Contradictions $\ldots$ $\ldots$ $\ldots$

This is a note for discussing contradictions and sharing facts, links opinions, info and more.

$\infty \times 0 = ?$
$Is \ \frac{1}{0} = \infty ?$
$0^{0} = 1, \ but \ why?$

Share more contradicting problems so we can discuss and debate upon their answer!

#Algebra

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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

Percy, here's my opinion on the value of $\frac{1}{0}$ , and here's my general opinions on working with $\infty$ (my comment, not the discussion).

David Stiff - 10 months, 4 weeks ago

$\frac{1}{0} \cancel{=} \infty$ as then $\Rightarrow$ $1 = 0 \times \infty$ $\Rightarrow$ $1 = 0$

This is true for all numbers divided by 0, so division by 0 is just undefined......

A Former Brilliant Member - 11 months ago

If $\frac{1}{0}$ "is" $= ∞,$ then we have an answer for $∞ × ,0$ which will be $\boxed{1}$

@Percy Jackson

Frisk Dreemurr - 11 months ago

But we know that $0 \times x$ is always $0$ with any value of $x$ .

@Hamza Anushath

A Former Brilliant Member - 11 months ago

@A Former Brilliant Member – @Percy Jackson

Only with finite values of $x$

Lâm Lê - 7 months ago

@Lâm Lê – Yes, that's also true...

A Former Brilliant Member - 7 months ago

@Lâm Lê – $x$ is a NUMBER and $\infty$ is NOT a NUMBER $\therefore$ $x$ alway have to be finite

Zakir Husain - 7 months ago

@Lâm Lê – And if you are talking about : $\boxed{\lim_{x\to\infty}(x\times 0)}$ then, $\lim_{x\to\infty}x\times 0 = 0$

Zakir Husain - 7 months ago

1/0 is undefined because you are basically asking what number times 0 or how many zeroes should you add so that you get 1, even if you add infinite zeroes the answer would be 0.

Siddharth Chakravarty - 11 months ago

@Hamza Anushath

If $\lim_{n\to 0}n^n=0\cancel{\Rightarrow}0^0=1$

The result will be different if you include complex numbers (something like $\lim_{a\to0}\lim_{b\to0}(a+bi)^{a+bi}$ or $\lim_{a\to0}(a+ai)^{a+ai}$ or $\lim_{a\to0}(ai)^{a}$ , etc ).

$\therefore0^0$ is undefined.

Also if $\lim_{x\to a}f(x)=k\cancel{\Rightarrow}f(a)=k$

Zakir Husain - 11 months ago

@Hamza Anushath see this if you don't believe me.

Zakir Husain - 11 months ago

@Zakir Husain – Link Isn't the topic still debatable?

Siddharth Chakravarty - 11 months ago

What about this? Assume $\frac{0}{0}$ is undefined. Then $\frac{1}{0} = \infty$ , but $0 \times \infty \neq 1$ , since to obtain this, we would have to multiply $\frac{1}{0}$ by $0$ , resulting in $\frac{0}{0} \times 1$ , which would then be undefined.

David Stiff - 7 months ago

But why $\dfrac{1}{0}$ must be $\infty$ ?, why not $\red{UNDEFINED}$

Zakir Husain - 7 months ago

Also $f(x)=\frac{1}{x}$ is discontinuous at $x=0$ so you can't give $f(0)$ any value

Zakir Husain - 7 months ago

@Zakir Husain – This is true, but only if we assume that there is both a positive and negative value of infinity. What if we have a single point of infinity where everything ends, similar to $0$ , where everything begins. Then we would get the situation I described in this discussion. Please note this is all speculation on my part. Definitely fun to think about!

David Stiff - 7 months ago

Percy, 0^0 is not equal to 1...

Shevy Doc - 9 months, 3 weeks ago

Yes that's just the*best *answer

Lâm Lê - 8 months, 1 week ago

Well, I just watched a video by Eddie Woo on $0^{0} = ?$ and he solves the question by using limits. He says that it is undefined, but it seems as if its one, so we have agreed upon that value, until we find a better solution. He shows that $\lim_{x \to 0} x^{x} = 1$ by calculating x^x for decimals and showing that is approaches 1.

A Former Brilliant Member - 7 months ago

$(\lim_{x\to a}f(x)=A)\cancel{\Rightarrow}(f(a)=A)$ $(f(a)=A){\Rightarrow}(\lim_{x\to a}f(x)=A)$

Zakir Husain - 7 months ago

Also you are looking only for Real limits not Complex limits

Zakir Husain - 7 months ago

Contradiction in Probability

Zakir Husain - 6 months, 3 weeks ago

Haha, nice @Zakir Husain!!!

A Former Brilliant Member - 6 months, 3 weeks 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}`	$\boxed{123}$

Contradictions \(\ldots\) \(\ldots\) \(\ldots\)

Comments

10=∞\frac{1}{0} \cancel{=} \infty01​=​∞ as then ⇒\Rightarrow⇒ 1=0×∞1 = 0 \times \infty1=0×∞ ⇒\Rightarrow⇒ 1=01 = 01=0

This is true for all numbers divided by 0, so division by 0 is just undefined......

$\frac{1}{0} \cancel{=} \infty$ as then $\Rightarrow$ $1 = 0 \times \infty$ $\Rightarrow$ $1 = 0$