- Ivan Niven, Numbers: Rational and Irrational. New York: Random House for Yale University (1961): 44.
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.
It is suggested by some sources that the symbol $\surd$ (a stylised r for radix) for the square root may have originated with René Descartes, but there is evidence that it may have been around a lot earlier than that.
The square root of $z$ is the $2$-valued multifunction:
where $\sqrt r$ denotes the positive square root of $r$.
- Principal Square Root
- Positive Real Numbers
- Positive Square Root
- Negative Square Root
- Square Root of $2$
- Square Root of $3$
- Square Root of $5$
- Square Root of $10$
- Square Root of $e$
- Square roots of negative and complex numbers
- Principal square root of a complex number
- Also known as
- In rings in general
- History of square root
- Square root Symbol on Mac
- Applications of the Square Root Symbol
- What is square root sign in Algebra?
- What does the square root symbol (√) represent?
- Who introduced the square root symbol?
- What are some practical applications of the square root symbol?
- How is the square root symbol (√) used in the Pythagorean theorem?
- What is the relationship between the square root symbol (√) and quadratic equations?
- How is the square root symbol (√) used in financial calculations?
- Nth roots and polynomial roots
- In integral domains, including fields
- Properties and uses
- Square roots of some small integers
- Negative Real Numbers
- Type square root symbol (√) in different devices –
- Square roots of positive integers
- As decimal expansions
- As expansions in other numeral systems
- As periodic continued fractions
- Geometric construction of the square root
- Square roots of some important constants
Principal Square Root
Let $z \in \C$ be a complex number.
The principal square root of $z$ is the principal branch of the $2$nd power of $w$.
- $-\dfrac \pi 2 < \arg w \le \dfrac \pi 2$
- 2008: Ian Stewart: Taming the Infinite … (previous) … (next): Chapter $8$: The System of the World: Calculus
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) … (previous) … (next): square root
- Apotome (mathematics)
- Cube root
- Functional square root
- Integer square root
- Nested radical
- Nth root
- Root of unity
- Solving quadratic equations with continued fractions
- Square root principle
- Quantum gate § Square root of NOT gate (√NOT)
A square root of a number $n$ is a number $z$ such that $z$ squared equals $n$.
Positive Real Numbers
The square roots of $x$ are the real numbers defined as:
- $x^{\paren {1 / 2} } := \set {y \in \R: y^2 = x}$
- $y = \pm \sqrt x$
is usually encountered.
Positive Square Root
The positive square root of $x$ is the number defined and denoted as:
- $\sqrt x := y \in \R_{>0}: y^2 = x$
Negative Square Root
The negative square root of $x$ is the number defined and denoted as:
- $-\sqrt x := y \in \R_{<0}: y^2 = x$
- Dauben, Joseph W. (2007). «Chinese Mathematics I». in Katz, Victor J.. The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: Princeton University Press. ISBN 978-0-691-11485-9.
- Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0-691-00659-8.
- Smith, David (1958). History of Mathematics. 2. New York: Dover Publications. ISBN 978-0-486-20430-7.
- .
Short description
: Number whose square is a given number
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.
Square Root of $2$
- $\sqrt 2 \approx 1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$
Square Root of $3$
- $\sqrt 3 \approx 1 \cdotp 73205 \, 08075 \, 68877 \, 2935 \ldots$
Square Root of $5$
- $\sqrt 5 \approx 2 \cdotp 23606 \, 79774 \, 99789 \, 6964 \ldots$
Square Root of $10$
- $\sqrt 10 \approx 3 \cdotp 16227 \, 76601 \, 68379 \, 33199 \, 88935 \, 44432 \, 71853 \, 3719 \ldots$
Square Root of $e$
The decimal expansion of Euler’s Number $e$ starts:
- $\sqrt e \approx 1 \cdotp 64872 \, 12707 \, 00128 \, 1468 \ldots$
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.
- 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.
- Replace x by the average (x + a/x) / 2 between x and a/x.
- Repeat from step 2, using this average as the new value of x.
