If the square root of x is x, then what value must x have?

The definition of a square root of x is a number y such that y^2 = x. Clearly,in this case x^2 = x. So x cannot be a negative number because x^2 must be positive or zero. It is clear also that both 0 and 1 are solutions, since 0 times 0 = 0 and 1 times 1 = 1. It is clear also that no other value will work since, if the value of x is not zero, then we may divide both sides of the equation x^2 = x by x to get the result x = 1.

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60 Comments

Chip Amos

Huge problem....the word "value" is used. "0" has no value.

drkegrossman

0 is not a number! Their question is BS!

Ray Brylski

Since zero is neither positive nor negative, its square root is undefined.

Jim Laughner

I knew there would be dispute here from people who do not understand basic arithmetic. What is 0X0=? What is 1X1=? It can't get any more simple than that. Zero squared is zero, therefore the square root of zero is zero. One squared is one, therefore the square root of one is one. I would say "Problem solved, but there is no problem to solve. Keep things simple and your life will be simplified, throw in altered definitions and other non-existent variables and you can complicate even the simplest of things.

Steve Edwards

I despair at all those who consider that 0 isn’t a number. It’s true that 0 was, at first, a placeholder (about 1000 years ago), but its usefulness as a number in its own right was quickly realised. It’s a nonsense to suggest that our “number line” has a gap in it where zero is because zero isn’t a number. Similarly zero IS a value. In the equation x+7=7, solution is x=0, which means that the value of x that makes the equation true is zero.

Jim Laughner

Ray Brylski, Zero is defined as "nothing" or "None". The number 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. If you multiply nothing by nothing you have nothing, therefore nothing squared, (or nothing times itself), is still nothing. From that, it stands to reason that the square root of nothing is nothing. Call it what you want but when you multiply it by itself or divide it by itself the answer is the same, NOTHING or ZERO, or ZERO, or ZILCH, or any other of other terms commonly used.

Phil Kennelty

0 times 0 does not equal 0.

Mary Elizabeth Mazi Ross

drkegrossman, I agree

Char

drkegrossman, exactly this isn't a math quiz anyway.Always putting others down for thier questions better than his to me

james parrillo

Jas Newman, Why? Thank you for explaining further. Christine, James' wife

Jas Newman

Peter Abrams, True. but that wasn't the question. In the case of the question, X cannot be -1.

Howard Groopman

I thought zero was not considered a number, and that anything divided by zero is infinity.

Glynis Rix

No clue what's happening!

Peter Abrams

Jim Laughner, yes the square root of 1 is 1, but it’s also -1.

earlepg

Al Breyette, I've already said the same thing. Except, as is true this time, I rarely notice the authors name until after I've already answered the question. I luckily anwered this question right.

Ron Gosling

0 not a number so wrong. Only one is answer.

Peter Abrams

reydlt, if 0 x A=0 then A can be 0. 0x0=0. Therefore the square root of 0 is 0. Division by 0 is undefined, yes. But 0x0 is 0. The square of 0 is not undefined.