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"Actually, it isn't that far off. The standard strategies work," the Sect Leader followed up with a shrug.
"For example-"
Two numbers: 1011 and 1001 appeared one on top of the other.
"If we are to add 11 and 9 in decimal digits, it's summing 1 and 9 in the unit's place with a 1 carrying over to the tens place, which gets added to a one. So, 20. In binary, you add the ones in the 2^0, the 1 and 0 in 2^1 and the ones in 2^3, so-"
The addition resolved to get: 2012.
"But since we can't have a number greater than one, we need to start carrying over. A 2 in 2^0 means that you can basically add a 1 to the 2^1, so-"
2020
"A 2 in the 2^2 is basically a 1 in the 2^2, so-"
2100
"A 2 in the 2^3 is 1 in the 2^4, so we can add another binary place/"
10100
"Now if you read this binary number, it should also give you 20."
"This just feels longer," Shuri mumbled.
"Maybe for you, sure," the Sect Leader responded. "But if you think about automatic systems, if a decimal place exceeds 1, then it is the same as flipping up a switch in the subsequent decimal place. You don't have to worry about excessive carrying over. All operations are just ones and zeroes, in the end."
"I guess that makes sense," Shuri admitted with a faintly sceptical tone.
"Let us check out the subtraction and the rest," the Sect Leader encouraged.
For Shuri, wrapping her head around this new method of digitally representing numbers was tougher. To that, the Sect Leader said, "There are many reasons it's difficult for you compared to the usual method. For one, you started learning and were taught primarily in another method. This has become ingrained so deeply that it's hard to regularise yourself with another. On top of that, this method isn't tractable for common mathematics that you and I use. It's meant for something else altogether."
"That's the thing, I can't see how this can be applied," Shuri said with a defeated sigh. "I feel like I'm missing something important."
"It will make sense once we start logic. With that in place, all this knowledge will come together into a neat bundle," the Sect Leader declared.
Shuri's enthusiasm rose once again as she anticipated the arrival of the crucial puzzle piece that would untangle the mess in her mind right now. But just like yesterday, the Sect Leader glanced at the time and said, "That's it for today."
"What? Why?" Shuri argued in disbelief. "We still have time!"
"Technically, sure. But I have other commitments too, you know," the Sect Leader said with a smile. "But I like your enthusiasm! Once again, let me remind you that it is inadvisable to read up on this topic beforehand."
He chuckled and added, "You're the first student I've told to specifically not read before coming to class. So... I guess, congratulations on being the first..."
After that, Shuri felt herself moving down a familiar chute that directly deposited her back into her body.
____
Following another night of strained sleeping, trying to hold back her urge to devour knowledge, Shuri found herself being siphoned down the dark chute.
"Today's topic is quite heavy, so please stop me if things become difficult or if we're moving too quickly," the Sect Leader prefaced.
"Binary numbers can either be 1 or 0, which has already been established. You can attach meaning to this 1 or 0, by saying that 1 means 'on' and 0 means 'off'. You can also say that 1 is 'true' and 0 is 'false."
"Every decision or action you do or take is preceded by a set of conditions that need to be met. For instance, eating food depends on the precondition of you being hungry. Scratching your skin is preconditioned on whether you are feeling itchy in that location. So, to define an analogy using the previous definitions, if for an action A, you have a precondition B, if B is true, then your brain will send a signal: 1 to execute the action A. On the other hand, if it is false, you get a signal: 0 to not execute it."
As the Sect Leader said this, a line formed in front of Shuri. On either side, there were two bulbous dots. Above the left dot, a B showed up, and above the right dot, an A did.
B o--------o A
"We can define the behaviour of this system with a truth table:"
| B | A |
| 0 | 0 |
| 1 | 1 |
"But what if you have two preconditions for the execution of action A?"
As he said this, the line branched:
B o-------o A
C o----/
"If both B and C need to be true for A to execute, then what do you do?"
Shuri thought for a while before saying, "It's obvious, right?"
"True, but you need to place something at the intersection that computes the AND in this precondition," the Sect Leader hinted. "In fact, what we put here is a gate called the AND gate."
At the point of intersection, a curved arrow took shape, like a large semi-circle. It had two lines going in from one side that started from the preconditions B and C, and from the other side, a single line exited ending at A.
"With the AND gate, both B and C need to be true for it to trigger:"
| B | C | A |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
"Instead, what if all that you needed was for one of the conditions to be true? Either B OR C?"
"Would you use an OR gate?" Shuri hypothesised.
"Excellent deduction!"
The AND gate was replaced with another curved arrow that looked like a triangle pointing to the right but with the base at the left arching inwards.
"Can you deduce what the truth table would be like?"
Shuri was presented with an empty table, and a pen materialised in her hand. Carefully, she started to jot down her calculations.
| B | C | A |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
The Sect Leader proceeded to go through a myriad of other gates that started with the NOT gate, which inverts the input signal.
| B | A |
| 0 | 1 |
| 1 | 0 |
Then the NAND gate, which basically inverts the output of an AND gate.
| B | C | A |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
The NOR gate inverts the OR gate.
| B | C | A |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
And finally, the XOR - exclusive-OR - gate.
| B | C | A |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
"Now, we are going to use most of these gates together to create an addition circuit called the 'Full Adder'."
Two XORs, two ANDs and one OR gate arranged themselves before Shuri, and lines started to move from connection point to connection point, creating an elaborate yet easy-to-understand circuit (for Shuri).
"As you can see, we have three input lines and two output lines. You have the inputs B and C which are binary digits and the Cin, which is the carry-over input from the addition of the previous digit. The outputs are the summation, and the carry-over output for the next digital addition."
"Would you like to try drawing the truth table for this one?" Sect Leader Larks challenged.
Shuri moved quickly, her eyes traced the lines, and the writing utensil in her hand started to fill in the table with ease.
| B | C | Cin | A | Cout |
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 |
"That's correct," the Sect Leader confirmed. "As a challenge, can you recreate the full adder with nine NAND gates instead?"
As he said this, six buckets materialised in front of her each with a unique logic gate inside it: AND, OR, NOT, NAND, NOR and XOR.
She picked nine NANDs from the respective bucket and placed them in front of her and they became suspended in mid-air. She pulled lines from the inputs and outputs and started to join all the logic gates into a circuit. Within five minutes, a full circuit was formed in place.
The Sect Leader moved forward and provided the input signals to the circuit, and, as expected, the outputs produced replicated the initial truth table.
"Can you do it again, but this time with nine NORs?"
Shuri agreed readily before tossing the arranged NAND gates into their original bin and picking out nine NORs in their place.
As Shuri worked on it, her mana started to bubble with excitement. From her memories, she started to form connections between the information regarding Automagy she picked up from the Yoruz Clan's texts and logic gates.
'Maybe if I could...'
Shuri's hand halted, and she ejected herself from the special space she was in.
"Is something wrong?" The Sect Leader asked worriedly.
"I... I need to go. I have something to do."
With that said, Shuri quickly shot up from her seated pose and rushed out of the office towards her room. She was struck with inspiration. The idea of building a logic circuit had opened up a plethora of possibilities for her. Her mind was building a burgeoning circuit that didn't show any sign of slowing down.
"I-If this works, then..." it would mark a completely new and possibly revolutionary pathway for Automagy.
She picked out a fresh mana core that lay on her table and prepared her inscription tool. She placed the core on the stand and then channelled her mana into the gem at its side, causing an ethereal projection to extend outwards from it.
Then with a calming breath, she started inscribing the circuit that had finished forming within her mind.
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