A strange inconsistency with square roots

In summary, we discussed the concept of taking the square root of an expression in which a constant is multiplied by an expression within a square root, such as ##4\sqrt{16}##. We saw that when the constant is positive, we can square it and push it into the square root, but when it is negative, this approach leads to a contradiction. We also cleared up some common misconceptions about square roots, such as the idea that taking the square root of a positive number yields two answers, and the misconception that ##\sqrt{-1} = \pm i##.
  • #1
ELB27
117
15
Hi,
I have a question that came into my mind while solving some problems. If I have a constant times an expression in a square root like ##4\sqrt{16}## I can square the constant and push it into the square root: ##4\sqrt{16}=\sqrt{4^2 16} = 16##. But what if the constant outside of the square root is negative? Then I will get a contradiction: ##-4\sqrt{16} = \sqrt{(-4)^2 16} = \sqrt{4^2 16} = 16 ≠ -16##. Why is this happening? Clearly I cannot square negative numbers the same way as I do with positive ones, but why?
Any comments will be appreciated!
 
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  • #2
You have to remember that when you take the square root of a positive number there are always two answers one positive and one negative.
 
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  • #3
ELB27 said:
Hi,
I have a question that came into my mind while solving some problems. If I have a constant times an expression in a square root like ##4\sqrt{16}## I can square the constant and push it into the square root: ##4\sqrt{16}=\sqrt{4^2 16} = 16##. But what if the constant outside of the square root is negative? Then I will get a contradiction: ##-4\sqrt{16} = \sqrt{(-4)^2 16} = \sqrt{4^2 16} = 16 ≠ -16##. Why is this happening? Clearly I cannot square negative numbers the same way as I do with positive ones, but why?
Any comments will be appreciated!
For starters, let's limit the discussion to the square roots of nonnegative numbers.

A property of square roots that you're using is this: ##\sqrt{ab} = \sqrt{a}\sqrt{b}##. This is valid only when a and b are nonnegative.

The problem your work is that ##\sqrt{(-4)^2} \neq -4 ##, so the first equality in your equation above isn't valid.
 
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  • #4
jedishrfu said:
You have to remember that when you take the square root of a positive number there are always two answers one positive and one negative.
I disagree. When you take the square root of a positive number, you get a single (positive) number. For example, ##\sqrt{9} = 3##. It's not ##\pm 3##. The square root symbol represents the principal (i.e., positive) square root of its argument.
 
  • #5
Mark44 said:
I disagree. When you take the square root of a positive number, you get a single (positive) number. For example, ##\sqrt{9} = 3##. It's not ##\pm 3##. The square root symbol represents the principal (i.e., positive) square root of its argument.

You're right, I was thinking of square root of x^2 where we don't know whether x is positive or negative.
 
  • #6
jedishrfu said:
You're right, I was thinking of square root of x^2 where we don't know whether x is positive or negative.
And it's the same here, meaning that you get one answer:

##\sqrt{x^2} = |x|##

If the value of x is not known, you can't say the following
##\sqrt{x^2} = x##

Also, the following isn't true
##\sqrt{x^2} = \pm x##
 
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  • #7
Mark44 said:
And it's the same here, meaning that you get one answer:

##\sqrt{x^2} = |x|##

If the value of x is not known, you can't say the following
##\sqrt{x^2} = x##

Also, the following isn't true
##\sqrt{x^2} = \pm x##

Okay got it. I keep forgetting the principal square root is the convention.

http://mathworld.wolfram.com/PrincipalSquareRoot.html

This must be one of the reasons I majored in Physics and not Math.
 
  • #8
if you define square root x1/2 as the inverse function of square x2, then yes, there will be two answers, and one will be the negative of the other.
However, if you remember your algebra teacher telling you about the "vertical line test"(if somewhere on the graph you have 2 outputs=y for one input=x), then square root is not a function at all, because it has 2 outputs for each input.In case your wondering, a justification for why a negative times a negative is a positive:

suppose the money in your bank account is increasing by x dollars per year.

then x*n=amount of money you made in n years.

if x was negative, then it would mean you are losing money
then x*n would be negative

Heres where this comes into play

If n=number of years was negative
and x=money per year was positve,
then x*n would be negative because it would show how much money you would lose if you went back in time(when you are actually gaining money as time moves forward).

so, if both x and n were negative?
then because x is negative, you are losing money,
but because n is negative, you are also going backward in time
therefore you are gaining money as you go backwards in time.

Its kinda like how if you play country music backwards the guy gets his truck fixed, gets married, goes sober, and his dog comes back to life. :p
 
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  • #9
DivergentSpectrum said:
if you define square root x1/2 as the inverse function of square x2, then yes, there will be two answers, and one will be the negative of the other.
No, this isn't right, either. The squaring function is not one-to-one, so doesn't have an inverse function.
DivergentSpectrum said:
However, if you remember your algebra teacher telling you about the "vertical line test"(if somewhere on the graph you have 2 outputs=y for one input=x), then square root is not a function at all, because it has 2 outputs for each input.
 
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  • #10
Thanks very much for the replies!
So if I understand correctly, I was unconsciously missing the step ##4\sqrt{16}=\sqrt{4^2}\sqrt{16}## even with the positive number and just focused on pushing it into the square root, and thus the flawed step with the negative case was ##-4\sqrt{16}≠\sqrt{(-4)^2}\sqrt{16}##?
 
