Principal Square Roots of Nonnegative Numbers

What is the principal square root of a nonnegative number? Remember that every positive real number has two square roots, a positive square root and a negative square root. So, the equation x^2 = 4 has two solutions. Certainly, x = 2 is a solution because 2^2 = 4, but also x = -2 is a solution because (-2)^2 = 4. We see that both 2 and -2 are square roots of 4, but only one is the principal square root, and that is 2.

The principal square root of a nonnegative real number is its nonnegative square root. The radical symbol, which we will write as sqrt() since we cannot write a real radical symbol here, refers specifically to the principal square root of a number. Thus, sqrt(16) = 4, even though 16 has two square roots, sqrt(16) = 4 because 4 is the principal square root. The negative square root is -4. Also, the principal square root of 0 is 0. Notice that 0 only has one square root.

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

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[佐藤広-qu]

It is bad as a strictly mathematical idea, but I understand it as follows. Elementary school students who haven't learned factorization look at the equation "x²=4", think that it's multiplied by the same numbers to get 4, so, 2×2=4 and the answer is 2. However, if you are a junior high school student or older who learned factorization, x²=4 ⇒x²-4=0 ⇒x²-2²=0 ⇒(x-2)(x+2)=0 ⇒x-2=0 or x+ 2=0 ⇒ x=2 or x=-2 and answer that the roots are 2 and -2. That is, x²=4=(2)²=(-2)². Then use √ to find x, that is what should be squared to get the number in √. Then, √4=√(2)²=√(-2)², so some people may mistake √4 for 2 and -2. But you should think about what will happen. If √4 is 2 and -2, how should you think about "-√4"? Is the positive and negative just reversed (+/-→-/+)? Now let's do a simple calculation. For example, if √25-√4, in addition to (5)-(2)=(3), but also (-5)-(2)=(-7), (5)-(-2)=(7), (-5)-(-2)=(-3) come out as a result. Furthermore, when the equations are longer or involve multiplication and division, simple numerical calculations turn into ridiculously complex calculations. It is natural to think that this is STRANGE. In other words, if the symbol √ represents only the positive side of the squared number in √, √4=2, -√4=-2, it will be solved and satisfied.

[nicklzz]

Thank you so much, this video helped me so much, none of the other videos helped me with this topic only your video did, thanks again!!!

[asalamkamal]

Why is root four only positive two simply because this is a simple arithmetic operation that doesn't make sense to have two outputs

[Goosedestroy]

The most helpful math vid on this subject, thanks for making it so simple to understand! :)

[saylemevelyn]

so what im getting is, the sq root CAN BE a negative, but the radical operation is specifically only referring to the positive answer

[williamstier]

Actually the postive principle root applies to negatives as well the only difference is you would do a 90° rotation around a 2D plane.

[iwillgotoiit]

i always had a doubt y x^2 = mod x now i got it...thanks bro alot

[ankitbasera]

Why √16 ! =(Not equal to) -4?
√-4×-4 = √-4 × √-4 = i√4 × i√4[i=√-1]
= - √4×4 = -√16
See contradictions

[annabellez]

Thank you so much. None of the other videos I saw fully explained it

[AbrahamLozadaabe]

In real variable, we don't consider multivalued function s. It has not the usefulness that has in complex variable. The square root is the inverse function for the positive reals of squaring them. By definition, its domain are the positive reals. To define a function you must specify its domain

[SolutionsHubYT]

Hi, I want to ask a question
If Square root refers to only positive value. Then how can
Square root of x^2 be equal to modulus of x i.e |x|.
As the absolute or modulus is a piecewise function defined for 2 values.?

[ranveer]

Then the value of √x² as well as √4 should also be a positive value and when we simplify this equation-
x²=4
→√x²=√4
→x=2

How do we get ±2?

[ochanrut]

my question is that if a is negative and n is an even natural number, we say square root of a of index n is not a real number, but we can say it is a real number if n is an odd natural number?

[RiteshkrRawat-dvdt]

sir plzz clear my doubt its very said root over 4 is only 2....but in quadratic eqtns we use + and - both...then what is the solution...plzz tell me sir

[tshepomotau]

Why are we allowed to say (-4)×(-4)=16 but we are not allowed to say -4 is the square root of 16? I fully understand the principal square root notion but would I be wrong to say the square root of 16 is also -4? Some tutors in you tube are saying the square root of 16 is 4 or -4.

[hetanshthakore]

Sir if we have to solve
X²=4
and we dont want to factor (and instead solve by taking roots on both sides) it will be
|x|= ± square root of 4
Right?

[aliatulatiqah]

Hi, can I know what the difference between square root 4 and square root 2^2. Does both have the same anws? Which were +2 and -2. Thanks for replying if you see this comment.

[pajikaur]

Do you mean we consider all solutions in case of equality and only principal solutions otherwise?

[Qwerty-lqop]

This isn’t actually answer my question :/
The thing that i still don’t understand is that why it only take the positive one? Why did the negative one is ignored? And why do we have to follow the principal square root?

[chazy]

Sir, can you please make a video on How to become Maths topper?