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Product Last Digit Consistency Law of Number Space

Articles, Mathematics, Unidigit mathematics

even multiplicand means even product last digit, Multiplicand Components of Specific Last Digits

- Consistency of Product Last Digits of Real Numbers:
Elementary arithmetic shows that since 2*4 = 8, then 12*14 (= 168) does also end in digit 8, as does 32*64 (=2048). And that since 7*3 = 21 ends in digit 3, then 17*3 (=51) also ends in digit 1, as does 27*33 (= 891). Also, 7*7 = 49 ends in 9, implies that 37*47 (= 1739) must end in digit 9. In other words, the last digit of the product XY of any two numbers X and Y however large or small, is always consistent with the last digit of the product of the last digit of X and the last digit of Y alone. Confirm this that for numbers ending in digits 7 and 9, their product will always end in 3 as in 63.

The last digits of products are shown below for each ending digit of any number. Note that for odd last digits, their products are odd or even as the other multiplicand is odd or even. Since even numbers always have even last digits, their products are always even, including the cases even when the other multiplicand is odd. Since we are here dealing with real numbers, and not unidigits, we include the digit zero (0) in our last digit values to cover multiplicands of 10.

Due to the inherent definition of our 10-digit numeral system, the last digits of products of a digit X and the last digits of products of digit 10-X would add up to 10 for each other multiplicand [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 0] in the product. The last digits in the products for each multiplicand ending in digits [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 0] are as follows:

For digit 1, product last digits are [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 0]
For digit 9, product last digits are [ 9, 8, 7, 6, 5, 4, 3, 2, 1, 0] That is the reverse mirror of that for digit 1, since 9+1 = 10.

For digit 2, product last digits are [ 2, 4, 6, 8, 0, 2, 4, 6, 8, 0]
For digit 8, product last digits are [ 8, 6, 4, 2, 0, 8, 6, 4, 2, 0] That is the reverse mirror of that for digit 2, since 8+2 = 10.

For digit 3, product last digits are [ 3, 6, 9, 2, 5, 8, 1, 4, 7, 0]
For digit 7, product last digits are [ 7, 4, 1, 8, 5, 2, 9, 6, 3, 0] That is the reverse mirror of that for digit 3, since 7+3 = 10.

For digit 4, product last digits are [ 4, 8, 2, 6, 0, 4, 8, 2, 6, 0]
For digit 6, product last digits are [ 6, 2, 8, 4, 0, 6, 2, 8, 4, 0] That is the reverse mirror of that for digit 4, since 6+4 = 10.

For digit 5, product last digits are [ 5, 0, 5, 0, 5, 0, 5, 0, 5, 0] Multiples of 5 always end in 5 or 0, the latter (0) being for even multiples of 5. No other last digit for multiples of 5 except 5 itself and zero. Last digit zero (0) comes up only for multiples of 5 and 10.

Multiplicand Components of Specific Last Digits:
From the above, observe that there are limited sets of multiplicands that yield a specific last digit.
Last digit 1. comes up only for multiplicands ending in 3 and 7 (as in 17*3 = 51) or 9 and 9 (as in 9*19 = 171) or 1 and 1 (as in 11*21 = 231).

Last digit 3. comes up only for multiplicands ending in 7 and 9 (as in 17*9 = 153) or 1 and 3 (as in 13*11 = 143).
Last digit 7. comes up only for multiplicands ending in 3 and 9 (as in 19*3 = 57) or 1 and 7 (as in 11*27 = 297).

Last digit 9. comes up only for multiplicands ending in 7 and 7 (as in 17*7 = 119) or 1 and 9 (as in 21*19 = 399).

Last digit 2. comes up only for multiplicands ending in 4 and 3 (as in 13*14 = 182), or  4 and 8 (as in 14*18 = 252) or 6 and 2 (as in 12*16 = 192), or 7 and 6 (as in 7*6 = 42) or 9 and 8 (as in 18*9 = 162) or 1 and 2 (as in 11*12 = 132).

