observations you need to make!

  • In our example below, since 34 is not a perfect square, we can try to find an approximation for the square root.
  • Perfect square numbers do not need estimation. You just need to know your multiplication table to find them. Many people know the perfect square numbers from 1 to 1000.
  • Notice that 34 is much closer to 36 than it is to 25. Therefore, we chose a number very close to 6 and that number is 5.8.

Estimate the square root

The example in the figure above is a quick and easy way to estimate a square root when it is not possible to find perfect square roots. You will learn two different methods in this lesson.

More examples showing how to quickly estimate the square root

We will illustrate the process with two more examples.

Estimate the square root of 17

We will find a whole number bigger than the square root of 17 and a whole number smaller than the square root of 17. 

      <    √17    <

Study carefully the procedure!

First, let us find the number that is bigger than  √17

√17     <     √18

√17     <     √19

√17     <     √20

√17     <     √21

√17     <     √22

√17     <     √23

√17     <     √24

√17     <     √25

Notice that the square roots of 18, 19, 20, 21, 22, 23, 24 are all bigger than the square root of 17.

However, only √25 will give a whole number, so this is the one we will choose.

√25 = 5 since 5 × 5 = 25

Second, let us find the number that is smaller than √17

√16   <   √17

Since  √16  is a whole number, this is the one we will choose.

√16   = 4 since 4 × 4 = 16

We get 4  < √17  < 5

The square root of 17 is between 4 and 5. We could estimate the square of 17 to be 4.1 for example.

Estimate the square root of 102

We will find a whole number bigger than the square root of 102 and a whole number smaller than the square root of 102. 

√101   <   √102

√100   <   √102

We will use square root of 100 since  √100   = 10

To find the number bigger than the square of 102, we will use a different strategy. It take too long to write down the square root of all these numbers. Some good observation will help us to solve the problem quickly.

Notice that the square root of any number between 103 and 120 is not a whole number.

However, square root of 121 is a whole number since 11 times 11 = 121.

Therefore, the square of 121 will give us the whole number that we need that is bigger than square of 102.

We get 10   < √102   < 11

The square root of 102 is between 10 and 11. We could estimate the square root of 102 to be 10.2.

Notice again that in our estimation, we chose a number close to 10 since 102 is much closer 100 than it is 121.

Another estimation method

Estimate the square root of 45

Pick a number you think √45 is close to. For example, pick 7 since √49 = 7. We did not pick 6 since √49 is closer to the answer than √36.

Find the difference between the square of √49 and the square of √45

(√49)2 — (√45)2 = 49 — 45 = 4

Divide the number you found in step 2 by twice the number you picked in step 1

4 / 2(7) = 4 / 14 = 0.2857

Since 7 is an overestimation, subtract 0.2857 from 7 and this is your estimation.

7 — 0.2857 = 6.714

6.714 times 6.714 = 45.0777 and as you can see 45.0777 is quite close to 45

Estimate the square root of 39

Pick a number you think √39 is close to. For example, you can pick 6 since √36 = 6. 

Find the difference between the square of √39 and the square of √36

(√39)2 — (√36)2 = 39 — 36 = 3

Divide the number you found in step 2 by twice the number you picked in step 1

3 / 2(6) = 3 / 12 = 0.25

Since 6 is an underestimation, add 0.25 to 6 and this is your estimation.

6 + 0.25 = 6.25

6.25 times 6.25 =  39.0625 and as you can see 39.0625 is quite close to 39

Estimate the square root FAQs

The easiest way to calculate square root is to use a calculator. This is a must if the number you are taking a square root of is not a perfect square and in this cae, teachers will let you use a calculator in most cases. When taking exams, it is not likely they will force you to use paper and pencil only if you are not dealing with perfect square roots.
If you are taking the square of numbers that are not perfect squares, you will end up with irrational numbers with a large number of digits after the decimal point. You can just round the answer to the tenths place or hundredths place.

√22 lies between 4 and 5. Since √22 is closer to 5, you could choose 4.7 as an estimate.

√23 lies between 4 and 5. Since √23 is closer to 5, you could choose 4.8 as an estimate.

