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

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

  • ×0=?\infty \times 0 = ?

  • Is 10=?Is \ \frac{1}{0} = \infty ?

  • 00=1, but why?0^{0} = 1, \ but \ why?

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

#Algebra

Note by A Former Brilliant Member
11 months ago

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Comments

Percy, here's my opinion on the value of 10\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

10=\frac{1}{0} \cancel{=} \infty as then \Rightarrow 1=0×1 = 0 \times \infty \Rightarrow 1=01 = 0

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

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If 10\frac{1}{0} "is" =,= ∞, then we have an answer for ×,0∞ × ,0 which will be 1\boxed{1}

@Percy Jackson

Frisk Dreemurr - 11 months ago

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But we know that 0×x0 \times x is always 00 with any value of xx.

@Hamza Anushath

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@A Former Brilliant Member @Percy Jackson

Only with finite values of xx

Lâm Lê - 7 months ago

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@Lâm Lê Yes, that's also true...

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

Zakir Husain - 7 months ago

@Lâm Lê And if you are talking about : limx(x×0)\boxed{\lim_{x\to\infty}(x\times 0)} then, limxx×0=0\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 limn0nn=000=1\lim_{n\to 0}n^n=0\cancel{\Rightarrow}0^0=1

The result will be different if you include complex numbers (something like lima0limb0(a+bi)a+bi\lim_{a\to0}\lim_{b\to0}(a+bi)^{a+bi} or lima0(a+ai)a+ai\lim_{a\to0}(a+ai)^{a+ai} or lima0(ai)a\lim_{a\to0}(ai)^{a}, etc ).

00\therefore0^0 is undefined.

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

Zakir Husain - 11 months ago

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

Zakir Husain - 11 months ago

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

Siddharth Chakravarty - 11 months ago

What about this? Assume 00\frac{0}{0} is undefined. Then 10=\frac{1}{0} = \infty, but 0×10 \times \infty \neq 1, since to obtain this, we would have to multiply 10\frac{1}{0} by 00, resulting in 00×1\frac{0}{0} \times 1, which would then be undefined.

David Stiff - 7 months ago

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But why 10\dfrac{1}{0} must be \infty?, why not UNDEFINED\red{UNDEFINED}

Zakir Husain - 7 months ago

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

Zakir Husain - 7 months ago

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@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 00, 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

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Yes that's just the*best *answer

Lâm Lê - 8 months, 1 week ago

Well, I just watched a video by Eddie Woo on 00=?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 limx0xx=1\lim_{x \to 0} x^{x} = 1 by calculating x^x for decimals and showing that is approaches 1.

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

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Haha, nice @Zakir Husain!!!

A Former Brilliant Member - 6 months, 3 weeks ago
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