why is the square root of a number always taken as the absolute value? eg. sqrt.16 = +4 and not -4? i'd like

Question

a proof for it.

what is absolute value? eg. |-4|

if the absolute value of all nos. is the +ve value, why then are we taking to cases to solve an equation like this:

x^2 + 2|x| + 5 = 0

we take two cases, one when x is +ve and the other when it is -ve?

isn't that contradictory?

thanks:)

Answer 1

The square root of a number is positive because it is defined to be that way. There is no proof.

|-4| = 4

We need to take two cases when we want to remove the absolute value sign. When x is positive, the absolute value of x is the same as x so we just remove the absolute value. When x is negative, the absolute value changes the value to a positive. A negative sign also will change it to a positive. So |x| = -x when x is negative.

Answer 2

Stick to the mathematical definition of square root ie. The square root of a number x is a number r such that r^2=x.

The square root of a number is not strictly positive, while it is true that every non-negative number has a unique positive square root called the principal square root.
The square root of 16 is both +4 and -4, where +4 is the principal square root. The radical symbol with no index is used to denote the principal square root. To indicate the significance of both square roots +/- is affixed to the symbol.

This concept is particularly significant when applied in solving equations.

Example: if x^2=16, then x=+/-sqrt(16), or 4 and -4. Both values check to the original equation.

The absolute value of a number is its distance from 0. Being defined as such, absolute value is +. 4 is 4 units away from 0, in the same way as -4. So |4|=4, |-4|=4
Do not be confused, when you are looking for an unknown value, you should take particular consideration to posibilities of its sign.
eg.

if |x|=4, x can be 4 or -4.

Answer 3

In normal everyday use people prefer positive numbers so they say 'the square root of 16 is 4'. However in mathematics it is very important to remember that the square root of a number can be positive or negative. The famous quadratic formula recognises this for instance ([-b +- sqr(b^2-4ac)]/2a where the +- recognises that the square root of b^2-4ac can be either +ve or -ve. Similarly if you have (x-1)^2 = 16 then x-1 = +4 or x-1 = -4 giving two answers (5 or -3)

Answer 4

sqrt(x) by definition means the non-negative square root of x when x is a non-negative real number. It is useful to have sqrt(x) be a function with a single value for each input. When a single symbol has two values, it is bad notation because each symbol we use in an expression should have an unambiguous meaning so that when we evaluate the expression we get a definite result. Every number (except 0) has exactly two square roots, and if we didn't work out some method to singe one of them out, we could not use square roots in expressions. When we start working with complex numbers, it is no longer possible to make the square root into a function in a totally satisfactory way, but for non-negative reals it works just fine.

Answer 5

-4 X -4 = +16 remember a -ve X a -ve is always +ve that is why the absolute value of a square root is +ve.

Answer 6

I don't know the answer to your second and third questions, but I can explain about absolute value.

Absolute value is neither positive nor negative. It just denotes number. In everyday reality a number would have to be either positive or negative, but mathematics has a number of abstract concepts, and absolute value is one of them.

Hope this helps.

Answer 7

Actually we take both +4 & -4 for sqrt. 16. In middle school for simplification only the positive value in taken while in high school maths both the values are taken!

Answer 8

It's used as the absolute value for simplicity. Because if it were +/- everywhere, root5+root5 could really be any of -2root5, 0, 2root5.

And x^2 is always positive (unless imaginary...)

Answer 9

It allm makes no sence to me...