The NAND gate is of grave importance in computing, both within Minecraft and outside of it.
To truly understand its importance, however, we need to first introduce a concept.
A logic gate is known as Functionally Complete if that same gate can be used to construct all the other gates. That may sound a little abstract at first, but there's a reason we're introducing this.
Of all the gates that we've discussed today, the NAND and NOR gates are the only ones that are functionally complete.
That means that using only NAND gates (or NOR gates), we can construct an AND gate, an OR gate, XOR gate, etc.
In other words, if we recognize the overlying idea that modern computers are simply just a humongous amalgamation of logic gates, we can arrive at the conclusion that, given the ability to construct a functionally complete gate (like NAND or NOR), we can build an entire computer!
This opens up a huge world of possibilities. Essentially, we've proven that since we can build a NAND or NOR gate in Minecraft, we can provably build an entire computer in it.
This is something computer scientists have been playing around with for years. A while ago, someone discovered that you could string together functional logic in Microsoft PowerPoint to create a functionally complete gate. He then released a paper about the functional, or Turing Completeness of Microsoft PowerPoint. Feel free to take a look at it here .
Computing as a field has progressed greatly over the years. During this time, many changes occurred that caused us to favor one functionally complete gate over the other.
At the outset, manufacturers found it much easier and cheaper to produce NAND gates. In CMOS, a common manufacturing process for circuit boards, this gate was much cheaper to produce than the NOR gate.
One should also remember that there is a certain cost-effectiveness to scaling. It is cheaper for a factory, in terms of specialized equipment, quality control, etc., to produce 1000 NAND gates than it is for them to produce 250 AND gates, 250 OR gates, 250 XOR gates, and 250 NOT gates. As such, pricing for the NAND gates dropped, as factories knew that they only really had to produce NAND gates.
Pretty soon, thanks to manufacturing infrastructure and surplus from aforementioned infrastructure, if one wanted to build an AND gate for example- it was easier to fashion it out of NAND gates than to fabricate a dedicated AND gate. As the computing industry scaled, the NAND gate had the versatility to take care of all the applications that logic gates were needed for. As such, it became known as the 'universal logic gate' for the industry.
Does this all mean that we're going to only build using NAND gates in Minecraft? Absolutely not. If we can save some space by building an XOR gate instead of piecing together a few NAND gates, we will.
Our resource of interest is not money or silicon- it's our wrists. When we want to optimize, we'll pick the design that takes the least effort to produce in Minecraft.
As such, our circuit design may not always reflect what actual engineering processes output, but you'll find that optimizing our creations for the unique limitations that Minecraft provides us will help you think like a digital logic designer in your own right.