9.65 RadixExpansion

It possible to expand numbers in general bases.

Here we expand 111 in base 5. This means

111::RadixExpansion(5)

\[\]

421

_{Type: RadixExpansion 5}

You can expand fractions to form repeating expansions.

(5/24)::RadixExpansion(2)

\[\]

0.00110‾

_{Type: RadixExpansion 2}

(5/24)::RadixExpansion(3)

\[\]

0.012‾

_{Type: RadixExpansion 3}

(5/24)::RadixExpansion(8)

\[\]

0.152‾

_{Type: RadixExpansion 8}

(5/24)::RadixExpansion(10)

\[\]

0.2083‾

_{Type: RadixExpansion 10}

For bases from 11 to 36 the letters A through Z are used.

(5/24)::RadixExpansion(12)

\[\]

0.26

_{Type: RadixExpansion 12}

(5/24)::RadixExpansion(16)

\[\]

0.35‾

_{Type: RadixExpansion 16}

(5/24)::RadixExpansion(36)

\[\]

0.7I

_{Type: RadixExpansion 36}

For bases greater than 36, the ragits are separated by blanks.

(5/24)::RadixExpansion(38)

\[\]

0.734312512‾

_{Type: RadixExpansion 38}

The RadixExpansion type provides operations to obtain the individual ragits. Here is a rational number in base 8.

a := (76543/210)::RadixExpansion(8)

\[\]

554.37307‾

_{Type: RadixExpansion 8}

The operation wholeRagitswholeRagitsRadixExpansion returns a list of the ragits for the integral part of the number.

w := wholeRagits a

\[\]

[5,5,4]

_{Type: List Integer}

The operations prefixRagitsprefixRagitsRadixExpansion and cycleRagitscycleRagitsRadixExpansion return lists of the initial and repeating ragits in the fractional part of the number.

f0 := prefixRagits a

\[\]

[3]

_{Type: List Integer}

f1 := cycleRagits a

\[\]

[7,3,0,7]

_{Type: List Integer}

You can construct any radix expansion by giving the whole, prefix and cycle parts. The declaration is necessary to let FriCAS know the base of the ragits.

u:RadixExpansion(8):=wholeRadix(w)+fractRadix(f0,f1)

\[\]

554.37307‾

_{Type: RadixExpansion 8}

If there is no repeating part, then the list [0] should be used.

v: RadixExpansion(12) := fractRadix([1,2,3,11], [0])

\[\]

0.123B0‾

_{Type: RadixExpansion 12}

If you are not interested in the repeating nature of the expansion, an infinite stream of ragits can be obtained using fractRagitsfractRagitsRadixExpansion.

fractRagits(u)

\[\]

[3,7,3,0,7,7‾]

_{Type: Stream Integer}

Of course, it’s possible to recover the fraction representation:

a :: Fraction(Integer)

\[\]

76543210

_{Type: Fraction Integer}

More examples of expansions are available in DecimalExpansionXmpPage , BinaryExpansionXmpPage , and HexadecimalExpansionXmpPage .