Number bases/Comparison

Notice: Incompletable. There are infinitely many number bases.

Binary

Advantages

As computers naturally count in binary, no computational effort is wasted on converting to and from binary.

There are no non-trivial rows in the multiplication table.

There is only one non-trivial sum in the addition table (1+1=10).

Disadvantages

As 10 is a prime number, only powers of the base will have terminating expansions.

The only other number that is usable is 11, with an alternating-sum divisibility test and a simple expansion (1/3=0.01).

The number base is small, thus making numbers harder to remember. This means that partial products in multiplication are difficult to do.

Some decimal 10x10 multiplications can become binary 111x111 multiplications.

Ternary

Advantages

There is only one non-trivial product in the multiplication table (2*2=11) which can be removed with balanced ternary (-1, 0, 1)

There are only 10 non-trivial sums (1+1=2, 1+2=10, 2+2=11) and one of them (1+2=10) can be removed with balanced ternary.

Disadvantages

As 10 is a prime number, only powers of the base will have terminating expansions.

There are only two other usable numbers, 2 and 11, having a sum and alternating-sum divisibility test. Their expansions are 0.1 and 0.02 respectively.

The small size of the number base is an issue as it results in longer, harder to remember numbers. A decimal 2x2 multiplication could become a ternary 12x12 multiplication.

Quaternary

Advantages

It is a square base, meaning that only two could be a maximally recurring prime reciprocal (which it isn't because 10 is even).

Minimal computational effort is used up on conversions to and from binary (one quaternary digit is two binary digits).

The only non-trivial multiplication rows are 2 and 3. As 10 divides 2, that row should be easy, leaving 3. 3 can be removed with signed quaternary (-2, -1, 0, 1, 2).

Quaternary can easily handle the first 3 primes:

2: Is the last digit even? 1/2=0.2

3: Do the digits add to a number divisible by 3? 1/3=0.1

11: Is the alternating sum divisible by 11? 1/11=0.03

Disadvantages

As 10 is a prime power, only powers of said prime will have terminating expansions.

The small size of the number base may still be an issue, as decimal 2x2 multiplications could become quaternary 10x10 multiplications.

Quinary

Disadvantages

As 10 is a prime number, only powers of the base will have terminating expansions.

The small size of the number base may still be an issue, as decimal 2x2 multiplications could become quinary 3x3 multiplications.

The 2- and 3-times tables can't be fully removed with signed quinary, as you'd still need to remember your 2-times tables.

Senary

Advantages

The only non-trivial multiplication rows are 2, 3, 4 and 5. As 10 divides 2 and 3, those rows should be easy, leaving 4 and 5. They can be removed with signed senary (-3, -2, -1, 0, 1, 2, 3).

It is the largest number base where all difficult rows can be removed.

Senary can easily handle the first 4 primes:

2: Is the last digit even? 1/2=0.3

3: Is the last digit divisible by 3? 1/3=0.2

5: Do the digits add to a number divisible by 5? 1/5=0.1

11: Is the alternating sum divisible by 11? 1/11=0.05

Disadvantages

The small size of the number base may still be an issue, as decimal 2x2 multiplications could become senary 3x3 multiplications.

Septenary

Disadvantages

As 10 is a prime number, only powers of the base will have terminating expansions.

The small size of the number base may still be an issue, as decimal 2x2 multiplications could become septenary 3x3 multiplications.

The 2-, 3-, 4- and 5-times tables can't be fully removed with signed septenary, as you'd still need to remember your 2- and 3-times tables.

Octal

Advantages

Minimal computational effort is used up on conversions to and from binary (one octal digit is three binary digits).

Disadvantages

As 10 is a prime power, only powers of said prime will have terminating expansions.

The small size of the number base may still be an issue, as decimal 2x2 multiplications could become octal 3x3 multiplications.

The 3- and 5-times tables can't be fully removed with signed octal, as you'd still need to remember your 3-times tables.

Nonary

Advantages

It is a square base, meaning that only two could be a maximally recurring prime reciprocal.

Disadvantages

As 10 is a prime power, only powers of said prime will have terminating expansions.

The small size of the number base may still be an issue, as decimal 2x2 multiplications could become nonary 3x3 multiplications.

The 2-, 4-, 5- and 7-times tables can't be fully removed with signed nonary, as you'd still need to remember your 2- and 4-times tables.

Decimal

Advantages

Decimal can easily handle the first 3 primes:

2: Is the last digit even? 1/2=0.5

3: Do the digits add to a number divisible by 3? 1/3=0.3

5: Is the last digit divisible by 5? 1/5=0.2

Disadvantages

The 3- and 7-times tables can't be fully removed with signed decimal, as you'd still need to remember your 3-times tables.

Undecimal

Disadvantages

As 10 is a prime number, only powers of the base will have terminating expansions.

The 2-, 3-, 4-, 5-, 6-, 7-,8- and 9-times tables can't be fully removed with signed undecimal, as you'd still need to remember your 2-, 3-, 4-, and 5-times tables.

Duodecimal

Advantages

As 10 is a superior highly composite number, this leads to a lot of easy multiplication rows.

Disadvantages

The 5- and 7-times tables can't be fully removed with signed duodecimal, as you'd still need to remember your 5-times tables.