  • #11
ELB27 said:
Thanks very much for the replies!
So if I understand correctly, I was unconsciously missing the step ##4\sqrt{16}=\sqrt{4^2}\sqrt{16}## even with the positive number and just focused on pushing it into the square root, and thus the flawed step with the negative case was ##-4\sqrt{16}≠\sqrt{(-4)^2}\sqrt{16}##?
Right.

We get questions about square roots like this fairly often, and there are a lot of misconceptions around this concept, that seem to lead to contradictions. One of these is this one:
##1 = \sqrt{1 * 1} = \sqrt{(-1)(-1)} = \sqrt{-1} \sqrt{-1} = i * i = -1 ##
which appears to show that 1 and -1 are the same number. This breaks down at the step ##\sqrt{(-1)(-1)} = \sqrt{-1} \sqrt{-1}## for reasons I already gave.

Another misconception that shows up a lot is that, for example, ##\sqrt{4} = \pm 2##. The root of this confusion, I believe, is that we're taught that every positive real number has two square roots. As already mentioned, though, by definition and common usage, the √ symbol means the principal, or positive square root.
 
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  • #12
im pretty sure if you put the numbers into polar coordinates
z=re
you don't need to be as careful with these steps.
so if
a=raea
and
b=rbeb
and i want to find
sqrt(a)*sqrt(b)
then i have
(raea)1/2(rbeb)1/2
then i can still use the property sqrt(a)sqrt(a)=sqrt(ab) to get:

(rarbeaeb)1/2

because e^(x)e^(y)=e^(x+y) i can change this to:

((rarb)ei(θab))1/2
then because (e^x)^y=e^(xy), i can say:

(rarb)1/2ei(θab)/2now z is positive if θ=0, and negative if θ=pi
so, if both are negative, then (θab)/2=pi
so we have eipi=-1
then because ra is just |a| and rb=|b|
we have -(|a||b|)1/2, which is correct for sqrt(a)*sqrt(b) where a and b are negative, even though i used sqrt(a)sqrt(a)=sqrt(ab)

to be honest, its a lot more work than is needed :p
 
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  • #13
DivergentSpectrum said:
then because (e^x)^y=e^(xy)

This is not true for arbitrary complex x or y. See wikipedia.
 
  • #14
pwsnafu said:
This is not true for arbitrary complex x or y. See wikipedia.
also i forgot to take the root of unity lol

i think the whole axbx=(ab)x thing can be reduced to the idea that if f(x)=a and b, it doesn't necessarily mean a has to equal b, but from what i read on the page it seems to be implying that the inconsistencies with
(e^x)^y=e^(xy)
can be explained and the real answer is
2f8b3e67ccc1d2ce6c6dfc4d84f0cdd8.png


while the inconsistencies with
ln(b^x)=x*ln(b)

cant be explained with a multivalued interperetation. pretty strange
 
  • #15
DivergentSpectrum said:
In case your wondering, a justification for why a negative times a negative is a positive:

suppose the money in your bank account is increasing by x dollars per year.

then x*n=amount of money you made in n years.

if x was negative, then it would mean you are losing money
then x*n would be negative

Heres where this comes into play

If n=number of years was negative
and x=money per year was positve,
then x*n would be negative because it would show how much money you would lose if you went back in time(when you are actually gaining money as time moves forward).

so, if both x and n were negative?
then because x is negative, you are losing money,
but because n is negative, you are also going backward in time
therefore you are gaining money as you go backwards in time.

Its kinda like how if you play country music backwards the guy gets his truck fixed, gets married, goes sober, and his dog comes back to life. :p

That's a pretty interesting "intuitive" explanation, but the fact does follow immediately from the distributive property.
 
  • #16
Square root always returns a non-negative number. It's also only defined on non-negative numbers. Not every step is reversible. When you square a number and then take its square root, you lose information about the sign of the number, since this will always return a non-negative number.
 
  • #17
Mark44: Edited by adding quote tags for the statements that statdad is citing.
nucl34rgg said:
Square root always returns a non-negative number.
If you are referring to the principal square root (commonly written [itex] \sqrt x [/itex]), but even in the realm of real numbers something like 25 has two square roots: 5 and -5.

nucl34rgg said:
It's also only defined on non-negative numbers.
If you are working in the realm of reals for domain and range, yes.

nucl34rgg said:
Not every step is reversible. When you square a number and then take its square root, you lose information about the sign of the number, since this will always return a non-negative number.
Yes (again, in the reals), but this is more clearly seen as a feature of the fact that [itex] y = x^2 [/itex] is not a one-to-one function. (The others are also, in sense).

But the point of all this is?
 
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What is a strange inconsistency with square roots?

A strange inconsistency with square roots refers to the fact that when taking the square root of a number, there are two possible answers: a positive and a negative. This is because a negative number squared also gives a positive result. This inconsistency can cause confusion and errors in calculations.

Why does this inconsistency occur?

This inconsistency occurs because of the definition of the square root function. The square root of a number is defined as the number that, when multiplied by itself, gives the original number. However, both positive and negative numbers satisfy this definition, resulting in the two possible answers.

How can this inconsistency be resolved?

To resolve this inconsistency, we can use the concept of absolute value. The absolute value of a number is always positive, so when taking the square root of a number, we can use the absolute value to ensure that we only get the positive root.

What are the implications of this inconsistency?

This inconsistency can have significant implications in various fields, such as mathematics, physics, and engineering. It can lead to incorrect solutions, especially when dealing with complex equations and problems.

Are there any real-life applications of this inconsistency?

Yes, there are real-life applications of this inconsistency, particularly in areas that involve measurements and calculations, such as construction, finance, and scientific research. It is crucial to be aware of this inconsistency and take necessary precautions to avoid errors.

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