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For example, decimal 365 (10) or senary 1 405 (6) corresponds to binary 1 0110 1101 (2) (nine bits) and to ternary 111 112 (3) (six digits). However, they are still far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary (base 9) and septemvigesimal (base 27).
Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented (this notation is used in this article). 1111011 2 implies that the number 1111011 is a base-2 number, equal to 123 10 (a decimal notation representation), 173 8 and 7B 16 (hexadecimal).
Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base b is equal to −r for some natural number r (r ≥ 2). Negative-base systems can accommodate all the same numbers as standard place-value systems, but both positive and negative ...
A multiplication by a negative number can be seen as a change of direction of the vector of magnitude equal to the absolute value of the product of the factors. When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The sign of the product is determined by the following rules:
It follows that a number is a decimal fraction if and only if it has a finite decimal representation. Expressed as fully reduced fractions, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are
One must multiply the leftmost digit of the original number by 3, add the next digit, take the remainder when divided by 7, and continue from the beginning: multiply by 3, add the next digit, etc. For example, the number 371: 3×3 + 7 = 16 remainder 2, and 2×3 + 1 = 7. This method can be used to find the remainder of division by 7.
The decimal digital root of any non-zero integer will be a number in the range 1 to 9, whereas the digit sum can take any value. Digit sums and digital roots can be used for quick divisibility tests : a natural number is divisible by 3 or 9 if and only if its digit sum (or digital root) is divisible by 3 or 9, respectively.
The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be sorted, and, except for the first two digits, all digits must be 7, 8, or 9. There are also additional restrictions on the first two digits.
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