Square root Definition and 383 Threads

In mathematics, a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 42 = (−4)2 = 16.
Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by





x


,


{\displaystyle {\sqrt {x}},}
where the symbol















{\displaystyle {\sqrt {~^{~}}}}
is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by





9


=
3
,


{\displaystyle {\sqrt {9}}=3,}
because 32 = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.
Every positive number x has two square roots:





x


,


{\displaystyle {\sqrt {x}},}
which is positive, and






x


,


{\displaystyle -{\sqrt {x}},}
which is negative. Together, these two roots are denoted as



±


x




{\displaystyle \pm {\sqrt {x}}}
(see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive x, the principal square root can also be written in exponent notation, as x1/2.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.

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  1. G

    Square Root in Special Relativity

    Can anyone explain what the square root does? I know 3 squared equals the square root of 9. I want to learn what is actually happening behind the math, if that makes sense. Consider the following: I’m calculating the wavelength of light, and traveling towards the source. Speed of light =...
  2. M

    Finding F'(x) for Square Root of x

    Homework Statement F(x)= square root of x what would F'(x)=? (F prime of x) Homework Equations F(x)= 3x^2 F'(x)=6x F(x)= x^3 F'(x)=3x^2 The Attempt at a Solution no clue. If the variable is a square root, it wouldn't have an exponent, right? confused.
  3. A

    Absolute value and square root

    what is /squared root sign4x+1 / what does this equal because I'm confused when it has the square root sign on it all and it's absolute value
  4. L

    Square Root of Complex Numbers

    Hi! I've got a question. There is a nice formula for finding square roots of arbitrary complex numbers z=a+bi: \frac{1}{\sqrt{2}}(\epsilon\sqrt{|z|+a}+i\sqrt{|z|-a}) where epsilon:=sing(b) if b≠0 or epsilon:=1 if b=0. I've just looked it up and it's nice to use it to find complex roots of...
  5. murshid_islam

    What is the square root of i^2

    what does \sqrt{i^2} equal to? is \sqrt{i^2} = i or \sqrt{i^2} = \pm i?
  6. U

    Is the square root of a prime number always going to be irrational?

    is the square root of a prime number always going to be irrational? just a random question.
  7. T

    Finding the Greatest Integer Square Root of an Integer

    Consider the square root operation. Suppose an integer numbers i > 0 as input variable. Design an algorithm which calculates the greatest natural number less than or equal to the square root of the input variable i. can smby pls explain to me what does this ques mean??if possible explain...
  8. P

    How can I simplify sqrt( 4t^2 + 4 + ( 1/t ) ) to (2t^2 + 1)/t?

    Couldn’t decide if I should put this in the calculus or general math forums but... I’m studying for a final that’s coming up this Wednesday and I’ve been looking at some past quizzes with the steps to finding the solutions that my instructor has posted online. Given the problem: 1. Compute...
  9. P

    Find the square root of a quantity in what unit

    Hey all, this is my first post here, I have just discovered the site, and it looks great :smile: I need some help on units, regarding square roots. When you find the reciprocal of a quantity, like say, mass, the new unit is kg-1. I'm not sure about when you find the square root of a...
  10. J

    How Do You Integrate the Square Root of Tan x?

    My class, teacher included, cannot seem to figure out the integral of the square root of tan x. Maybe someone here can help? thanks, jeremy
  11. C

    Sequence based on sequential square root function

    Express each term of the sequence \{\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, ...\} as a power of 2. I found \{2^{\frac{1}{2}}, 2^{\frac{3}{4}}, 2^{\frac{7}{8}}, ...\} but I can't get the formula for it so I can find it's limit.
  12. N

    Is the Book Wrong? - Square Root of 4

    hey guys on my textbook, it says that square root of 4 equals to 2 but not negative 2. The book is wrong right?
  13. D

    Another simple square root problem

    why is the square root of 25 just 5 but when the question x^2 = 25, the answer becomes +/- 5? whats the logic here? i don't quite get it. thanks.
  14. D

    Can You Multiply Square Roots of Negative Numbers?

    suppose you have square root of -16 times square root of -25 (in separate square root symbols). would the answer be: a) no solution - can't have negative square roots b) 4i times 5i which equals 20i^2 which is -20 c) -16 times -25 which is +400, then the square root of that is 20...
  15. L

    Irrationality of the square root of a prime

    I saw a proof saying the root of a prime is always irrational, and it went something like this: sqrt(r) = p/q where p/q is reduced r = p^2/q^2 r*q^2 = p^2 therefore, r divides p so define p = c*r r*q^2 = (c*r)^2 = c^2*r^2 q^2 = c^2*r therefore, r divides q also since r divides p...
  16. N

    Prove the Square Root of 2 is irrational

    This is Algebra 2 question... I have to prove that the square root of 2 is irrational... First we must assume that sqrt (2) = a/b I never took geometry and i don't know proofs... Please help me. Thank you.
  17. T

