# Shift of Base

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Related KMR pages:

• Numbers and their Digits in different Bases.
Representation and Reconstruction of Numbers.
Shift of Basis (in general).

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Other sources of related information:

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Experiment here with computing the Digits of a Number with respect to different Bases.

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The Base-Shift formula for Numbers

It is often useful to have a symbolic name for the operation of shifting the $\, B_{ase} \,$ of a $\, n_{umber} \,$ from $\, {B_{ase}}_1 \,$ to $\, {B_{ase}}_2 \,$. This operation will be called $\, B_{aseShift} \,$ and its action on a given $\, n_{umber} \,$ will be written in three different ways:

i) $\, {[B_{aseShift}]}_{{B_{ase}}_2}^{{B_{ase}}_1} \;$, or

ii) $\, {[n_{umber}]}_{{B_{ase}}_2}^{{B_{ase}}_1} \;$, if the $\, B_{aseShift} \,$ operation is clear from the context, or

iii) $\, {[\,\,\,]}_{{B_{ase}}_2}^{{B_{ase}}_1} \;$, if both the $\, n_{umber} \,$ and the $\, B_{aseShift} \,$ operation are clear from the context.

A Base-Shift Diagram for Numbers:

Numbers are naturally related to Music:

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