Is there a numeral system that makes both addition and multiplication easy? Decimal positional notation, the system for writing numbers we all use every single day, makes addition very easy by transforming it from a computation to a repeated operation on individual digits (long hand addition). A more formal description would be that if two numbers $a$ and $b$ are presented in decimal notation, long hand addition is a constant-memory, logarithmic time algorithm to obtain the decimal notation of the number $a+b$.

Multiplication (that is, long multiplication) is not so easy in decimal notation, requiring doubly logarithmic memory and logarithmic time while relying heavily on long hand addition as a "subroutine".

It is of course straightforward to invent a numeral system where the roles of the two operations are reversed: Just use the decimal notation of $\log(a)$ to denote the number $a$. Multiplication then becomes as simple as long hand addition in standard notation, because $\log(ab)=\log(a)+\log(b)$ (this is the principle behind the slide rule). Unfortunately, the price one has to pay for this convenience is that addition is suddenly very difficult.

Is there a numeral system that provides a balance between these two extremes, that is, a system for writing numbers in such a way that both addition and multiplication can be done by hand with less effort than decimal long multiplication, but perhaps greater effort than decimal long addition? (Note the focus on still being a system that is of practical use for manual computations; I doubt anyone would be able to execute the Schönhage–Strassen algorithm by hand, even it it has better asymptotic complexity than long multiplication.)
Maybe using primoradic(see :https://oeis.org/wiki/Primorial) . Very difficult to answer here.

Addition : We use the remainders and quotients of Euclidean divisions. The quotients become R.
$Example:4096+1024+512+256+2+1=5891$
$$primes... :11-7-5-3-2-1$$
$$R:1-2-1-2$$
$$4096=(1-8-3-2-2-0)$$
$$1024=(0-4-6-0-2-0)$$
$$0512=(0-2-3-0-1-0)$$
$$0256=(0-1-1-2-2-0)$$
$$0002=(0-0-0-0-1-0)$$
$$0001=(0-0-0-0-0-1)$$


$$5891=(2-6-0-1-2-1)$$

Multiplication :$137\times43=5891$
Here, first I will convert to base 2:
$$137\times43=(10.001.001)_2\times(101.011)_2=(1.011.100.000.011)_2=4096+1024+512+256+2+1=(1:8:3:2:2:0)+(4:6:0:2:0)+(2:3:0:1:0)+(1:1:2:2:0)+(1:0)+1=(2:6:0:1:2:1)$$according to the example given for the addition.
A better example to illustrate multiplication would be calculating the powers of two in primoradic, first by hand, then using a spreadsheet up to $2 ^ {16384}$ for example, or the powers of ten, or the powers of any integer, for example three($1,3,9,...$). See : I&#39;m trying to generalize some simple results about $ 2 ^ n $. It&#39;s useful to write them in primoradic (see stub OEIS).. Let us take three. With a pen and a sheet of paper, old fashioned way, using Charles-Ange Laisant's notation in his article of 1888 : 1=(1); $3=3\times1=3\times(1)=(11)$; $9=3\times3=3\times(11)=(111)...$



Remark : 4 = (2:0) = ((1:0):0)...
If you haven't heard of the duodecimal system (or its variations) you might give it a look over. Some (obvious) examples which operate wrt the duodecimal system...


Standard timekeeping.


60 s in a min.
60 min in an h.
24 h in a day.

The imperial system.


A thou = $\frac{1}{12000}$'.
An inch = $\frac{1}{12}$'.
A yard = $3$'.
etc.



1 of the reasons why the duodecimal system is desirable, &amp; why people think that it simplifies hand calculations, has to do w/ $12$ having so many factors $\left(1\;2\;3\;4\;6\;12\right)$. It really shines wrt to division/ratios/proportions, e.g., for a carpenter to perform calculations based on the partitioning of a log into 3 equal pieces, he must consider their lengths to be...


$0.\overline{3}$ $\left[\mathrm{m}\right]$, if the log length was $1$ $\left[\mathrm{m}\right]$.
$4$ $\left[\mathrm{ft}\right]$, if the log length was $12$ $\left[\mathrm{ft}\right]$.