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 root symbol or square root sign looks like √ and it is a mathematical symbol that people had been writing from the times when ASCII encoding was developed anciently. This sign is well-known as radical, in words. And you can simply type it right from your keyboard very easily. In this article we’ll show you how to do it by using different ways depending on your Operating System and tastes.
Square root symbol copy
√
From time to time we are faced with the need to add a certain character to the text, but it is absent on the keyboard. It doesn’t matter, because the symbol table is used for the icons. Symbols, in turn, can be written by pressing certain keys on the keyboard. Today we will learn to print/write the square root sign.
For example – Square root of 3 is √3 = 1.73205080757
- History of square root
- Type square root symbol (√) in different devices –
- SQUARE ROOT SYMBOL IN WINDOWS
- Square root Symbol on Mac
- Applications of the Square Root Symbol
- What is square root sign in Algebra?
- Conclusion
- What is square root sign in Algebra?
- What does the square root symbol (√) represent?
- Who introduced the square root symbol?
- What are some practical applications of the square root symbol?
- How is the square root symbol (√) used in the Pythagorean theorem?
- What is the relationship between the square root symbol (√) and quadratic equations?
- How is the square root symbol (√) used in financial calculations?
- Applications of the Square Root Symbol
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This article is under construction.
Please do not rely on any information it contains.
The square root
of a number
is the number that multiplied by itself gives
. For example, the square root of 44100 is 210, since 210 × 210 = 44100. Actually, positive numbers have two square roots, one positive, one negative; e.g., –210 × –210 = 44100. The square root of a negative number is an imaginary number.
Theorem SQRT1. The square root of a positive integer is either a positive integer or an irrational number, but never a non-integral rational number.
Proof. Take the set of all positive integers
and square all its members, label the resulting set
. Clearly
is the set of all positive integers that have integer square roots. Obviously, these integer square roots are rational numbers, as they can be expressed as
, where
.
Since
, we can write
. Multiply by
to get
. This means that
is divisible by
, and therefore
(the value of
is not necessary for this proof). So,
and thus
. Dividing both sides by
we obtain
. This means that
is also divisible by
. But we established that
is also divisible by
, contradicting the assertion that
and
are coprime, and therefore
is not a rational number. ¿¿¿IS THERE A HOLE IN THIS PROOF REGARDING COMPOSITE NUMBERS???
In summary, if
, then
, but if not, then
as specified by the theorem. □
Corollary. Much of the foregoing can be said for negative numbers with only small adjustments. For convenience, let’s say that the function
returns a real value, that is to say,
, not
. Then, if
, either
or
.
This proof is essentially a generalization of proofs for the square roots of specific integers. Perhaps it would be more elegant to first prove the fundamental theorem of algebra and then derive not only this result but also the similar results for cubes, biquadrates, etc.
However it is proven, this result can be used to prove the irrationality of some other numbers involving square roots quite easily. For example:
Theorem SQRT23. The number
is irrational.
See A135611 for the decimal expansion of
, and A089078 for the continued fraction expansion.
Theorem SQRT2.25. The square of rational number that is not an integer is another rational number that is not an integer either.
For example,
.
Corollary. The square root of an integer may be an integer or it may be an irrational number, but it may not be a non-integral rational number, as that would obviously contradict what we have just proven.
The converse is not always true: the square root of a rational number that is not an integer may be an irrational number. This is the case with the reciprocals of most integers, e.g.,
, which is clearly irrational.
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
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:
- Counterexample for the principal square root: z = −1 and w = −1
This equality is valid only when - Counterexample for the principal square root: and z = −1
This equality is valid only when - 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
Also known as
Because square roots (and in particular, positive square roots) are so much more commonly encountered in mathematics than any other sort of root, $\sqrt x$ is frequently just called root $x$
In translations of Euclid‘s The Elements, the word side can be found, often in quotes to emphasise the awkward nature of the language available to the Ancient Greeks.
In the words of Euclid:
- If an area be contained by a rational straight line and the first binomial, the «side» of the area is the irrational straight line which is called binomial.
