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# +/- Square Roots watch

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1. Hi

I do not understand why sometimes we need to take the positive and negative square roots and other times we only use the positive square roots.

The latest example is a STEP question

Q3, STEP I, 2009

x = (a-x)^1/2

In the markscheme they say by convention we only consider the positive square root.

WHY!!!!!!!!!

I have never really understood this and would appreciate it if someone could it explain it to me

thanks
2. (Original post by bonbon)
Hi

I do not understand why sometimes we need to take the positive and negative square roots and other times we only use the positive square roots.

The latest example is a STEP question

Q3, STEP I, 2009

x = (a-x)^1/2

In the markscheme they say by convention we only consider the positive square root.

WHY!!!!!!!!!

I have never really understood this and would appreciate it if someone could it explain it to me

thanks

Well, an example of why we ignore the negative would be if we were finding the length of a line, as a line can't have a negative length, so that value could just be discarded.
3. When I saw the title of this thread I thought it read "Square Robots". Upon clicking, I was disappointed.
4. Sometimes the negative value doesn't make any sense, like for example in lengths, times, (thermodynamic) temperatures ... anything in which a negative value just doesn't make sense.

Usually in maths, though, at least in Higher (Scottish equivalent to A-level I think) you take both the positive and negative values since it's pure math and not applied to specific application. Actually in many maths questions it's extremely important to consider the negative square root as well!
5. i understand in certain contexts not to take the negative square root...

but why not in the STEP question ha!
6. The function x^2 is not a bijection (basically it's not one to one) so an inverse function doesn't exist: the naive definition of sqrt(x) as both square roots of x doesn't give you a function out the end (as functions are commonly regarded as single-valued).

Hence, we restrict the definition of sqrt(x) to the +ve square root only, since this is the one we use in physical applications and it's a faff writing - signs everywhere.
7. (Original post by around)
The function x^2 is not a bijection (basically it's not one to one) so an inverse function doesn't exist: the naive definition of sqrt(x) as both square roots of x doesn't give you a function out the end (as functions are commonly regarded as single-valued).

Hence, we restrict the definition of sqrt(x) to the +ve square root only, since this is the one we use in physical applications and it's a faff writing - signs everywhere.

ah ok thanks

but sometimes we do consider both roots...
is this only when dealing with numbers rather than functions?

if so surely this means that we miss some solutions to functions?
8. Okay. When you are GIVEN a formula which contains a square root, you use what has been stated. In this case the positive square root is implied as the equation is positive.

When YOU take a square root of something, then you consider both plus and minus.

Depends on what this particular question is asking though.
9. (Original post by COMMUNIST)
Okay. When you are GIVEN a formula which contains a square root, you use what has been stated. In this case the positive square root is implied as the equation is positive.

When YOU take a square root of something, then you consider both plus and minus.

Depends on what this particular question is asking though.

thanks that makes a lot of sense
10. (Original post by bonbon)
but sometimes we do consider both roots...
It is as simple as this. When we write "f(x)", for some function f, we want this answer to be "well-defined" - a fairly specific term for a very simple concept, i.e. we don't want to get different answers when we feed in the same x. So, for example, the function f sending x to x^2 is well-defined: when we input 4, we very definitely always get 16, and don't ever get anything else. The function cos is well-defined: when we input 3pi, we get -1, always, no question. Square rooting is more difficult, so by convention, we always take the positive square root*. Because we don't want to input 16 and sometimes get 4 but sometimes get -4. Similarly, cos-1 (otherwise known as arccos) is defined to always give you a value between 0 and pi. So, even though cos(3pi) = -1, cos-1(-1) = pi; likewise, (-4)^2 = 16, but sqrt(16) = 4.

Sometimes, however, we want to solve equations looking like x^2 = 16. Simple; x = 4 is a solution, and since (-1)^2 = 1, x = -4 is another solution. We haven't "taken positive and negative square roots" (though it's sometimes referred to as that); we've taken the square root, and then noticed that giving it a minus sign also works.

* so long as 'positive' makes sense - so we always do this when working with real numbers, but it's not so obvious with complex numbers; is -1+i 'positive' or 'negative'?
When I saw the title of this thread I thought it read "Square Robots". Upon clicking, I was disappointed.
When I saw the title of the thread I thought I was being summoned.
12. (Original post by bonbon)
thanks that makes a lot of sense
Makes sense, but sadly isn't terribly accurate.
13. (Original post by generalebriety)
It is as simple as this. When we write "f(x)", for some function f, we want this answer to be "well-defined" - a fairly specific term for a very simple concept, i.e. we don't want to get different answers when we feed in the same x. So, for example, the function f sending x to x^2 is well-defined: when we input 4, we very definitely always get 16, and don't ever get anything else. The function cos is well-defined: when we input 3pi, we get -1, always, no question. Square rooting is more difficult, so by convention, we always take the positive square root*. Because we don't want to input 16 and sometimes get 4 but sometimes get -4. Similarly, cos-1 (otherwise known as arccos) is defined to always give you a value between 0 and pi. So, even though cos(3pi) = -1, cos-1(-1) = pi; likewise, (-4)^2 = 16, but sqrt(16) = 4.

Sometimes, however, we want to solve equations looking like x^2 = 16. Simple; x = 4 is a solution, and since (-1)^2 = 1, x = -4 is another solution. We haven't "taken positive and negative square roots" (though it's sometimes referred to as that); we've taken the square root, and then noticed that giving it a minus sign also works.

* so long as 'positive' makes sense - so we always do this when working with real numbers, but it's not so obvious with complex numbers; is -1+i 'positive' or 'negative'?

thanks

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Updated: February 9, 2010
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