Last digit 4. comes up only for multiplicands ending in 3 and 8 (as in 13*8 = 104), or  4 and 6 (as in 14*16 = 224) or 2 and 2 (as in 12*22 = 264), or 9 and 6 (as in 19*6 = 114) or 8 and 8 (as in 18*8 = 144) or 1 and 4 (as in 11*14 = 154).

Last digit 6. comes up only for multiplicands ending in 3 and 2 (as in 13*2 = 26), or  6 and 6 (as in 9*16 = 96) or 4 and 4 (as in 14*24 = 336), or 9 and 4 (as in 19*4 = 76) or 8 and 2 (as in 18*2 = 36) or 1 and 6 (as in 11*16 = 176).

Last digit 8. comes up only for multiplicands ending in 3 and 6 (as in 13*6 = 78), or  4 and 2 (as in 14*22 = 308) or 4 and 7 (as in 14*7 = 98), or 9 and 2 (as in 19*12 = 228) or 6 and 8 (as in 18*6 = 108) or 1 and 8 (as in 11*18 = 198).

The above sets of last digits of products exhaust the space of all last digits for integer multiples without approximations of either multiplicand in each product of two real number integers.

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Unidigit behaviours of Prime numbers

Articles, Mathematics, Unidigit mathematics

7, number seeds

- Unidigit behaviours of Prime numbers:
A prime number is that divisible only by itself and one 1 without leaving a remaining fraction. That excludes all even numbers and multiples of 2 and 3 and so on. Ordinarily, this leaves only the numbers 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 127, 131 …179, 181, 203…307 and so on as prime numbers. Their sequence has always fascinated mathematicians, if only because prime numbers present arithmetic difficulties in many practical calculations involving division, thereby forcing the approximations of results.

The numbers 1,2,3,5 as prime numbers are easy enough to handle, since 1 is a simple identity; 2 is even and yields 0.5 in division; five in turn yields simple 0.2 in division; and 3 yields a consistent 0.333333 ad infinitum in division. It follows that 1, 2 and 5 complement each other digitally in fractions, in that 1/5 = 0.2 and 1/2 = .05 are simple enough to handle operationally. By itself, 3 is not only simple to handle as number, but quickly merges into the simplicity of 6 or 9 in multiplication and division.

Note however that multiples of 7 and therefore its exponents are all simple to handle just like other numbers. Yet, the one single digit prime number that presents some issue in operational manipulation is 7. Note that 1/7 = .142857142857142857142857 …. which repeats the sequence 142857 ad infinitum. This sequence has exactly six digits since 7-1 = 6. In comparison, the value of 1/13 = .0769230769230…. in a sequence of 12 self-repeating digits since 13-1= 12.

Among the unidigits, uniproducts of unidigit 7 are a mirror reverse sequence of the uniproducts of 2 as already shown. Note too that the 7th exponent (X^7) or septempower of unidigits follow their own unique sequence : 1 2 9 4 5 9 7 8 9 which is not similar to the sequence of other exponent powers.

Note that from 7 onwards, prime numbers are adjacent to a multiple of 6. For 11 and 13 are 12 minus and plus 1, and 17, 19 are 18-1 and 18+1. The reason for this is that between adjacent prime numbers, multiples of the other numbers 2,3,5, prevent the incidence of the primacy condition (division by another number). Hence through division operations, prime numbers do generate infinite number sequences.

This makes them crop up as seeds of various physical constants, and as seeds for generating random numbers (for encryption codes for example).

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Unidigit behaviour in Inverse Operations – Fractions

Articles, Mathematics, Unidigit mathematics

odd and prime number inverse sequences,

- Unidigit behaviour in Inverse Operations – Fractions:
The inverse of a number X is 1/X, one divided by the number; or what is the same thing X to exponent -1 (minus 1). The first difficulty with inverse functions is that division by some digits yields continuous fractions that make exact unidigits of the result tough to handle. Some approximation is therefore required which is less neat and robust than foregoing analysis that proved to be universally valid.

