Amazon.com Widgets

# Can the square root of 1 be -1

Can the square root of 1 be -1
• ## The answer is what you make of it.

The radical symbol on its own means nothing; it is a shape of lines only.
The phrase "square root" also doesn't mean anything based on the words making it up.

A message conveyed between two people through a medium such as a language of words or symbols does not have intrinsic meaning derived from the way it is represented in the medium. The sender and receiver interpret what the message is SUPPOSED to mean, and that's all a meaning is.

The letter A does not mean anything except a component of words or an article to most people, but if a teacher sends the letter A to a student on a sheet of paper which was previously submitted by the student as an assignment, the teacher and the student both take the letter to mean that the student has done well.

If one who does not speak English is told that his shirt is on backwards (just giving a random example here), he might think it was a friendly compliment and feel happier for the day. It didn't mean the same thing to the non-English-speaker as it meant to the English-speaker.

This can be applied to the concept of the radical sign and the term "square root". It doesn't always mean the same thing to different people. An authority figure could say that it means something to him, and this meaning could be widely accepted, but it might not be universally accepted. The meaning of the operation is what it is thought to be by the person using it.

Therefore, if someone believes that the square root operation means only the principal root (the positive answer), and that this operation is therefore a function, he would be correct in saying it means that to him. If someone believes that the square root operation means the inverse of squaring, so it means the positive and negative root, he would also be correct in saying it means that to him.

But no one can say it means something by virtue of itself. It represents a concept to those who interpret it to do so. On an alien planet, square root might not mean anything at all. On an alien planet, square root might mean an obscenity. It can mean whatever it is believed to mean.

I believe the principal square root and the other square root are two DIFFERENT operations, therefore, and depending on context and what I think the writer of the operation intended, I could take "square root" or the radical symbol to mean either operation.

I put this under the YES column only because it had to be under a column, and because it only asked if the square root CAN be -1. It CAN mean that. That doesn't mean the NO column people are wrong.

• ## -1 x -1=1

When asking this question, you need to consider what a square root is. Well, the square root of a number is whatever is multiplied by itself to get that number. Basically, the sq. Rt. Of a = b*b. 1= -1*-1 is a true statement, so -1 is a square root of 1. It doesn't matter if we want it to be a function. -1 is still a solution to the square root of 1

• ## Really Makes Sense

Negative One Multiplied By Negative One Equals To One So Ummmmmmm Ya....... I Guess, I Have Never Of This But When i Hear It. What Pops in my mind is that its Correct, So My Question Is The Square root of 4 Possible to Be -2?
I'm really Confused..... But I support it!
:)

• ## This is a mathematical fact.

Since -1 squared equals 1, it is necessarily true that -1 is a square root of 1. The other square root is 1. It would be most accurate to say that -1 is "a square root of 1" rather than it is "the square root of 1". I think that would be the only room for debate here, except the question as asked is "can the square root of 1 be -1". It can be, but it can also be 1.

• ## Is this a serious question?

Squaring a number is simply taking a number and multiplying it by itself. So when you find a square root of an number you are looking for a the same number that can be multiplied together to get what is being square rooted. So when you take the number 1 and find the square root of it you can either have 1 times 1 or -1 times -1. The reason you can have a a negative for the square root of one is because the two negatives cancel one and other out which creates a one.

Posted by: CFS
• ## Guys I don't think any of you know what you're even talking about

First of all, the square roots are the inverses to squares. If a^2 = c, then √c = a.

We define √c to be equal to a, where a^2=c. It's just that √c has multiple solutions, a and -a.

So √1 = 1, and √1 = -1. There's no denying it. Both 1 and -1 are solutions to the equation x^2 = 1. In this case, 1^2 = 1 and (-1)^2 = 1. Nothing more, and nothing less.

The reason why we only use the positive values, is not because there's anything mathematical about it, but because it's just our human convention. We just like positive values.

• ## Guys I don't think any of you know what you're even talking about

First of all, the square roots are the inverses to squares. If a^2 = c, then √c = a.

We define √c to be equal to a, where a^2=c. It's just that √c has multiple solutions, a and -a.

So √1 = 1, and √1 = -1. There's no denying it. Both 1 and -1 are solutions to the equation x^2 = 1. In this case, 1^2 = 1 and (-1)^2 = 1. Nothing more, and nothing less.

The reason why we only use the positive values, is not because there's anything mathematical about it, but because it's just our human convention. We just like positive values.

• ## In our math books, the definition of a square root of any number x is any number y such that y×y=x

So, since -1×-1=1 then -1 is a square root of 1.
Also 1 is a square root of 1.
The radical sign is reserved for the positive square root only.
So , in our curriculum, there is a big difference between the square root of a number and the radical of a number.

• ## It's very simple

The square root of 1 is -1 and also 1 as 1x1 is 1 and -1x-1 is 1. It is very possible to have two square roots of a number as all positive numbers have two square roots... I need more words so i am writing out this unnecessary sentence.

• ## This is a simple topic.

Yes, if you divide 1 by -1, you get -1. In the same manner, -1 times -1 equals 1. In squares, you multiply numbers together to make a product, in this case, -1 times -1. Negative times negative = positive. -1, a negative, times itself, thus, equals a total of 1.

