# 1 = -1 !

1. Aug 5, 2009

### 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??

2. Aug 5, 2009

### lukaszh

Re: 1 = -1 !!!!!!

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

3. Aug 5, 2009

### Dragonfall

Re: 1 = -1 !!!!!!

Your third line is wrong, for complex numbers.

4. Aug 5, 2009

### Tac-Tics

Re: 1 = -1 !!!!!!

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

5. Aug 5, 2009

### johncena

Re: 1 = -1 !!!!!!

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

6. Aug 5, 2009

### johncena

Re: 1 = -1 !!!!!!

Actually,what is the use of complex numbers ?

7. Aug 5, 2009

### CRGreathouse

Re: 1 = -1 !!!!!!

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.

8. Aug 5, 2009

### arildno

Re: 1 = -1 !!!!!!

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

9. Aug 5, 2009

### Newtime

Re: 1 = -1 !!!!!!

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

Last edited: Aug 5, 2009
10. Aug 5, 2009

### arildno

Re: 1 = -1 !!!!!!

Nope.

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

11. Aug 5, 2009

### Tac-Tics

Re: 1 = -1 !!!!!!

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.

12. Aug 5, 2009

### negitron

Re: 1 = -1 !!!!!!

So as to distinguish it from current.

13. Aug 5, 2009

### arildno

Re: 1 = -1 !!!!!!

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

14. Aug 5, 2009

### Дьявол

Re: 1 = -1 !!!!!!

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

If a=b then:

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

so that

$$c=|a|$$

Regards.

15. Aug 5, 2009

### arildno

Re: 1 = -1 !!!!!!

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

16. Aug 6, 2009

### johncena

Re: 1 = -1 !!!!!!

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

17. Aug 6, 2009

### snipez90

Re: 1 = -1 !!!!!!

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.