Maths Imaginery Numbers?

Question

i=underoot -1,
i=underroot -1/1,
i=underroot 1/-1,
i=underroot 1/underroot -1,
i=1/underroot -1,
i=1/i,
i square=1,
which is wrong
can someone please tell where is the mistake.
what step is wrong.
is changing signs of denom. and num. under the root is not allowed.
if yes then why not

Answer 1

You want to claim that

√(1/-1) = √1 / √(-1).

However, this treats √x as a well-defined function on the real numbers; it isn't. Given a real number x, if y satifies

y² = x,

then

(-y)² = x.

Hence, there are two square roots of each non-zero real number. The question you might want to consider is 'What is meant by underroot -1?'.

***

EDIT You asked me to elaborate. A mistake comes in the very first line:

i = underoot -1 = √(-1).

It is truer to say that

√(-1). = ± i,

ie. that we have two choices for the square root of minus one. As I've said above, √ is not a well-defined function on the real numbers. Similarly, we can't really write

√(1/-1) = √1 / √(-1),

because what we actually have is that

± i = ± 1 / ± i = ± i.

This does not empower us to say that

i = 1 / i = -i,

which is what you're trying to say in the line

"i=underroot 1/underroot -1".

Answer 2

The 4th line is wrong. In passing from the 3rd to the 4th line you assumed -1=underroot -1, that is false, because underroot -1 is i, and i is different from -1.
I try to understand why you made this mistake: you replaced -1 with sqrt(-1). It is not correct to replace x with sqrt(x). You can do it only if x=1 or x=0. Even without going into imaginary numbers, think of x=4. Would you replace 4 with sqrt(4)? In other words, 4=2? Hope I've been of help for your homework.

Answer 3

a million) (5-3i) / (a million-i) rationalize the denominator with the help of multiplying with (a million+i)/(a million+i) (5 - 3i)(a million+i) /(a million-i^2) (5 - 3i + 5i - 3i^2) / a million-(-a million) 5 + 2i - 3(-a million) / 2 =>(8 + 2i) / 2 = 4 + i 2) x^2 - 4x + 5 = 0 x^2 - 4x + 4 + a million = 0 (x -2)^2 + a million = 0 (x-2)^2 = -a million (x-2) = +/- sqrt(-a million) x - 2 = + i or - i x = 2 + i or 2 - i

Answer 4

The mistake is at step 4. (1/-1)^0.5 is not the same as (1^0.5)/(-1^0.5).

Answer 5

i squared = -1
^2 means squared
i^2=(1/i)^2
-1=1^2/i^2
-1=1/-1
-1=-1
i see no problem except for i^2=-1

Answer 6

i is the square root of -1
i squared is -1 (square root of -1xsquare root of -1)
i cubed is -i (-1xi)
i to the 4th power is 1 (-1x-1)
i to the fifth power is i (ixi)
i to the 6th power is -1 (1x icubed which is -1)

Who was your Algebra 2 teacher?