    Unraveling the Mystery of Solving Square Roots

    I'm just wondering about something, look at this: \[ \begin{array}{l} \sqrt {3 + x} + \sqrt {2 - x} = 3 \\ \left( {\sqrt {3 + x} } \right)^2 + \left( {\sqrt {2 - x} } \right)^2 = 3^2 \\ 3 + x + 2 - x = 9 \\ 5 = 9 \\ \end{array} \] Obviously something isn't right...
  18. E

    Draw a Line of Length √3: Tips from an Old Man

    An old man gives you a set square (http://www.buchhandlung-umbach.de/pbs/geodreick.gif ) and then asks you to draw a line of exactly the length \sqrt{3} . How would you do it?
  19. K

    F(t) = square root of 3t+2 / 4t-3

    Need to find the derivative of: f(t) = square root of 3t+2 / 4t-3 please help.
  20. C

    Can a Triangle Have Sides of 1m and Hypotenuse of √2m?

    Square root of 2... Is it actually possible to produce a right angled triangle with sides exactly equal to 1m and 1m? Because this would produce a hypotenuse with length "square root of 2" m, which has no exact length. Thanks in advance. :smile:
  21. C

    Solving the Mystery of i & j: The Square Root of -1

    Square root of -1... We say that the square root of -1 is equal to i ( or j ), and that this is therefore not a real number - but what is this fact actually useful for? Why over complecate things with something not on the number line - why is it so useful to treat the square root of -1 as a...
  22. T

    I'm sorry, I'm not sure what your question is. Could you clarify?

    Hi, I'm trying to find this integral: \int \frac{x}{(x+1)\sqrt{1-x^2}}\ dx Because 1-x^2 has two different real solutions, I can write \sqrt{ax^2 + bx + c} = \sqrt{-a}(x_2 - x)\sqrt{\frac{x-x_1}{x_2 - x}} so \sqrt{1-x^2} = (-1 - x)\sqrt{\frac{x - 1}{-1 - x}} = (-1 -...
  23. A

    Square Packing Solutions for 24 Integer Squares

    \begin{array}{c} {{A_n}={\sqrt{\sum _{z=1}^{n}{z^2}}} } \\ {{A_1}=1 } \\ {{A_{24}}=70}\end{array}\ Is there a proof that only for n =1 or n=24 that An is an integer quantity?
  24. A

    Understanding Imaginary Roots in Complex Numbers

    hi i was trying to determine the sqrt of -1,and here is wat i have done. sqrt of -1=-1^2/4,this further gives [[-1]^2]^1/4, which further gives 1^1/4 therefore the sqrt of -1 is 1^1/4 which is 1 have u any objections,let me know
  25. D

    What is the Simplest Way to Find the Square Root of i?

    Does i, the imaginary number, have a square root? This was bothering me for a while, then I thought I happened upon a simple solution, but have since forgetten. \sqrt{i}=?
  26. M

    Find the square root of negative numbers

    I don't get the use of imaginary numbers. To find the square root of negative numbers but it does not exist and it is not a real number. Can u please explain it to me.
  27. T

    Square Root of x^9 ≠ x^4.5: Exploring the Difference

    Why is it that the square root of x^9 for some x isn't always the same as x^(4,5) ? I tried to do this with x=12 on my TI-89 and it comes very close, the difference comes after like the 8th decimal or something, but shouldn't it be excactly the same?
  28. L

    The irrationality of the square root of 2

    I saw this proof in class today to prove the square root of 2 is irrational: 1. Assume that √2 is a rational number. Meaning that there exists an integer a and b so that a / b = √2. 2. Then √2 can be written as an irreducible fraction (the fraction is shortened as much as possible) a...
  29. A

    Find Square Root: 5+12i Calculation

    does anyone know how to find the square root of 5+12i ?
  30. C

    Proof of limit involving square root

    Hello all I am having trouble proving the limit of the following: lim sqrt(( n+1) - sqrt(n)) * sqrt(n+ 1/2 ) = 1/2 n --> 00 I tried using the fact the the limit of the first factor as n approaches infinity is 0. Then I tried expressing the first factor as 1 /...
  31. A

    Is the Square Root of 0 Undefined?

    Is the square root of 0 undefined? I was recently told that the square root of 0 is undefined because the limit of a square root didn't exist at 0. The reason is that from the negative direction you have i and from the positive direction you don't. At first, i agreed with this. It made enough...
  32. O

    Simple square root factoring question

    Hi, is \sqrt{a^2-a^2\sin^2{x}} = a\cos{x}? If not, what should it be? :confused: Appreciate the help! :smile:
  33. I

    Square Root of Pi: Is it Possible?

    The square root of pi is 1.777245... I know. But my math teacher says its impossible to determine a square root of an irrational number. Can anyone shed any light on this? is it or is it not possible to determine the square root (or any root) of an irrational number?
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