Prop. $\mathbb{Q} \left( \sqrt {2}\right)$ under usual properties, i.e., $\mathbb{Q} \left( \sqrt {2}\right)$ is a field.  Proof. (Step of multiplicative inverse of $\mathbb{Q} \left( \sqrt {2}\right)$)  Let $a+b\sqrt {2}$ $\in$ $\mathbb{Q} \left( \sqrt {2}\right)$. We will find a',b' $\in$ $\mathbb{Q}$ such that $\dfrac {1} {a+b\sqrt {2}}$= $a'+b'\sqrt {2}$.  It suffices to show that if a+b$\sqrt {2}$ $\neq$ then $a^{2}-2b^{2}\neq 0$. Suppose not, say $a+b\sqrt {2}\neq 0$ but a^{2}-2b^{2}=0. Then 0 = $a^{2}-2b^{2}$= $(a+b\sqrt {2})$ $(a-b\sqrt {2})$. As $a+b\sqrt {2}$ $\neg$, so $a-b\sqrt {2}$=0. But then $\sqrt {2}$=a/b $\in$ $\mathbb{Q}$ provided by b $\neg$ 0. So b=0. However, (final step) if b=0 then a=0 as $a-b\sqrt {2}$. And this yields $a+b\sqrt {2}$=0. Contradiction.  My question is that is ''if b=0 then a=0 as $a-b\sqrt {2}$ $\equiv$ if $a-b\sqrt {2}$ then a=0 and b=0''? Because my proof is that (final step) if $a-b\sqrt {2}$=0 then a=0 and b=0. And this yields $a+b\sqrt {2}$=0. Contradiction
I have been reading through my notes and I came across this example and I found it hard to understand so I need some help in explaining how the inverse of this is found.  The set $\mathbb Q(\sqrt 2)=\{a+b\sqrt2$: $a,b$ in $\mathbb Q \}$ is a field. The inverse of an element $a+b\sqrt 2$ in $ \mathbb Q(\sqrt 2$) is  $$\frac{1}{a+b\sqrt{2}}=\frac{a}{a^2-2b^2}+\frac{-b}{a^2-2b^2}\sqrt{2} $$  Can anyone explain  to how the inverse was found? Appreciate your help.
(Disclaimer: I've read the  and I do understand that this is bordering on off-topic since it's about consumer devices. However I hope that I've managed to make enough on-topic by asking about why it is the way it is, and what is the explanation from an electromagnetic point of view - in other words, what laws of electricity and what practical considerations have gone into these decisions).  I recently bought a power tool and in its manual I found a few peculiar safety rules. I cannot see why they would be there, although reading up more it seems that they are commonplace. My understanding of electricity is limited (just about enough to be dangerous), so I'm hoping to expand it. :)  The rules are:   Don't use an extension cord longer than 25m. Considering this is an outdoors tool, this limitation is severely restrictive. The best explanation I've seen is that longer cables would simply have too high of a resistance and the tool wouldn't be able to operate at full power. But this argument doesn't convince me. The tool is rated at 2.4kW, and it's meant to be used with 220V AC power source, which means it uses about 11 amps of current. All the extension cords I've ever come across have at least 16A limitation, so there is quite a reserve still. In addition, the power has to travel much, much further to get to my house, and even after the meter it still takes probably close to those same 25m to get to the power outlet. Why only 25m after the outlet? The resistance of the cables isn't that big, is it? Don't daisy-chain extension cords. I suspect this is a variation of the above - don't exceed 25m; every connection adds additional contact resistance; water in the outdoors connections can be a problem - in general, it just adds risk. However, apart from the water in the contacts, the rest seems to me to be practically negligible. Is there really a significant contact resistance? Assuming that the extension cords are in a good shape. Make sure that the power cord is entirely unrolled. This is another odd one. My suspicion is that it has to do with inductive resistance. But is it really that strong? Wherever I've seen people actually wanting to create an electromagnet, it's pretty much hard work. You need a lot of neat, tightly-wound loops of wire. An extension cord has a lot of insulator around it, spacing the wires pretty far apart, and the loops are very chaotic. In addition, if that was a problem, why do  exist? They're used especially when you need a very long extension cord, and you don't need to unroll them completely every time you want to use them. Solved: OK, I was off the mark. The correct answer (also a ) has to do with the wire heating up from being used close to its limits. When coiled together, the temperature can go pretty high and melt the cables. Not fun. Ever since upgrading to Windows 10 1903 the other day, my CPUs have all been pegged at 100%.  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What I fail to understand here (and I am sure I am wrong), is nothing can travel faster than light. I also read about Relativity where Einstein stated that every event happening for you depends on how fast the light gets from where the event is happening to you.  This is my particular problem with this Moon shadow experiment. You cannot make a shadow travel faster than light, because in order for the "disappearance" of light to get to the Moon, it has to travel at light speed.  Here is one way to understand my point. Consider we shine a flashlight on Moon, like the guy in the linked video did, we move the finger across the face of the flashlight, before we moved the finger it was casting the shadow at Point A on Moon, after we finish moving our finger, the shadow is at Point B on the Moon.  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