[Math] Converting Between Non-Decimal Bases

number-systems

How can a number in an arbitrary base be converted to another random base – for example from 9.2E9B03 base 17 to base 5? I would prefer a generalized procedure avoiding an intermediary base 10 conversion – which I already know how to do.

Best Answer

You can use the same process you use for converting to/from base 10 but you will have to do arithmetic in another base. $17_{10}=32_5$, so $0.2E9B03_{17}=\frac{2\cdot 32^5+30\cdot 32^4+14\cdot 32^3+21\cdot 32^2+3}{32^6}_5$ Now multiply out the numerator and do long division in base $5$ to get the part right of the fraction point. For the whole numbers you do the same thing without the division.

Related Solutions

Number Systems – How to Change a Number Between Arbitrary Bases

Let $x$ be a number. Then if $b$ is any base, $x \% b$ ($x$ mod $b$) is the last digit of $x$'s base-$b$ representation. Now integer-divide $x$ by $b$ to amputate the last digit.

Repeat and this procedure yields the digits of $x$ from least significant to most. It begins "little end first."

EDIT: Here is an example to make things clear.

Let $x = 45$ and $b = 3$.

x   x mod 3
45    0
15    0                (integer divide x by 3) 
 5    2
 1    1

We see that $45 = 1200_3$. Read up the last column to get the base-3 expansion you seek. Let us check.

$$1\cdot 3^3 + 2\cdot 3^2 + 0 + 0 = 27 + 18 = 45.$$

I hope this helps you.

[Math] Negative decimal to other base conversion

The $-$ sign is not something special to base $10$. In general if $83.5_{10} = 1010011.1_2$ then $-83.5_{10} = -1010011.1_2$. What you are doing here is an extension of two's complement. Two's complement is a method to represent both positive and negative numbers in binary without using a $-$ and is used in computing. Also, it only works for integers. This is how fractions are represented.

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Number Systems – How to Change a Number Between Arbitrary Bases

[Math] Negative decimal to other base conversion

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