A Mathematical Investigation: -1 = 1?

[johncena]

1 = -1 !

$$\sqrt{}a$$.$$\sqrt{}b$$ = $$\sqrt{}ab$$
When a = b = -1,
$$\sqrt{}-1$$.$$\sqrt{}-1$$=$$\sqrt{}-1*-1$$
i*i = $$\sqrt{}1$$ = 1
i² = 1
-1 = 1

Had i gone wrong anywhere??

 

[lukaszh]

This is not true in general:
$$\sqrt{x}\sqrt{y}=\sqrt{xy}$$
!

 

[Dragonfall]

Your third line is wrong, for complex numbers.

 

[Tac-Tics]

Algebra fail!

You may also enjoy these. The basic idea of all of them is the same. They all rely on a rule that has a subtle restriction (such as the nonnegative requirement on the root-multiplication distributive property), and then ignore the restriction.

http://en.wikipedia.org/wiki/Invalid_proof

 

[johncena]

Algebra fail!

You may also enjoy these. The basic idea of all of them is the same. They all rely on a rule that has a subtle restriction (such as the nonnegative requirement on the root-multiplication distributive property), and then ignore the restriction.

http://en.wikipedia.org/wiki/Invalid_proof

I was searching for a site like this ...thanx

 

[johncena]

Actually,what is the use of complex numbers ?

 

[CRGreathouse]

Actually,what is the use of complex numbers ?

They're very useful in many capacities. Electrical engineering uses them extensively (though they call the complex unit j for some reason). Many/most proofs of, e.g., the fundamental theorem of algebra use complex numbers.

 

[arildno]

Actually,what is the use of complex numbers ?

They are very useful; you may, for example use them as Christmas tree decorations.

 

[Newtime]

Isn't the main error the 4th line? sqrt(1) = +/- 1, and in this case, it would be -1.

 

[arildno]

Isn't the main error the 4th line? sqrt(-1) = +/- 1, and in this case, it would be -1.

Nope.

The square root is defined as a FUNCTION, and can therefore have only one value for each argument.

 

[Tac-Tics]

Actually,what is the use of complex numbers ?

Complex numbers are a natural extension of the real numbers. Just as real numbers were created ("invented" in a sense) to solve the issue of the square root of a length (a non-negative number), the complex numbers were invented to solve the issue of the square root for any real number.

The complex numbers have the amazing property, embodied by the fundamental theorem of arithmetic that all polynomials of complex numbers (of degree 1 or more) ALWAYS have zeros.

Complex numbers are a little different from real numbers in important ways, too. The complex numbers cannot be field ordered.

Complex numbers are isomorphic to vectors in R^2.

 

[negitron]

They're very useful in many capacities. Electrical engineering uses them extensively (though they call the complex unit j for some reason).

So as to distinguish it from current.

 

[arildno]

Complex numbers are isomorphic to vectors in R^2.

That's why they work so well as Christmas tree decorations.

 

[Дьявол]

$$c=\sqrt{a} * \sqrt {b}=\sqrt{a*b}$$

If a=b then:

$$c=\sqrt{a} * \sqrt {a}=\sqrt{a^2}$$

so that

$$c=|a|$$



Regards.

 

[arildno]

Only if "a" is a non-negative number.

 

[johncena]

Why is degree measure not used in the polar form of complex numbers?
What is the advantage in using radian measure?

 

[snipez90]

In general, radian measure is more "natural" in a variety of settings. The number of degrees in a circle (360), is more or less arbitrary, I think. One of the reasons 360 was chosen is because many numbers divide it (it's not nice to work with 100/3 degrees, for instance).

Before one encounters a rigorous development of trigonometry in calculus or analysis, trig functions are often extended to obtuse angles (or rather, any real number) via points on the unit circle. Here, radian measure is very natural, since any angle that cuts off an arc has the same measure as the arc itself.