In rings in general
History of square root
The radical first appeared in 1525, when Christoff Rudoff, a German mathematician, made use of the symbol to refer to the square root in his textbook on algebra, entitled “Coss.”
For the year 1637, the famous French philosopher and physical mathematician Rene Descartes makes use of the symbol and adds an upper bar in geometry.
Origin of the Radical Symbol
The square root symbol is represented. The Initial use of this sign is originate in the work of Leonardo of Pisa, “Practical Geometry” commencing the year 1220.
1637 – Nicolas Chuquet
1525 – Christoff Rudoff
The square root
In the eleventh century, the French mathematician Nicolás Chuquet, wrote his thesis on algebra where he refers to an exponential data, printed as: RU, where R is the square root, This opens the entry for new studies and better representations, but above all to give the symbol a character of stability and worldwide existence.
Other names for root symbols √
- Radical symbol
- Radical sign
- Root symbol
- radix
- Deaf
Square root Symbol on Mac
To type the root symbol in Mac OS X, all you would like to try to to is clicking Option+V
√ – Option+V
Navigate to the file where you would like the root symbol inserted in. you’ll use this advance in any Mac program that permits typing including an internet browser.
Tap the situation where you would like to feature the symbol.
Click ⌥ Option+v. it’ll bring the Square root symbol Easily.
Option + V key
Applications of the Square Root Symbol
- Standard Deviation in Statistics
- Electrical Engineering and Complex Numbers
- Financial Calculations and Risk Management
Basically, the square root is used in mathematics, physics to simplify complex power functions, inequalities and equations. Even in ancient times, people just needed to calculate the square root. Many were engaged in agriculture and dividing the area into squares, they could not calculate anything without a root. Therefore, the root sign was introduced by human necessity, since knowing the area, people in the sixteenth century needed to calculate the side of the square.
That is why the square root was introduced, which we use to this day. And now, we often need it. Let’s remember the Pythagorean Theorem, What do we know about it? The square of the hypotenuse is equal to the sum of the squares of the legs in a right-angled triangle. And if we need to calculate the side of a square-shaped plot of 100 square meters, what do we use? The square root, of course. And then it will be easy for us to do it. The square root of 100 is 10. So our side of the site is 10 meters.
What is square root sign in Algebra?
The definition of the arithmetic square root does not add clarity, but it is worth memorizing it:
The arithmetic square root of a non-negative number “m” is a non-negative number whose square is equal to a.
The definition of the square root can also be represented in the form of formulas:
√m = x
x 2 = m
x ≥ 0
m ≥ 0
To understand why this is exactly the case, let’s look at an example.
Let’s try to find the root of √-81
Here it is logical to assume that 4, but let’s check: 9 * 9 = 81 – does not converge.
If – 9, then -9 * -9 = 81, (minus by minus always gives plus).
It turns out that no number can give a negative result when squaring it.
The numbers under the root sign must be positive.
Based on the definition, the value of the root should also not be negative.
Reasonable questions may arise here, why, for example, in the example x 2 = 81, x = 9 and x = -9.
The square root symbol (√) is a mathematical symbol used to represent the square root operation. It is widely used in mathematical equations and expressions to indicate the principal square root of a non-negative number. The symbol resembles a checkmark or a horizontal line with a diagonal line extending from its top left to its bottom right. Different devices and platforms, such as computers, smartphones, and word processing software, offer various methods to write the square root symbol, including keyboard shortcuts, HTML entities, LaTeX commands, and ASCII representation.
What does the square root symbol (√) represent?
The square root symbol (√) represents the mathematical operation of taking the square root of a number. It denotes finding a value that, when multiplied by itself, equals the given number.
Who introduced the square root symbol?
The square root symbol (√) is credited to the ancient Greek mathematician, Heron of Alexandria, who introduced it in his work “Metrica” in the 1st century AD.
What are some practical applications of the square root symbol?
The square root symbol (√) is widely used in various fields, including geometry, algebra, statistics, physics, engineering, computer science, and finance. It finds applications in solving equations, analyzing data, calculating standard deviation, manipulating complex numbers, and option pricing, to name a few.
