kalkmat asked in Science & MathematicsMathematics · 1 decade ago

# why square root of 4 is 2 not -2?

yes?

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• Anonymous

Grab a caluclator. Read what it says when you enter the square root of 4. It gives you 2... 2 only, and not -2.

A square root is a function, meaning it can only have one value. A bunch of fellow math geeks a long time ago decided the non-negative one was better.

It's kind of like inverse trig functions. The sine of a 0° angle is 0. The sine of a 180° angle is 0. So are the sines of 360°, 540°, -180°, and so on. When it comes to the arcsine, though, arcsin(0) could equal -180°, 0°, 180°, 360°, or any of a host of others. To make the Arcsin a function, the math geeks picked one, Arcsin(0) = 0°.

Yes, (2)(2) = 4. Yes, (-2)(-2) = 4. But 4 has only one square root, and it's 2. People answering otherwise are thinking of solving the equation:

x² = 4, which does have two solutions... x = 2 or -2.

x² = 4

SqRt(x²) = SqRt(4) [Take the square root of both sides.]

|x| = 2 [Simplifying. Note on the right side, the square root of 4 is 2. It's a function, and can only have one value. On the left, taking a square root of a variable that would leave an odd power (x^1) means the expression must be an absolute value.]

x = 2 or -2. [It's solving the absolute value part of this problem that yields plus or minus 2.]

An equation [like x² = 4, or sin(a)=0] can have many solutions.

A function [like the square root of 4, or Arcsin(0)] can have only one value.

Source(s): B.S. in Mathematics and M.S.T. in Math Education. I've taught this stuff every semester for the past six years.
• Anonymous

The value 2 is the absolute value of the square root of 4. An absolute value is the distance, in points, from the origin (or value 0); the absolute value is a natural number value, so it can't be a negative number. The square root of 4 can be -2, or 2. Either is a real number point along a number line.

The building next door is a three story building. On the roof is the number 4. On the third floor--the function "square root". On the second floor--the absolute value of the square root of 4; value 2. On the first floor-- ...

• 1 decade ago

actually, the square root of 4 is both positive and negative 2 because 2*2=4 and 2*2 = 4. However, in most mathmatical conventions, the primary square root (2) is used, since negative numbers are a hard concept...

• Anonymous

because 2 times 2 = 4

• 1 decade ago

The square root of 4 is plus OR minus 2.

• 1 decade ago

The problem is semantics. Both 2 and -2 are "square roots" (plural) of 4, because 2*2 = 4 , and (-2)*(-2) = 4. The question is asking, not about square roots, but about "THE square root" (singular) referring to the principle square root. The principle square root is defined as the nonnegative square root. In this way the principle square root, referred to as the square root, is single valued, and becomes a function. It is this principle square root that is denoted by the radical. As the two square roots of any number are opposites, once you can find the principle square root you automatically know both square roots.

• 1 decade ago

Whoa, glaring ignorance in some of your answers. The square of any real number is positive. The square root is two numbers, one positive, one negative. So:

sqrt(4) = +/- 2

No magic, it is the same as, for instance, factoring 12

-3*-2*2 = 12

so factors could be: (2,2,3) (-2,-2,3) (2,-2,-3) (3,4) (-3,-4) (2,6) (-2,-6)

so the original question could be written as the factor of four (2,2) (-2,-2)

It is the 'principal square root' that is defined strictly positive. And the 'negative square root' negative. Perhaps the semantics is the origin of confusion.

Hope this helps.

• 1 decade ago

When you say "square root", you generally mean the positive one. Also, if you look at the graph of y = sqrt(x), you see that the square root of 4 is 2, not -2. But why such confusion? I'll tell you why. The inverse of the graph y=x^2 does not exist!!! If you try to come up with a single inverse, you are doomed. There are actually 2 inverses, y=sqrt(x) and y=-sqrt(x). Mathematicians have developed this method to distinguish between the positive and negative square roots.