CSE 30 -- Lecture 2 -- Oct 1

Polynomial representation of a number: in base 10, the number n = 524398, denoted 524398₍₁₀₎ equals 5 * 10⁵ + 2 * 10⁴ + 4 * 10³ + 3 * 10² + 9 * 10¹ + 8 * 10⁰. Suppose I want to express this same number in base 2. The ``ones'' binary digit (or bit) can be determined as follows: we write n = m₁ * 2 + n₀. We get n₀ = 0, and m₁ = 262199. We write m₁ = m₂ * 2 + n_{1, and get n1 = 1 and m2 = 131099. Repeated division by the base and writing the remainder down (in reverse order) gives 1000 0000 0000 0110 1110 (2).}

That was the general algorithm for getting the representation of a number in a different base. If the original and destination bases are related, however, we can do better. Suppose we have
n = d₂ * B² + d₁ * B¹ + d₀ * B⁰
where d_i are the digits of the number n in base B. Suppose we have (lowercase) b where B = b², and we wish to write down n in base b. Well, each digit d_i is in the range 0, ..., B-1, and can be written as a two base-b digit number c_2i+1 * b + c_2i. Thus,
n = d₂ * B² + d₁ * B¹ + d₀ * B⁰
= (c₅ * b + c₄) * B² + (c₃ * b + c₂) * B¹ + (c₁ * b + c₀) * B⁰
= (c₅ * b + c₄) * b⁴ + (c₃ * b + c₂) * b² + (c₁ * b + c₀) * b⁰
= c₅ * b⁵ + c₄ * b⁴ + c₃ * b³ + c₂ * b² + c₁ * b¹ + c₀ * b⁰

Thus, each base B digit turned into two base b digits. Doing the reverse conversion, of course, would involve grouping two base b digits together to form a base B digit.

Assignment: perform the following base conversions.

1. 387₍₁₀₎ = ?₍₃₎
2. 78354₍₉₎ = ?₍₃₎
3. F786₍₁₆₎ = ?₍₈₎
4. 12112₍₃₎ = ?₍₉₎ There was a typo here.

Due before class next Wednesday. Expect to see a question on this on the midterm.

Missing from this web page: design philosophy behind RISC. Time to complete instructions. Clock cycle times. Simple instructions finish fast, complex ones take longer. Some CISCy computers (e.g. VAX) had instructions that no compiler knew how to generate. Profiling programs to get dynamic execution traces. Optimal use of chip real estate. More registers in RISCs. Instruction density and memory bandwidth.

bsy@cse.ucsd.edu, last updated Sun Oct 26 23:14:20 PST 1997.