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

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In summary, expressing a number in base b refers to writing a number in the form of (b0b1...bm.b(m+1)...b(m+k))c, where b is the base and c is the number given in base 10. This involves representing the whole part and decimal part separately. An example of this would be a binary string, such as (110.0101)2, which represents the number 6.325 in base 10.
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iceblits
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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...
 
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Oh it's just referring to writing a number in a different base...
 
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iceblits said:
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...bm.b(m+1)...b(m+k))c

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

Which represents the expression:

1)Whole Part: bm+b(m-1).c+ b(m-2)c2+

...+bmcm

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.
 
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1. What is base-b expansion of a number?

Base-b expansion of a number is a method of representing a number in a different base, or number system. It involves writing the number as a sum of multiples of powers of the new base, with the coefficients being the digits of the new number system.

2. How is base-b expansion different from the decimal system?

The decimal system, or base-10 system, uses 10 digits (0-9) to represent all numbers. Base-b expansion uses a different number of digits depending on the base, and the value of each digit is determined by its position in the number.

3. Why is base-b expansion useful?

Base-b expansion allows us to easily convert numbers between different number systems. It is also useful in computer programming, cryptography, and other fields that involve working with different number bases.

4. Can any number be represented in base-b expansion?

Yes, any number can be represented in base-b expansion. However, some numbers may have repeating or never-ending expansion in certain bases, such as 1/3 in base-10.

5. How do I convert a number from base-b expansion to decimal?

To convert a number from base-b expansion to decimal, you can use the formula:
decimal number = (first digit * b^(n-1)) + (second digit * b^(n-2)) + ... + (last digit * b^0)
where b is the base and n is the number of digits in the expansion. For example, to convert 1101 from base-2 to decimal, we have: (1 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 8 + 4 + 0 + 1 = 13.

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