Help me: Where am I wrong??

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

$\Rightarrow e^{2i\pi} = (-1)^2$ (Squaring both sides)

$\Rightarrow e^{2i\pi} = 1$

$\Rightarrow e^{2i\pi} = e^0$

$\Rightarrow 2 \times i \times \pi = 0$

But how can this be???

$2 \neq 0$ AND $i \neq 0$ AND $\pi \neq 0$

Then how can $2 \times i \times \pi = 0??$

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

The error is in saying that $e^x = e^y$ implies that $x=y.$ This is true for the real exponential function, but for the complex exponential function, this is very false.

Patrick Corn - 1 year, 6 months ago

$f(a) = f(b) \implies a = b$ is only true for monotonic functions. For oscillatory functions (such as complex exponentials), it doesn't hold. For example, $\sin(0) = \sin(2 \pi)$ but it doesn't follow that $0 = 2 \pi$

Steven Chase - 1 year, 6 months ago

Hi there, the 2π here actually refers to how many radians around the unit circle we go. So e^(0i) means we have no rotation, so we're at 1. e^(πi) means we're halfway round the circle, so we're at -1. e^(πi/2) means that we're at +i. e^(3πi/2) means that we're at -i. And lastly at a full revolution, back to the start we're at e^(2πi), so back at one. This means that e^(2nπi) = 1 also for all n in the natural numbers.

Cameron Chandler - 1 year, 6 months ago

Thanks for clearing my doubt.

Nishant Ranjan - 1 year, 6 months ago

2iπ ∈ ln(1) and 0 ∈ ln(1)

∈ means "element of set" because there is a solution set. It also may in this context mean "is a solution of."

ln(1) has multiple solutions, which are all the 0iπ, 2iπ, 4iπ, -2iπ... AFAIK. To check for all real solutions or all complex solutions, we can use algebra. But to prove for "all" points, I think you would have to account for ∞, -∞, ∞i, -∞i, and probably these "quaternion" numbers, there would be just so much. We can at least check for "all real or complex solutions" and that may be good. Very clever spinoff of a classic paradox.

chase marangu - 1 year, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$