With some approximation in mind, the inverse of ordinary digits are as follows.
For the digit X = 1, 2, 3,   4, 5, 6,   7,   8, 9
Inverse = 1/X = 1, 5, 3+, 7, 2, 7+, 5+, 8, 1+.

Graphically, it will be observed that the unidigit inverse peaks for even unidigits 2, 4, 6, 8, but dips for the odd unidigits 1, 3, 5, 7, and 9. This has to do with the approximations for the odd unidigit inverses, except for unidigit 1.

Note that for once, unidigit 9 here succumbs from its frequent domination and leaves an inverse of 1.111111 ad infinitum which we approximate to 1. Similarly, unidigits 3 and 6 have inverses 3.333333 ad infinitum (approximated to 3) and 6.666666666 ad infinitum (approximated to 7) roughly here. The only odd unidigit with a well rounded full unidigit inverse is 5 with inverse unidigit equal to 2.

The inverse of number 7 = .142857142857142857 ….. in which the six-digit sequence 142857 repeats itself ad infinitum. Note that 7 is a prime number, and that the recurring sequence in its inverse has exactly 7-1 = 6 digit sequence. Note also that the inverse of 7 is in the constant pi* for converting linear radius to circular dimensions in geometry as in calculating the periphery and area of a circle or the volume of a sphere. Without detailed information, it is noted that 7 as a prime number here conforms to a known behaviour of prime numbers generally, and that 7 is the only pure prime number unidigit.

Of course, every number except the trio unidigits 3, 6 and 9 have some component prime numbers. For example, 13 and 67 are prime number component members of unidigit 4, as 23 and 41 are prime number component members of unidigit 5, just as 61 and 97 are of unidigit 7, while 53, and 89 are of unidigit 8.

Without going into further details here, it is noted that many (not all) multiples of 6 are adjacent in unidigit value to some prime number. Thus prime numbers 11 and 13 are adjacent to 12 : a multiple of 6, as are 19, 23, 29, 31, 37, 41, 47, 53, 59, 61 and so on.

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Unidigit Cubes and Higher Unidigit Exponents

Articles, Mathematics, Unidigit mathematics

Unicubes Uninovems 1 8 9, Unihex 1 9, Uniquadricals Unidecems 1 4 9 7, Uniquinciliacs 1 7 9 4, Uniseptems

- Unidigit Cubes and Higher Unidigit Exponents:
This covers the nth Exponential (X^n) of unidigits i.e. the series X-squared, X-cubed, ..for unidigit X to power as n goes from 2 (square) to 10 (power 10). We shall now pay attention to their graphical and numerical unidigit values and useful symmetries.

For n = 1, X^n gives the simple unidigits themselves as already discussed earlier on. And for n = 2, X^n gives the unisquares of unidigits as discussed in the here preceding section. Note again the symmetry between the unisquares for unidigit pairs 1,8,  2,7, 3,6, and 4,5. One may infer that analysis of just the unidigits 1,2,3,4 would say something about the likely behaviour of the other unidigits – 8,7,6, and 5.

For n = 3, X^n gives the Unicubes of unidigits for all component members of all unidigits. The corresponding values of such cubes are :
For Unidigit X = 1, 2, 3, 4, 5, 6, 7, 8, 9.
Unicube = X^3= 1, 8, 9, 1, 8, 9, 1, 8, 9,

That is, 1 8 and 9 are the only true unicubes.

It is noteworthy that unicubes and the 9th exponents of unidigits show similarity in unidigit end results for their corresponding component members. For n = 9, X^n gives the Uninovem powers of unidigits for all component members of all unidigits. The corresponding values of such Uninovems are  also:
For Unidigits  X = 1, 2, 3, 4, 5, 6, 7, 8, 9.
Uninovem = X^9= 1, 8, 9, 1, 8, 9, 1, 8, 9,

That is, 1 8 and 9 are the only true uninovems. In both cases (cubes as well as 9th or novem power functions), the resulting unidigit values are 1 or 8 or 9 and no other values for all component numbers however large or small.

