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
• ## Obviously, This is a fact

(most)every number has two square roots, One positive and one negative. -1*-1=1, So yes. It is a square root. 1*1=1 as well, So the two square roots of 1 are -1 and 1. Come on people, I am 12 years old. This is not that hard to understand because one negative number times another negative number equals a positive number.

• ## √1 equals both 1 and -1.

The use of the square root is to find the one number that when multiply to each other, Gets the number under the radical symbol.
√25=5, -5 because 5(5)=25 and -5(-5)=25.
When doing all square roots or any radicals with an even index, It is important to include the plus or minus (±) symbol.
So by definition, √1=±1. Most graphs and calculators only focus on the principal root, Where they only deal with the non-negative numbers. But according to algebra, Especially in quadratics, The ± is very important. For example:
(x^2)-4=0
(x^2)=4
x=±2

(x^2)-4=0
(x+2)(x-2)=0
x=-2, X=2

By all these examples, √1 can equal to -1.

• ## Ahh, Algebra Skills

The square root of 1 can be -1 because (-1)(-1)=1, And so does 1. But if we are talking about principal root, Then the answer is 1 only. Remember, Any radical with an even index will have a positive and negative answer. It is just the basis of algebra.
Note: Most graphing calculators only graph principal roots, As it is much more easier to deal with only the positive than the negative.

• ## 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.

• ## 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.

• ## First, Understand what "square root" is defined as

First, You must know what you're talking about. The square root of any number, By definition, Means the positive number, Which when multiplied by itself, Gives the number you want the root of. In fact, In mathematics, The modulus function (the function that gives the positive value of any number) is defined as root of square of the number which means that square root gives only the positive value. Of course x^2=1 has 2 solutions, 1 and -1, But that doesn't mean that -1 is the square root of 1, It is the negative of the square root of 1.

• ## It does not equal -1

A square root simply cannot be negative.

Alright this is just random stuff so the word count is reached. Well actually the people on the Yes side is LEGIT STUPID as they think the square root of 1 equals to -1. I wonder if they dropped out of school when they were in kindergarten.

• ## 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/