What is the base-b expansion of a number and how is it calculated?

[iceblits]

Has anyone come across how to find "the base-b expansion" of a number? I don't think its tricky or anything I just don't know what it's referring to...

 

[iceblits]

Oh it's just referring to writing a number in a different base...

 

[Bacle2]

Has anyone come across how to find "the base-b expansion" of a number? I don't think its tricky or anything I just don't know what it's referring to...

AFAIK, expressing a number in base c it refers to expressing a number, usually given in base 10-- in the form:

(b₀b₁...b_m.b_(m+1)...b_(m+k))_c

(let's assume for simplicity the decimal expansion is finite)

Which represents the expression:

1)Whole Part: b_m+b_(m-1).c+ b_(m-2)c²+

...+b_mc^m

2)Decimal Part: b_(m+1)c^-1+b_(m+2)c^-2+


...+b_(m+k)c^-k+...


An example I think most would be familiar with would be a binary string, say:

(110.0101)₂

Which stands for:

1')Whole Part:

0.2⁰+1.2¹+1.2²=2¹+2²=6

2')Decimal Part:

0.2^-1+1.2^-2+0.2^-3+1.2^-4=

1/4+1/16=5/16

So the string 110.0101

Represents , in base 2, the number 6.325 in base 10.

Or, like (13.2)₁₀ represents 3.10⁰+1.10¹ in the

whole part, and 2.10^-1 in the4 decimal part.