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Square Root of Decimals

Square root of decimals is carried out in the same way as for whole numbers. Square root of a number and squaring a number are inverse operations. The Square of a number is the value of power 2 of the number, while the square root of a number is the number that is multiplied by itself to give the original number.

In this article, let’s learn how to find the square root of decimals using solved examples and practice questions.

What is Square Root of Decimals

Square root of decimal is the value of a decimal number to the power 1/2. For example, the square root of 24.01 is 4.9 as (4.9)2 = 24.01. The square root of a decimal number can be calculated by using the estimation method or the long division method.

In the case of long division method, the pairs of whole number parts and fractional parts are separated by using bars. And then, the process of long division is carried out in the same way as any other whole number.

How to Find the Square Root of Decimals?

Square Root by Estimation Method

Estimation and approximation is a reasonable guess of the actual value so as to make calculations easier and realistic. This method also helps in estimating and approximating the square root of a given number. We just need to find the nearest perfect square numbers to the given decimal number to find its approximate square root value.

Let’s find the square root of 31.36.

  • Step 1: Find the nearest perfect square numbers to 31.36. 25 and 36 are the perfect square numbers nearest to 31.36.
  • Step 2: √25 = 5 and √36 = 6. This implies that √31.36 lies between 5 and 6.
  • Step 3: Now, we need to see if √31.36 is closer to 5 or 6. Let us consider 5.5 and 6.
  • Step 4: 5.52 = 30.25 and 62= 36. Thus, √31.36 lies between 5.5 and 6 and is closer to 5.5.

Thus, the square root of 31.36 is close to 5.5.

Square Root by Long Division Method

Long Division method is mainly used in case we need to dividing large numbers into steps or parts, breaking the division problem into a sequence of easier steps. We can find the exact square root of any given number using this method. Let us understand the process step by step, with an example. Let’s find the square root of 2.56.

  • Step 1: Place a bar over every pair of digits of the given decimal number starting from the unit’s place. We will have two pairs, i.e. 2 and 56.
  • Step 2: We start with the whole number part and divide it by the largest number whose square is less than or equal to that number. Here, the whole number part is 2 and we have 1 x 1 = 1. So, the quotient is 1.

Square root of decimal number by long division step 1

  • Step 3: Bring down the number, that is the pair of the fractional parts under the next bar to the right of the remainder(that is 1).
  • Step 4: Add the last digit of the quotient to the divisor, which is 1 + 1 = 2. To the right of the obtained sum( that is 2), find a suitable number that, together with the result of the sum, forms a new divisor for the new dividend (that is, 156) that is carried down. Also, add decimal after 1 in the quotient as we move to the fractional part.

Square root of decimal number by long division step 2

  • Step 5: The new number in the quotient will have the same number as selected in the divisor, thus the divisor will now be 26 and the quotient will be 1.6, as 26 x 6 = 156. (The condition is the same — as being either less than or equal to the dividend).
  • Step 6: Continue this process further using a decimal point and adding zeros in pairs to the remainder.

Square root of decimal number by long division

  • Step 6: The quotient thus obtained will be the square root of the number. Thus, the square root of 2.56 is 1.6.

Related Articles on Square Root of Decimals:

Given below is the list of topics that are closely connected to the square root of decimals.

FAQs on Square Root of Decimals

What Is the Square Root of Decimals in Math?

Square root of decimals refers to the value of a decimal number to the power 1/2. For example, the square root of 12.25 is 3.5 as (3.5)2 = 12.25. It is calculated by estimation as well as long division method. It is easy to find the exact square root of any given number using the long division method.

Which Method is Used to Find the Square Root of Non-Perfect Square Numbers?

In Math, a non-perfect number is considered the one that is in decimal form. The square root of a non-perfect square number can be calculated by using the long division method.

What are the Applications of the Square Root of Decimals?

There are various applications of the square root of decimals.

  • Used in algebra and geometry. It helps in finding the roots of a quadratic equation.
  • Make calculations for the area, volume, and other measurements easy.
  • Widely used by engineers.

How To Find Square Root of Decimals by Estimation?

To find the square root of decimals by estimation, look for the perfect square numbers close to the given decimal numbers. Find their square roots to have an approximate value for the square root of the given decimal number. For example, the square root of 11.56 is 3.4.

