Unusual Bases for Number Systems

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Sections

• 12-1 Base $-2$
• 12-2 Base $-1+i$
• 12-3 Other Bases
• 12-4 What Is the Most Efficient Base?

Problems

1. Schroeppel’s formula for converting from base $-2$ to binary has a dual involving the constant 0x5555555. Can you find it?

2. Show how to add 1 to a base $-2$ number using the arithmetic and logical operations of a binary computer. For example, 0b111 becomes 0b100.

3. Show how to round a base $-2$ number down, in the negative direction, to a multiple of 16 using the arithmetic and logical operations of a binary computer. For example, 0b10 becomes 0b110000.

4. Write a program, in a language of your choice, to convert a base $-1+i$ integer to the form $a+bi$, where $a$ and $b$ are real integers. For example, if you give the program the integer 33, or 0x21, it should display something like $5-4i$.

5. How would you convert a number in base $-1+i$ to its negative? Extract its real part? Extract its imaginary part? Convert it to its complex conjugate? The complex conjugate of $a+bi$ is $a-bi$.