All component member numbers of unidigit 1, 4, and 7 have a unicube 1. All component member numbers of unidigit 2, 5, and 8 have a unicube 8. In line with expectations from the earlier-mentioned overwhelming influence of 9 (see Unidigit Multiplication Table), all component members of unidigit 3 have a unicube and a 9th exponent with unidigit value equal to 9.

For n = 6, X^n gives the * powers of unidigits for all component members of all unidigits. The corresponding values of such 6th powers are :
For Unidigit X = 1, 2, 3, 4, 5, 6, 7, 8, 9.
Unihex power = X^6= 1, 1, 9, 1, 1, 9, 1, 1, 9.

This shows that all numbers have a unihex exponent (6th power) equal to unidigit 1, except for component members of 3 and all its multiples for which the hexpower is 9. You can confirm this with a calculator. For 1^6 = 1. And 2^6 = 64 (unidigit 1). 4^6 = 4096 (unidigit 1). 5^6 =15625 (unidigit 1). And 7^6 = 117649 (unidigit 1). And 8^6 = 262144 (unidigit 1). Without going into further details, this unique result for exponent to power 6 is a stabilising* property made possible by the unique factors 2 and 3 in the simple digit 6.

For n = 4, X^n gives the Uniquadricals* of unidigits for all component members of all unidigits. The corresponding values of such 4th powers are :
For Unidigit X = 1, 2, 3, 4, 5, 6, 7, 8, 9.
Uniquadrat = X^4= 1, 7, 9, 4, 4, 9, 7, 1, 9.

It is noteworthy that uniquadricals (X^4) and the 10th exponents (X^10) of unidigits show similarity in unidigit end results for their corresponding component members. For n = 10, X^n gives the Unidecem powers of unidigits for all component members of all unidigits. The corresponding values of such Unidecems are also:
For Unidigits  X  = 1, 2, 3, 4, 5, 6, 7, 8, 9.
Unidecem = X^10= 1, 7, 9, 4, 4, 9, 7, 1, 9.

Note again the nth power symmetry between the unidigit pairs 1,8,  2,7, 3,6, and 4,5. One may infer that analysis of just the unidigits 1,2,3,4 would say something about the likely behaviour of the other unidigits – 8,7,6, and 5 in the quadrical and decempower spheres.

For n = 8, X^n gives the Unioctopowers of unidigits for all component members of all unidigits. The corresponding values of such cubes are :
For Unidigit  X  = 1, 2, 3, 4, 5, 6, 7, 8, 9.
Unioctopowers = X^8= 1, 4, 9, 7, 7, 9, 4, 1, 9.

Note that these are the same unidigit results earlier found for unisquares. This surprising parallel between unisquares and unioctopowers for any number in the universe, is a new dimension of unidigit analysis to be treated in a later work. Note again the symmetry between the unidigit pairs 1,8,  2,7, 3,6, and 4,5. One may infer that analysis of just the unidigits 1,2,3,4 would say something about the likely behaviour of the other unidigits – 8,7,6, and 5 in the octopower sphere.

For n = 5, X^n gives the * of unidigits for all component members of all unidigits. The corresponding values of such cubes are :
For Unidigit  X  = 1, 2, 3, 4, 5, 6, 7, 8, 9.
Uniquinciliacs = X^5= 1, 5, 9, 7, 2, 9, 4, 8, 9.
For once, here in 5th powers are exponent unidigit powers that show no direct symmetry in the power values for unidigits.

For n = 7, X^n gives the Uniseptem powers of unidigits for all component members of all unidigits. The corresponding values of such 7th powers are :
For Unidigit  X  = 1, 2, 3, 4, 5, 6, 7, 8, 9.
Uniseptem powers = X^7= 1, 2, 9, 4, 5, 9, 7, 8, 9.
With 7 a prime number, septem exponent unidigit powers show no direct symmetry in the power values for unidigits.