How is the square root symbol (√) used in the Pythagorean theorem?
In the Pythagorean theorem, the square root symbol (√) is used to calculate the length of the hypotenuse in a right-angled triangle. By taking the square root of the sum of the squares of the other two sides, the length of the hypotenuse can be determined.
What is the relationship between the square root symbol (√) and quadratic equations?
The square root symbol (√) plays a crucial role in solving quadratic equations, particularly in determining the nature of their roots. By calculating the discriminant, which involves the square root symbol (√), we can determine if the equation has real roots, repeated roots, or complex roots.
How is the square root symbol (√) used in financial calculations?
In finance, the square root symbol (√) is used to calculate the standard deviation.
Nth roots and polynomial roots
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.
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.
Properties and uses
The graph of the function f(x) = √x, made up of half a parabola with a vertical directrix
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,
- (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 some small integers
* With a different offset.
A note on square roots of positive integers: we can write
where
is squarefree. Then
is given by A000188(n), which we can call the «inner square root» of
, while
is given by A007913(n), and
is the «squarefree kernel» of
, given by A007947(n);
, the «outer square root» of n, is given by A019554(n). For example,
.
Negative Real Numbers
Then the square root of $x$ is defined as:
- $\sqrt x = i \paren {\pm \sqrt {-x} }$
where $i$ is the imaginary unit:
- $i^2 = -1$
Type square root symbol (√) in different devices –
- Using the Equation Editor:
- Go to the “Insert” tab.
- Click on the “Equation” button.
- In the Equation Editor, choose the “Square Root” symbol from the available options.
- Using the Symbol menu:
- Go to the “Insert” tab.
- Click on the “Symbol” button.
- Select “More Symbols” from the drop-down menu.
- In the “Symbol” dialog box, choose “Mathematical Operators” from the “Subset” drop-down menu.
- Look for the square root symbol (√) in the list and click on it.
- Click the “Insert” button to add it to your document.
- Using keyboard shortcuts:
- For newer versions of Microsoft Word (2010 and above) with the Equation Editor enabled, you can use the shortcut
Alt + =
to open the Equation Editor and then type\sqrt
followed by a space to insert the square root symbol. - For older versions of Microsoft Word (prior to 2007), you can use the shortcut
Alt + 251
on the numeric keypad to directly insert the square root symbol (√).
- For newer versions of Microsoft Word (2010 and above) with the Equation Editor enabled, you can use the shortcut
Understand In More Detail
#1 How to type square root symbol
- You will not see the root sign on the main keyboard of your computer or laptop, but this does not mean that it cannot be specified.
- Turn on the numeric keypad that is on the right side of the main keyboard with the Num Lock key.
- Then press the Alt key and, while holding it, type 251 on the numeric keypad and finally Release the Alt key.
- If everything is done correctly, you will see the root symbol.
- By the way, if the right Alt doesn’t work, use the left Alt.
- If you did the whole thing properly, a root sign will appear on the screen. It looks like this: √ (you can simply copy it from here)
#2 Square root symbol in word
- In case you don’t have a numeric keypad or this method will be more convenient for you.
- Press Win + R on your keyboard to bring up the Run window.
- Add charmap.exe command and click OK.
- If everything is done correctly, you will see a symbol table in front of you.
- In the list of symbols, find the root sign by clicking on Advanced view then type square in search box, then click on it and alternately click on the “Select” and “Copy” buttons.
- Then paste the character where you want it on the line.
- Done.
- And yet, the first method seems to us more convenient – provided that the main keyboard is endowed with a numeric keypad (on laptops, for example, it is not so common).
- If you did all in the approved manner, a root sign will appear on the screen. It looks like this: √ (you can simply copy it from here). So you yourself can write many other different emoticons in Word and other text editors.
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.
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.
Geometric construction of the square root
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.
Square roots of some important constants
As before, these are given to 20 decimal places, truncated.
The square root of –1 is the imaginary unit
, and the square root of the imaginary unit is
.