• ## Square root is a function

Square root is a function taking nonnegative to nonnegative reals. So NOPE. It is not true. There are two roots of a quadratic equation, but square root is a function. Only idiots dont know this. Are you an idiot. Are you. Really. Only idiots dont know this. Are you an idiot. Are you. Really

• ## Not a possibility.

In order for a square root to function normally, you cannot come out with a negative answer. And as for all of the people on the "yes" side saying that since -1*-1=1 that means the square root of 1 is negative one, there are different rules for negative numbers the square root of any negative number is i. I would also like to point out that 1*1 also equals 1 which is why we get the real number 1 as the proper answer to the square root of 1.

• ## Don't forget the definition of a function!

Assuming that the question is asking for the function for square rooting, you can not state that a single input produces two outputs. In the 1800's the Mathematics Association of America came to a conclusion that the square root of a positive number will always output a positive number. Whereas, inputting a negative number outputs an imaginary number.

Also, let's go on a calculus rant. Continuity is very important in a function. Going from 1 to 1.0000...1 is a continuous function. But jumping from 1 to -1.0000...1 is not. The square root function assumes that the output will either be positive or negative; NOT BOTH!

But really, both sides of the argument are correct. It just depends on how familiar you are with functional algebra. In this case, the more advanced your math experience, the more the answer becomes no.

• ## NO at all

Square root of root 1 could not be - 1 because math is based on the reversibility which means if 2 + 2 is 4 then - 2 should be also 2 not 3 . Therefore according to reversibility root minus 1 square is not 1 but it is only and only i.

• ## No, by definition.

As others have pointed out, the square root returns the positive number that when multiplied by itself gives the input.
This is the one and only meaning of square root.
One can easily see that (-1)*(-1)=1, but this does not mean that the square root of 1 is negative one, since, by definition, the square root returns a positive number (It is a function, only 1 output for an input).

• ## √ is defined as the positive square root.

Before I answer, I'd like to say that in math you should be aware of the difference between the underlying truth, and the symbols/definitions we use to express it. The former is true no matter who you are, but the latter depends on the conventions you want to use.

Some concepts:

It's pretty common to say that the square roots of x are any values y that solve y² = x. By this definition, both -1 and 1 are square roots of x.

The radical symbol √ used in the question is a function. Functions must give exactly one value wherever they are defined. So, by convention, we define √ on the nonnegative real numbers to be:
√x = "the positive value y that solves y² = x"

By this definition √1 = 1. It is not true that √1 = -1.

But this is only a convention, because it's usually the most useful thing to use. Functions are useful for computers to use and for quick notation, but they must be single valued, so we need to pick one of the square roots. By convention we picked the positive one. But you could equally define your own function to do something else.

But remember, the underlying truth here is that there are two square roots of 1.

• ## The radical symbol √ is usually defined as the positive square root.

Before I answer, I'd like to say that in math you should be aware of the difference between the underlying truth, and the symbols/definitions we use to express it. The former is true no matter who you are, but the latter depends on the conventions you want to use.

Some concepts:

It's pretty common to say that the square roots of x are any values y that solve y² = x. By this definition, both -1 and 1 are square roots of x.

The radical symbol √ used in the question is a function. Functions must give exactly one value wherever they are defined. So, by convention, we define √ on the nonnegative real numbers to be:
√x = "the positive value y that solves y² = x"

By this definition √1 = 1. It is not true that √1 = -1.

But this is only a convention, because it's usually the most useful thing to use. Functions are useful for computers to use and for quick notation, but they must be single valued, so we need to pick one of the square roots. By convention we picked the positive one. But you could equally define your own function to do something else.

But remember, the underlying truth here is that there are two square roots of 1.

• ## No it is not

No by definition
Square root of 1 is the positive x such that x*x = 1.
If you don’t agree don’t do math. You can say that i means you, but if you do, please don’t speak english with anyone, it won’t work.
I thibk this is relevant :
https://xkcd.Com/1860/

• ## The Dunning Kruger effexor is strong in here

Mathematics requires precise definitions and notation. Then conventions are built around those definitions. The convention for there term square root is that it is a function which returns the positive solution to the equation sqrt(x)^2 = x. There are two solutions but again the conventional definition is that the square root refers to he positive one.

There is nothing wrong with defining a different function which returns the negative solution, we usually call this the "negative square root." Heck you could even say no I wanna call it "the square root" I'm gonna be different but then if you want anyone to understand you you need to make this definition explicit and clear. And then it only invites confusion as you now have two distinct definitions for the same word. There are already too many over uses words in mathematics is numerous definitions (e.G. See "normal") which rely on context to properly interpret.

So for this reason the only way the square root of 1 could be -1 is if you make clear that by square root you do not mean what everyone else in the mathematical community means, you mean something else. In which case people just tell you you have the wrong definition of square root why are you making up needless complications when there is already a term for your function which is the negative square root.

This is on the same level as if someone demanded that the symbols for + and × should be swapped, not for any good reason but because they fell like it.

• ## THE square root

A square root of 1 is definitely -1. The other one is 1. But if we're talking about THE (unique) square root, we mean a continous function from [0,∞[ to [0,∞[. There are other square root functions which are continous in a subset of the complex plane and which yield -1 though.