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If adding is the sum of two numbers, Subtraction is the removal of a number from another.

Multiplication is $x$ lots of $y$.

Division is splitting $x$ between $y$.

Powers are multiplying a value by itself.

What is a square root?

To be clear, I'm asking what function a square root actually performs?

I can't seem to come up with an answer without the concept of powers to refer to.

For example; how would you go about solving the square root of $121$ without prior knowledge that $11^2$ is $121$?

Is it as blind as testing every value up to or past an answer until we find a solution then doing bisection until we get a result?

Edit: I think my question was still unclear, how could I perform a square root on a value without reverse engineering the question into a powers question. E.g. square root $123456$, you do not know if the root is whole or decimal.

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NB your definitions of multiplication, division, and powers only make sense as stated for nonnegative integers. If you're happy to extend them all to rational values, then the square root map is just raising to the power $\frac{1}{2}$. – Travis Mar 7 at 13:29
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For that matter, how does one divide (i.e. "split $x$ between $y$") without "prior knowledge that $x = yz$"? "Is it as blind as testing every value up to or past an answer until we find a solution then doing bisection until we get a result?" – Travis Mar 7 at 13:31
    
Take $x$ lots of $x$ till you get $y$ ... may be? – Win Vineeth Mar 7 at 13:32
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Well, I don't think you can avoid falling back to powers, because the square root is shorthand for a particular power (specifically, one with an exponent of 0.5). So, it seems to me like the core of your question would be answered with an answer on how to calculate powers with real exponents. See this: math.stackexchange.com/questions/280901/calculating-powers – Theodoros Chatzigiannakis Mar 7 at 15:58
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They used to teach how to compute square roots manually to arbitrary precision in grade school. It looks superficially like long division. – Matt Samuel Mar 8 at 3:42
up vote 28 down vote accepted

As I don't fully understand your question, I will divide my answer into three parts: the definition of a square root, how to understand it, and algorithms to calculate it.

DEFINITION OF A SQUARE ROOT

To cite Wikipedia,

In mathematics, a square root of a number $a$ is a number $y$ such that $y^2=a$, in other words a number $y$ whose square $(y \times y)$ is $a$. For example, $-4$ and $4$ are both square roots of $16$ because $(4)^2=(-4)^2=16$.

Every non negative real number $a$ has a unique non-negative square root, called the principal square root, which is denoted by $\sqrt{a}$ where the $\sqrt{\square}$ sign is called a radical sign or a radix. For example, the principal square root of $9$ is $3$, denoted $\sqrt{9}=3$, because $3^2=9$ and $3$ is non negative. The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example $9$.

HOW TO UNDERSTAND IT

A way to approach this is thinking by reversal. What you do, someone can undo.

A $\color {blue}{\text{summation}}$ is the reverse of $\color {blue}{\text{subtraction}}$. Indeed, $\color {blue}{\text{subtraction}}$ can 'undo' a $\color {blue}{\text{summation}}$, and $\color {blue}{\text{summation}}$ 'undoes' a $\color {blue}{\text{subtraction}}$.

$\color{green}{\text{Division}}$ is the reverse of $\color{green}{\text{multiplication}}$. $\color{green}{\text{Division}}$ undoes a $\color{green}{\text{multiplication}}$, $\color{green}{\text{Multiplication}}$ can be 'undone' with a $\color{green}{\text{division}}$.

Think of a $\color {orange}{\text{square root}}$ as a reversal of a $\color {orange}{\text{square}}$: you can "undo" a $\color {orange}{\text{square}}$ with a $\color {orange}{\text{square root}}$, and a $\color {orange}{\text{square}}$ can "undo" a $\color {orange}{\text{square root}}$. This may lead to some confusion, as $x^2$ is not invertible. To clarify, think of the square root of $16=4^2=(-4)^2$. A square root is not a function. Thus taking the square root of $16$ undoes the square, yielding two values- it becomes $4$, $-4$.

So to summarize $$\color {blue}{\text{summation}} \leftrightarrow \color {blue}{\text{subtraction}}$$$$\color{green}{\text{Division}} \leftrightarrow \color{green}{\text{multiplication}}$$ $$ \color {orange}{\text{square root}} \leftrightarrow \color {orange}{\text{square}}$$

ALGORITHIMS TO FIND IT

If you want to determine if it is whole or irrational, if a integer is between two squares, then it is decimal. In your example, since $351^2=123201<123456<123904=352^2$, it is irrational.

Of course, there are more clever arguments. For example, if $3$ divides $n$ but $n$ is not divisible by $9$, this implies $n$ is not a square. This is true in case of $n=123456$. Other examples include $$n \equiv 2 \pmod 3 \Rightarrow n \quad \text{is not a square}$$ $$n \equiv 2,3 \pmod 5 \Rightarrow n \quad \text{is not a square}$$ But note these methods are not always easy-and sometimes require luck.

A operation for finding (or approximating) square roots exists, not to different from long division. A simple one can be seen here.

Of course, the quickest way to approximate it is using the sequence {$x_{n}$} which is defined as follows.

$$x_{n+1}=\frac{1}{2}(x_{n}+\frac{c}{x_{n}})$$

It is not to hard to see that $\lim_{x \rightarrow \infty}x_{n}=\sqrt{c}$. This sequence actually converges pretty quickly, so this is useful, which is equivalent to Newton's Method in $x^2-N$.

