# Can 1 = 2 ?

1. Sep 4, 2006

### navneet1990

can 1 = 2 ??

is this possible
1 = 2
??

2. Sep 4, 2006

### waht

Consider the equation,

x + 1 = x + 2

And you are asked to solve for x.

By looking at it, you can say there is no solutions.

Subtract x from both sides, and you get

1 = 2

But what if you could subtitute 1 = 2 in one eqation

you get x + 1 = x + 1

or x + 2 = x + 2

That makes more sense now.

3. Sep 4, 2006

### CRGreathouse

Sure, if the underlying assumptions are inconsistent.

4. Sep 4, 2006

### navneet1990

i didnt get it
for any value of x how can x+1 = X+2
the assumption itself is wrong
right??

5. Sep 4, 2006

This really has nothing to do with math.

6. Sep 4, 2006

### arildno

Sure 1=2 in some number systems.
In a number system that denies the axiom $0\neq{1}$, for example, 1=2 will be a true statement.

7. Sep 4, 2006

### StatusX

1 and 2 are just symbols. By their standard definition, they correspond to two distinct real numbers, and so the statement '1=2' is false. If you want to use these symbols in a different way, then the statement '1=2' can be true, false, or meaningless, depending on how you define them.

8. Sep 5, 2006

### loom91

1 is not equal to 2 in any of the standard formulations of natural numbers (Peano axioms, set theory etc). However you can very easily define two symbols 1 and 2 in a number system such that they are equal.

9. Sep 5, 2006

### navneet1990

cant we use complex numbers to prove it
like :

i = i

root -1 = root -1

hence ,

root -1/ root 1 = root -1 / root 1

hence,
whole root [-1/1] = whole root [ -1/1]

hence,
whole root [-1/1] = whole root [ 1/-1]

[ -5/4 can also be written as 5/-4...cant it??? i mean -1/1 is the same as 1/-1 right??]

hence,
root -1 / root 1 = root 1 / root -1

hence,

i / 1 = 1 / i

i square = 1

hence,
-1 = 1

3/2 + ( -1 ) = 3/2 + 1

3/2 - 1 = 3/2 + 1

2/ 2 = 4/2

therefore,

1 = 2

can this be possible

10. Sep 5, 2006

### J77

Nope.

You can't take roots on both the top and bottom like that.

$$i=e^{i\pi/2}$$

$$1/i=e^{-i\pi/2}$$

With these 1=2 things, there's always a mistake/trick.

11. Sep 5, 2006

### navneet1990

um...
i kinda understood a little
but i didnt understand
the

i = e raised to (-i Pie/ 2)
what is that

12. Sep 5, 2006

### chroot

Staff Emeritus
He's expressing complex numbers as complex exponentials.

$$i=e^{i\pi/2}$$

is the value on the complex unit circle corresponding to an angle of pi/2. It is equal to i.

His point is that 1/i and i/1 are quite different numbers, on opposite sides of the unit circle, so your "proof" contains an error.

- Warren

13. Sep 5, 2006

### navneet1990

ohk
thank you

14. Sep 5, 2006

A rational number is, by definition, a number of the form $$\frac{p}{q}$$, where $$p \in Z$$ and $$q \in N$$. So, you can't write -5/4 as 5/-4.

15. Sep 5, 2006

### arildno

Eeeh, wherever do you have this limitation from??
Not saying you might not be right, but I really don't see the necessity of this limitation.

16. Sep 5, 2006

### jnorman

i remember my father showing me a proof once that 1=2, but cant quite recall it. but, if you start with the equation:
x^2 -1 = 0, you can factor x^2 - 1 into (x+1)(x-1)=0
then divide both sides by x-1, and get x+1=0.
for a value of x=1, you have shown that 2=0.
:-)

17. Sep 5, 2006

### chroot

Staff Emeritus
You should tell your father to take more math classes. You cannot divide both sides by x-1, when x=1, because that is equivalent to division by zero. Division by zero is not a "legal" mathematical operation.

- Warren

18. Sep 5, 2006

### mattmns

That is NOT the definition of a rational number

http://mathworld.wolfram.com/RationalNumber.html

A rational number is a number p/q where p and q are Integers and $q \neq 0$

So, you can write (-5)/4 as 5/(-4)

19. Sep 5, 2006

### CRGreathouse

How quaint.

20. Sep 5, 2006

### HallsofIvy

Staff Emeritus
A rational number is, by definition, a number which can be written in the form $\frac{p}{q}$ where $p \in Z$ and $q \in N$. Whether or not the number is written that way is irrelevant.

Yes, you can write -5/4 as 5/-4 just as you could write it as -1.25.