help me find the error in this math that "proves" i= -1 or 1 and 1 = -1

[The_Tao_s_Revenge]

Here’s a train of thought I had. The logic produces a wrong result, so I’m wondering what I did wrong. Here it is QED style:

i can be shown to be either 1 or -1

Start with basic idea i as the square root of negative one.
i² = -1

therefore:

i⁴ = i² * i² = -1 * -1= 1

therefore

i⁴= 1

take a fourth root of both sides remembering it produces both a positive and negative result:

[4]√(i⁴) = [4]√(1)

i,-i = 1, -1

result:

i is either 1 or -1, QED
Which using the orignal starting point of i² = -1 could be used to show 1 =-1

If i = 1 and i² = -1, and 1 * 1 is 1 there -1 =1 QED

Or if i = -1 and i² = -1 and -1 * -1 = 1 therefore -1 = 1 QED

All negatives are positive, war is peace, freedom is slavery, cats and dogs living together, etc.

Where did I go wrong? I suspect the error is in taking a 4th root of each side, but don’t know why. Is this the i version of trying to divide by zero?

[Terr]

Why go through all the jumps?

1 = 1

take square root of each side

-1,1 = -1,1

from that, if you use your logic, 1 = -1

when taking a square root of both sides of an equation, the rule would be to take the “Principal Value” of a square root on both sides - that is, the positive one. Otherwise you get erroneous results, obviously, since a positive result cannot be equal to the negative result.

[Giles]

Generally, a real or complex number (apart from 0) has four 4th roots – and, more generally, for any positive integer n, any non-zero number has n nth roots. So i[sup]4[/sup] and 1 both have four 4th roots, i.e., 1, i, -1 and -i.

[Corresponder]

Yes, to say i**4=1 does not mean i is equal to any of the 4th roots of 1 (i,-1,1,-1), it means it is equal to ONE of them, namely, to i. This is like saying that since George W. Bush’s father is George H.W Bush, and Jeb Bush’s father is also George H.W. Bush, then George W. must equal Jeb.

[chowching]

I think because the premise is i = sq.root of -1 and you tried to arrive at a conclusion which contradicts your premise.
You also missed the other fourth roots of 1. Aside from 1 and -1, in complex numbers, i and -i are also fourth roots of 1.

[The_Tao_s_Revenge]

@Corresponder

Yes, to say i**4=1 does not mean i is equal to any of the 4th roots of 1 (i,-1,1,-1), it means it is equal to ONE of them, namely, to i. This is like saying that since George W. Bush’s father is George H.W Bush, and Jeb Bush’s father is also George H.W. Bush, then George W. must equal Jeb.

@Giles

Generally, a real or complex number (apart from 0) has four 4th roots – and, more generally, for any positive integer n, any non-zero number has n nth roots. So i[sup]4[/sup] and 1 both have four 4th roots, i.e., 1, i, -1 and -i.

Mystery solved, thanks!

[Napier]

@chowching

I think because the premise is i = sq.root of -1 and you tried to arrive at a conclusion which contradicts your premise.

I think this is a perfectly legitimate method. You could use a premise to try to prove something contradictory to the premise. If it works, I think you have proven the premise is wrong (but not necessarily anything else). Isn’t this the “reduction to absurdity” method?

I mean, it didn’t work in this case, but that just means reduction to absurdity can’t prove i isn’t sqrt(-1) in this case, it doesn’t mean reduction to absurdity as a general concept is invalid.

Comments?

[Derleth]

@Napier

I think this is a perfectly legitimate method. You could use a premise to try to prove something contradictory to the premise. If it works, I think you have proven the premise is wrong (but not necessarily anything else). Isn’t this the “reduction to absurdity” method?

Indeed it is, and it has been a foundation of mathematical logic since at least Classical Greece. It is commonly called reductio ad absurdum, Latin for ‘reducing to absurdity’, or abbreviated to reductio.