There are, of course, other algorithims to calculate the square root. And there are some rather exclusive ones as well. For example, $$\lim_{n \rightarrow \infty}{\frac{F_{n+1}}{F_{n}}+\frac{F_{n}}{F_{n+1}}}=\sqrt{5}$$ Where $F_{n}$ is the Fibonacci Sequence. But not converge quite as quickly as the one above.

More discussion on the definition, and the algorithims can be found in the links provided.

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It is not quite true that a square root is the undoing of the operation of squaring. The reason is that squaring is not an invertible function. (-5)^2 = 25 but the square root of 25 is +5. – DavidC Mar 7 at 16:24
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I hate when MathJax is used to render normal text. – kamil09875 Mar 7 at 16:31
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@DavidC but you are wrong that the square root of $25$ is $5$-that is incorrect. It is both $5$ and $-5$. – MXYMXY Mar 7 at 16:31
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Bear in mind that about 5-10% of the male population is colorblind to one degree or another. If you must use colors, blue is pretty safe, red not quite so and that bown color you use is a really poor choice. – Rick Decker Mar 7 at 19:09
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@MSYMXY, If you take the view that the square root is a function (or an operation), as I do, then there can only be a single element from the co-domain for any given element of the domain. (Expressed differently, there can only be one output for any input.) Am I missing something? – DavidC Mar 7 at 21:47

I think you're confusing the definition of a square root with the algorithm needed to compute it. In mathematics, there are many concepts that we can define, but where it may be hard to actually find some object satisfying the property.

A square root of a number $n$ is defined as any number $x$ that satifies this property: $$x^2 = n.$$

For example, we can verify that $2$ and $-2$ are square roots of $4$.

Often we talk about the square root, which is an abbreviation for the term principal square root. By that we mean the simple convention that $\sqrt{4} = 2$ (and not $-2$). But that is just a convention.

Actually finding a square root is more complex. Many algorithms will use somewhat higher mathematics. But the gist of it is: take a number $a$ whose square is less than $n$ and another number $b$ whose square is bigger than $n$ and then you know that the square root of $n$ must be between those two. Repeat until you have enough accuracy as needed (you will rarely ever be done, since most square roots are irrational).

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In other words, yes, it’s basically blind-testing until you get there. (Right?) – Janus Bahs Jacquet Mar 7 at 23:59
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It's not so much "blind testing" as using an algorithm that is guaranteed to get you closer and closer to the answer the more often you apply it. But in essence, it is some sort of "advanced guesswork"; only, you can be clever about it, see MXYMXY's answer for details. – Fryie Mar 8 at 0:01

If you think at the square power $l^2$ as the area of the square of side $l$, than the square root is to find the side of a square of given area. This seams the more intuitive interpretation of the square root for positive real numbers.

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You are asking "Is it as blind as testing every value up to or past an answer until we find a solution then doing bisection until we get a result?", "How could I perform a square root on a value without reverse engineering the question into a powers question?"

A very similar situation occurs with division, which is defined by reference to multiplication. How do I known if $\dfrac{123456}{643}$ is whole or decimal ? There's nothing more mysterious about [square] roots.

As there are numerical algorithms that allow you to perform divisions, there are some for roots. You can also resort to logarithms/antilogarithms.

And, yes, extracting a root requires to somehow solve a power equation, in the same sense that performing a division requires to solve a multiplicative equation.

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I like This way, which is a cute tool for mental square roots (a bit approximate but it may works for daily use).

Call $N$ the number you have to root and think about the nearest perfect square $s^2$. Then you can always write

$$N = s^2 \pm q$$

There is the approximation:

$$\sqrt{N} = \sqrt{s^2 \pm q} \approx s \pm \frac{q}{2s}$$

Example

Suppose we want to find an approximation of the square root of $87$ (which is $9.3274..$). Then the nearest perfect square could be $81$:

$$\sqrt{87} = \sqrt{81 + 6} \approx 9 + \frac{6}{18} = 9 + \frac{1}{3} = 9.3333333...$$

On the same way on could take $100$ as the nearest perfect square:

$$\sqrt{87} = \sqrt{100 - 13} \approx 10 - \frac{13}{20} = 10 - 0.65 = 9.35$$

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To answer in the same fashion you asked:

To square a number is to convert a line into a square--for instance a line of 10 blocks made "Square" will create a square of 100 blocks.

To take a "Square" of 25 blocks and reduce it to a line would give you 5 blocks.

There are many better answers here, but I didn't see one explaining it in the same way you asked..

Edit: I just realized I could have said it more clearly with better wording:

Squaring is the operation used to find the area of a square from the length of a side.

Square root is finding the length of a side from the area of a square.

This is why we use the name "Square".

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You need square root of x. What you are doing can be explained in a simple way.Say you have x kg of wheat and you are breaking this into n groups of n kg each.Now we have n groups of n kg implying ${n}^{2} $ kg which is equal to x kg. Hence, we find square root of x is n.So, I would say that square root would be to divide the number x in n equal parts such that each group has value n.This is a relatively simple and qualitative argument.Hope this helps you in understanding.

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I'm not 100% sure what you are asking but I offer up the following two thoughts in case they help. Considering the positive integer x:

1) Add up odd numbers until the sum is greater than or equal to x. Let n be the number of addends (the number of odd numbers you summed). If you hit x exactly, then the square root is n. If you overshot, then the square root is between n and n-1. This is sort of a computational approach.

2) If you want to make a square with area x, then square roots tell you how long the sides need to be. This is sort of a functional approach.

My guess is that you want to explain the concept to someone who isn't particularly inclined to math so I'm not being very formal. Again, I hope this is of some help.

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