I have $$3x=0$$ equation. I divided both sides of it by x and got: $$\frac{3x}{x} =\frac{0}{x}$$ $$3 = 0$$
I want to ask, how is that possible? What did I do wrong? Did I break any rule of math?
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5What if $x=0$? You cannot divide both sides by $0$... – Milly Oct 30 '14 at 19:33
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You cant divide by $x$ unless you know that $x \ne 0$. Are you in this case? – FormerMath Oct 30 '14 at 19:37
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If we are talking about an equation over a field like $\Bbb R$ or $\Bbb C$, then $x=0$. If two things multiply to zero in a field like $\Bbb R$ or $\Bbb C$, then at least one of them is $0$. And $3$ isn't zero, so... – rschwieb Oct 30 '14 at 19:41
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See something similar in my answer here. – Eff Oct 30 '14 at 20:33
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Yes, math broke today. :(
On a more serious note: The only solution to your equation $3x = 0$ is $x=0$. Now, since we cannot divide by zero, the operation
$$\frac{3x}{x} = \frac{0}{x}$$
is illegal. Does that make sense?
naslundx
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The reason is that you're assuming that $\frac{1}{x}$ exists. This is not a valid assumption since $x = 0$.
Remember that you really are multiplying b $x$ inverse, but $0$ has no inverse.
TN888
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Daniel Goldman
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I do not agree with the previous comments. There are no mistakes in the Ty221's post. Indeed, he shows that the equation $3x=0$ has no solutions in $\mathbb{R}\setminus \{0\}$.