  • Step 1: √11.56 is close to the perfect squares, 9 and 16.
  • Step 2: √9 = 3 and √16 = 4.
  • Step 3: Check for 3.52. 3.52 = 12.25. This implies √11 lies between 3 and 3.5.

What Are the Steps to Find Square Root of Decimals by Long Division Method?

  • Step 1: Place a bar over every pair of digits of the number, whole number part as well as fractional part.
  • Step 2: Divide the left-most number, the whole number part by the largest number whose square is less than or equal to the number in the left-most pair.
  • Step 3: Bring down the number under the next bar to the right of the remainder. Add the last digit of the quotient to the divisor. To the right of the obtained sum, find a suitable number which, together with the result of the sum, forms a new divisor for the new dividend that is carried down.
  • Step 4: The new number in the quotient will have the same number as selected in the divisor. The condition is the same — as being either less than or equal to the dividend.
  • Step 5: Then, continue this process further using a decimal point and adding zeros in pairs to the remainder.
  • Step 6: The quotient thus obtained will be the square root of the given decimal number.

how can I create a method that returns the sqrt of a given nunber?

user2314737's user avatar

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asked Jun 16, 2010 at 8:11

Java program to find out square root of a given number
without using any Built-In Functions

public class Sqrt
{ public static void main(String[] args) { //Number for which square root is to be found double number = Double.parseDouble(args[0]); //This method finds out the square root findSquareRoot(number);
}
/*This method finds out the square root without using
any built-in functions and displays it */
public static void findSquareRoot(double number)
{ boolean isPositiveNumber = true; double g1; //if the number given is a 0 if(number==0) { System.out.println("Square root of "+number+" = "+0); } //If the number given is a -ve number else if(number<0) { number=-number; isPositiveNumber = false; } //Proceeding to find out square root of the number double squareRoot = number/2; do { g1=squareRoot; squareRoot = (g1 + (number/g1))/2; } while((g1-squareRoot)!=0); //Displays square root in the case of a positive number if(isPositiveNumber) { System.out.println("Square roots of "+number+" are "); System.out.println("+"+squareRoot); System.out.println("-"+squareRoot); } //Displays square root in the case of a -ve number else { System.out.println("Square roots of -"+number+" are "); System.out.println("+"+squareRoot+" i"); System.out.println("-"+squareRoot+" i"); } }
}

mega6382's user avatar

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answered Jul 18, 2013 at 22:17

John Joshua Austero Lipio's user avatar

This version uses Newton’s Method, the most common method of calculating sqrt, and won’t check that the input is actually an integer, but it should solve your problem just fine.

int num = Integer.parseInt(input("Please input an integer to be square rooted."));
while(0.0001 < Math.abs(guess * guess - num)){ guess = (guess + num / guess) / 2;
}
output(Integer.toString(guess));

The second line checks how close the current guess is to the true result, and if close enough breaks the loop. The third line uses Newton’s Method to get ever-closer to the true value of the sqrt. I hope this helps. 🙂

answered Feb 12, 2013 at 6:19

democidist's user avatar

1 silver badge6 bronze badges

Here’s something to think about:

To find a square root, you simply need to find a number which, raised to the power of 2 (although just multiplying by itself is a lot easier programmatically 😉 ) gives back the input.

Depending on your need of precision and/or performance, there are of course lots of ways. The solution hinted at in this post is in no way the best one in either of those categories, but it gives you a clue on one way to go.

answered Jun 16, 2010 at 8:21

Tomas Aschan's user avatar

Tomas Aschan

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One that I invented (or reinvented if the case may be) is this:

Next Guess = ( ( Guess2) + N ) / ( 2 × Guess )

Square root of 10, first guess is, let’s say, 10:

Guess1 = (100+10)/20=5.5
Guess2 = (30.25+10)/(2*5.5)= 3.6590909090...
Guess3 = (13.3889+10)/(3.65909090*2)=3.196005082...

This is actually a simplified version of my original method

Guess + ( ( N + Guess2 ) / ( 2 × Guess ) )

which looks an awful lot like Bakhshali’s method.

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answered Apr 7, 2011 at 0:31

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