So far we have covered the unidigit behaviour of the exponents of unidigits for all powers from 1 to 10. With that, we have covered all full digit exponents of all numbers in the universe. It is even more revealing to view these exponents in graphic form.
[Unidigit Exponent Graphs by Ayo ] *** yet to fill in ***

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Unidigit Product Graphs and symmetry pairs

Articles, Mathematics, Unidigit mathematics

mirror sequences, Odd and Even Uniproduct Sequences, swing 5

Unidigit Product Graphs:
The beauty of this numerical symmetry is best viewed graphically by plotting the two row pairs of product values for unidigits that are unisquare roots of the same unisquare. The graph will show that for each symmetry pair, as the one product rises to a peak, the other sinks to a low point. In other words, each symmetry pair defines an equivalent band of product rise and fall functionally.

For the symmetry pair 1 and 8, as the graph for unidigit 1 rises evenly upwards, that of unidigit 8 falls just as evenly downwards. Similarly for the pair 2 and 7.
[Unidigit Product Graphs by Ayo ]*** yet to fill in ***

For 2 [ 246813579 ] the graph is reverse of that for unidigit 7 and peaks (to 8 ) for unidigit 4.
For 7 [ 753186429 ] the graph peaks (to 8 ) for unidigit 5 and minimises to 1 at unidigit 4.
That is, with unidigit 9 as the plane of a mirror, the uniproducts for unidigits 2 and 7 are a mirror sequence of each other. For uniproducts, the sequence is rising for unidigit 2, as the sequence is decreasing for unidigit 7; yet such that for each unidigit the two corresponding uniproducts sum up to 9.

For 4 [ 483726159 ] the graph is reverse of that for unidigit 5 and peaks (to 8 ) for unidigit 2.
For 5 [ 516273849 ] the graph peaks (to 8 ) for unidigit 5 and minimises (to 1) at unidigit 2.
That is, with unidigit 9 as the plane of a mirror, the uniproducts for unidigits 4 and 5 are a mirror sequence of each other.

For 1 [ 123456789 ] the graph reverses the diagonal fall in unidigit 8 to peak (to 8 ) for unidigit 8.
For 8 [ 876543219 ] the graph diagonally falls from 8 for unidigit 1 to minimum (1) at unidigit 1.
That is, with unidigit 9 as the plane of a mirror, the uniproducts for unidigits 1 and 8 are a mirror sequence of each other. For uniproducts, the sequence is rising for unidigit 1, as the sequence is decreasing for unidigit 8; yet such that for each unidigit the two corresponding uniproducts sum up to 9.

For 3 [ 369369369 ] the graph is reverse of that for unidigit 6 but same peaks at unidigits 3, 6.
For 6 [ 639639639 ] the graph dives to minimum thrice for unidigits 2, 5, 8.
That is, with unidigit 9 as the plane of a mirror, the uniproducts for unidigits 3 and 6 are a mirror sequence of each other. For uniproducts, the sequence rises and falls for unidigit 3, as the sequence decreases and rises for unidigit 6; yet such that for each unidigit the two corresponding uniproducts sum up to 9.

Needless to say that the corresponding values for zero and 9 are respectively zero and 9 for all product columns. Showing that zero is a product destroyer or an empty placeholder or annihilator, and 9 is a rigid band. These and other characteristics of individual digits can be shown in more detail later.

Due to the symmetry between the unidigit pairs 1,8,  2,7, 3,6, and 4,5, one may infer that analysis of just the unidigits 1,2,3,4 would say something about the likely behaviour of the other unidigits – 8,7,6, and 5.

Odd and Even Uniproduct Sequences:
Note that for each unidigit, the set of uniproduct values starts with an odd or even value according as the unidigit itself  is odd or even because the first value corresponds to unidigit 1.

Note that the uniproduct value set for the lower unidigits from unidigit 1 (1,2,3, all below 5) show rising values, featuring odd and even uniproduct patterns. For example, for unidigit 2, the even increasing sequence is followed by the odd increasing sequence.

The uniproduct values for unidigits above 5 (6,7,8) show decreasing values featuring odd and even uniproduct patterns. For example, for unidigit 7, the odd decreasing sequence is followed by the even decreasing sequence. For example, for unidigit 2, the even increasing sequence is followed by the odd increasing sequence.

