Explain how does $1 = i\;$?

Asked Mar 21 '23 at 18:14

Active Mar 21 '23 at 18:33

Viewed 96 times

Start with the expression
$$\sqrt[4]{81\mathrm{p}^3\mathrm{q}^5}$$

Simplify to
$$3q\sqrt[4]{\mathrm{p}^3q}$$

Assume $p = i, q = i$. Evaluate.

$$\sqrt[4]{81\mathrm{p}^3\mathrm{q}^5} = 3$$

$$3q\sqrt[4]{\mathrm{p}^3q} = 3i$$

Therefore, $3 = 3i, 1 = i$.

I suspect something related to absolute value, but can't figure it out at this moment.

edited Mar 21 '23 at 18:33

Angelo

asked Mar 21 '23 at 18:14

Ian Q.

1

Consider a simpler example, $\sqrt{(-1)^2}$. By your logic, this would be equal to $-1$. However $\sqrt$ is defined as the positive square root, and is equal to $1$. The correct way to pull something out of a square root is $\sqrt{a^2b}=|a|\sqrt{b}$. The same goes for $\sqrt[4]$ as well. – Wolfuryo Mar 21 '23 at 18:18
4

Exponentials and roots with complex numbers don't quite work the same as real valued stuff. – Sean Roberson Mar 21 '23 at 18:18
This is a very common mistake – FShrike Mar 21 '23 at 18:22
1

$i^4=1$ and $1^4=1$. This does not mean that $i=1$. They are different numbers on the unit circle. – M. Wind Mar 21 '23 at 19:07
Welcome to Mathematics Stack Exchange. Cf. this question – J. W. Tanner Mar 21 '23 at 19:41

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