Quaternary random bit, photonics and new electronics
The bit are the basis of today electronics but perhaps we can change electronics in a very smart way. If we use bit not binary but quaternary perhaps the velocity of calculation can improve more more, but we can change also Boolean logic which until today is the bit logic. We can also use ternary logic but quaternary is better because it's possible to make a vectorized logic and perhaps the logic of quantum physics electronics! Now we see how changes the logic with quaternary logic and then we use it for quantum physics. When you use binary basis, you use only the number 0, or 1, and so the Boolean logic of true and false. When you use the quaterbit you must use the 0,1,2 and 3 and some different logic rules. If we see table of the truth of the sum we can see the differences.
Sum |
0 |
1 |
2 |
3 |
0 |
0 |
1 |
2 |
3 |
1 |
1 |
2 |
3 |
10 |
2 |
2 |
3 |
10 |
11 |
3 |
3 |
10 |
11 |
12 |
Where the number underlined indicates the second quaterbit. We can see that the table is very different from the sum of bit :
Sum |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
10 |
if we put only the underlined digit in the sum table we have OR logic port but if we make the same in the first table what does we have? I think now is difficult to answer but after some consideration we can answer that we have the logic of quaterbit. Now we explain some operations in bit logic and then we extend rules to the quaterbit logic. If we make a sum with the three bits and a bit for the sign we have:
0_011+ 0_100 = 0_111 but if we have 0_110 – 0_101 = 0_001 but also we can complementate to 1 or to 2 the second number and 0_110+ 1_010 +0_001= 0_110 + 1_011= 10_001 and ignoring the underlined bit = 0_001. It's possible to extend this rules of complementation but the complementation is little different in quaterbit logic:
Complementations |
0 |
1 |
2 |
3 |
Complementation to 1 |
3 |
2 |
1 |
0 |
Complementation to 2 |
10 |
3 |
2 |
1 |
0_3-0_2=0_3+1_1+0_1= 0_3+1_2=1 0_1
ignoring the underlined 1 we have the correct answer.
Now if we want to use all the information of quaterbit of sign we can add this rule if quaterbit is 0 the number in positive, if it is 1 the number is negative but if it is 2 the number is positive but along y axis and if it is 3 is negative along y, 0 and 1 obviously for x axis. In this manner we can use the quaterbit becomes a number which has 2 dimensions vector properties. But the amusing property is that you can write a number without a particular order, i.e. if we write 3_01+ 1_32 = 1_32+ 3_01 or
3_01 1_32 = 1_32 3_01 without the + sign because the numbers are real numbers or better : complex number!! Now, you can understand the word “random” in the name quaterbit you can write them in the order you like. Obviously we must rebuild better this algebra, because with complex numbers we make the algebraic sum, the product and division so let us rebuild complex algebra.
Sum:
Where the signseparates the sign from the other digit so the sum among different axes is commutative it's easy and I think it's easy demonstrated all properties but we can have some accuracy with symbol for the digit so :
if modulus of :
For imaginary axes:
Whereindicates the zero. So these are the rules for the sign for the modulus the rules are the same for real number when we sum 2 different axes they are indifferent one to other:
and so on because the axes are orthogonal and so they are independent or better indifferent among themselves. Now solved the sum, we must solve the rules for multiplication and division it is useful for the solution the rules of Algebraic Couples of William Rowan Hamilton about 1840 before the his Real Quaternions born on 16th October 1843, under the sign of the Libra . It is simple to deduces them you must multiply the complex numbers:
we change a+ ib in ib+ a = because it's possible, with this notation, changing the order without losing information so if divide numbers in sign and modulus or mantissa so for the signs:
the same for viceversa, then
for the other axes:
now the rules for the mantissas :
Complex Conjugation and Division
If we have a complex number the conjugation changes the number in number with the same real part and the same imaginary part but with opposite sign in our notation:
if we define the commutation of the signs :
the real commutation changes sign to real part, the imaginary commutation is conjugation 2 commutations give the same sign, so we can define the division using the rules of W.R. Hamilton, for the sign the same rules
where ^{ }it's real commutation which changes the sign of the product bd or dd.
The algebra is closed now but how store it in an electronic memory ? Normally, the complex are two ordinated binary numbers but with these bit code, which is possible to encode with 2 simple binary bits e.g. 00;01;10;11 instead of 0,1,2,3 like in my notation, it' s possible to store in a random fashion code, in fact we can put first the real part and then the imaginary part, but also in the reverse sense, and also only the real part and only the imaginary part, because it's useless or better: not efficient to store like today, now we see it in details.
Quite Random Storing
Suppose we store first a complete number, to mark the beginning of a different number, we put after the first the four bit 0000, after we can store another number, not necessary complete but also pure real or pure imaginary and then the sequence 0000 or 1111.So the storing of second number can have different length and this more efficient because we store more numbers using less bits. We can use the rest of bits for others bits shifting them back. But how we can make a quaterbit ? For Example we can use the atomic orbital levels of atom suppose four excited level and a ground level, so we can have four type of 1 photon emissions and so we can call this type 0, for 0->1, 1 for 2->1, and so on but we can use also rotational levels or vibrational levels so in theory it's possible managing the atoms in new devices, but the logic ? It' possible using Aristotelian logic if the Heisenberg's Principle don't work against the “long time storing” or if we can correct in some manner the error due to non stimulating emission with some mechanism like control bit but realized for this new devices, we don't analyze this now.
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