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Uniproduct Patterns and Symmetry pairs of unidigits

Articles, Mathematics, Unidigit mathematics

1 8, 2 7, 3 6, 4 5, swing centre 5,

- Uniproduct Patterns and Symmetry pairs of unidigits:
Note that in the Unidigit Uniproduct Table, the value set for uniproducts for each unidigit except for unidigits 3, 6, and 9, include all digits 1 to 9. The uniproducts value sets for 3, 6, and 9 only contain unidigits 3,6,and 9. With the unidigit 9, all other unidigits form a uniproduct equal to unidigit 9 itself.

Note also that certain pairs of unidigits have similar uniproduct value sets, even if in reverse orders. For example, uniproducts for 2 and 7 follow reverse sequence of each other. The uniproducts for 1 and 8 follow the reverse trends of each other, and those of unidigits 4 and 5 also reverse mirrors of each other. Note too that in all these symmetry pairs, this observed symmetry of uniproducts, the two unidigits in each pair both sum up to 9.

Note too that the symmetry values revolve on the value for unidigit 5. Unidigit 5 is thus called a swing centre for the uniproduct operation. Note also that the unidigit values for each symmetry pair are mirror patterns centred on the uniproduct value of 9 for unidigit 9. See examples below.

Unidigits 3 and 6 as Uniproduct Symmetry pair bounded in 9.
The supreme power of unidigit 9 is traced to the stabilising nature of its unisquare root 3. This for example accounts for triangular stability in the physical world.

For unidigit 3, the products are [ 3, 6, 9, 3, 6, 9, 3, 6, 9].  (see row 3 of Ayo’s Unidigit Product Table)
That is, 3, 6 and 9 are the only three unidigit products ever formed by unidigit 3. In other words, unidigit 3 and all its component members (e.g. 3, 21, 93, 66, 75) do not yield product members of 1 nor 2, 4, 5, 7, 8. Moreover the product of any two unidigits of which 3 is a member of both, will always have unidigit 9 since by being in both product factors, 3 becomes squared into 9.

The products of unidigit 6, (row 6 of Ayo’s Unidigit Product Table) are 6, 3, 9, 6, 3, 9, 6, 3, 9.
That is, 6, 3, and 9 are the only three unidigit products ever formed by unidigit 6. In other words, unidigit 6 and all its component members (e.g. 15, 24, 96, 168, 75) do not yield product members of 1 nor 2, 4, 5, 7, 8. This is similarly true for unidigit 3. Since unidigit 6 is a uniproduct of unidigit 3 (as 2*3=6), it follows that the product of any two unidigits of which 6 is a member of both, will always have unidigit 9 since by being in both product factors, 3 as factor unidigit becomes squared into 9.

Note in the two rows for unidigits 3 and 6 in Ayo’s Unidigit Product Table that for each column, the products for both unidigit 3 and 6 add up together to 9 through out. That is, where the one has as product 3, the other has 6, and otherwise they are both have 9. In other words as symmetrical product pair, 3 and 6 have corresponding products summing up to 9.

This is much like the symmetry earlier indicated between 4 and 5 as unisquare roots (of 7), and like the symmetry between 2 and 7 as unisquare roots (of 4), and the symmetry between 1 and 8 as unisquare roots (of 1). Verify this wide symmetry in the respective rows 4 and 5, then rows 2 and 7, and rows 1 and 8. For each of these symmetry pairs, their corresponding uniproducts row for row, each sum up to 9 in each column.

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Exponential extensions of Ayo’s Unidigit consistency Law:

Articles, Mathematics, Unidigit mathematics

symmetry unisquares-1 4 7 9 unsquare-roots, supremacy of unidigit 9

Exponential extensions of Ayo’s Unidigit Law:
As in multiplication, unidigits show interesting behavioural regularities that even run between specific higher levels of exponentiation. Ordinarily, for a number X, the exponents (or powers) X^2, X^3, X^4 … to X^10 can be examined as the series of squares, cubes, .. etc for X as X varies from 1 upwards. Similar exponential operations with unidigits enables us to exhaust the behavioural range for all full digit exponents of numbers. It turns out that the range of these exponents is quite limited not only in values, but also in their characteristics as follows.

Unidigit Squares and Unidigit Square Roots:  

See table diagonal**  for unidigit squares 1497
As already shown, the law holds for X*Y = Z. The special case of Y=X makes it valid for X*X = square of X = Z which gives the unidigit of squares. The specific unidigits of the squares are clearly along the diagonal descending from left to right on the unidigit products table, namely 1, 4, 9, 7, 7, 9, 4, 1 and the final 9.

Observe the symmetry* of the series 1, 4, 9, 7, 7, 9, 4, 1 that centres on the 4th and 5th unidigits. That is, specifically, the unidigits of all squares in the universe are component members of 1, 4, 9, or 7. Outside of these four unidigits, full squares are not observable. This implies that the unidigits 2,3,5,6 and 8 and their component members have no full squares. Unidigit 9 is a unique unidigit square as will be seen here below.

The observed centre of symmetry also implies that unidigits 4 and 5 have the same unidigit square namely 7. For 4*4 = 16 , a component member of unisquare* 7, and 5*5 = 25, also a component member of unisquare7. That is, 4 and 5 are therefore ‘square symmetrical’ unidigits. They 4 and 5 are unisquare roots* of 7. Note also that the sum of these two symmetrical unisquare roots 4+5 = 9; a very special square in itself. That 14*14 = 196 and 13*13 = 169 both confirm 13 and 14 to be unisquare roots of unisquare 7. Other unisquare roots of 7  include 49, 58, 77, 68, 12865 and 78467, for example.

Symmetry among Unisquare Roots:
From the observed symmetry, note also that for the unisquare 1, the unisquare roots are unidigits 1 and 8. And for their component members 19 and 17, for example. That is, 17*17 = 289 with unidigit 1, and 19*19 = 361 with unidigit 1. Note that 1 and 8 as ‘symmetrical unisquare roots’ of the same unisquare 1, their sum (8+1=9) is unidigit 9. Remember from the preceding paragraph that unidigits 4 and 5 are both unisquare roots of the same unisquare 7, and that both 4+5 do sum up to unidigit 9. This again shows one of the special characteristics of the unidigit 9 for unisquares 1 and 7.

This special characteristic of unidigit 9 can also be shown for the unisquare 4. For the square 7*7 = 49 which a component member of unidigit 4, and the square 2*2 = 4 with the same unisquare 4. And here too, the two ‘symmetrical unisquare roots’ 2 and 7 of the same unisquare 4, both add up (2+7 = 9) to unidigit 9. Of the four unisquare digits 1,4,9, and 7, we need only now confirm the unisquare roots of 9 itself.

Observe that for the unidigits 3 and 6, their squares 3*3 = 9 and 6*6 = 36 both have unidigit 9 as
their unisquare. Moreover, their sum 3+6 = 9 as for other symmetrical square roots pairs of the same unisquare. And their unisquare happens to be the very special unidigit 9. This concludes the case for symmetrical unisquares 1, 4, 9 and 7 in their natural order, and for the component members of their unisquare roots.

Supremacy of unidigit 9 as overwhelming uniproduct, and unisquare root:
Above all that symmetry of unisquares is 9 as unidigit, but also as unisquare of itself (for 9*9 = 81, which is a component member of 9). From Ayo’s Unidigit Product Table, it will be observed in the lowest row and in the last column that all products of 9 and any digit always result in unidigit 9. In other words, digit 9 and all its component members absorb all other numbers into their unidigit 9. This makes 9 the supreme inflexible component unidigit with respect to mathematical operations multiplication, square and whole digit exponents of which it happens to be member in any form. The appropriate descriptive term for this behaviour of 9 is supreme stability capture – unidigit 9 captures all numbers in any product mode involving the unidigit 9 and stabilises the result in itself as 9.

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