sc lecture 8

Key Concepts

Complement numeral systems were created to make additions faster and easier.
One decimal number can be subtracted from another by adding the difference of the subtrahend from all nines and adding back a carry. This is called taking the nine’s-complement of the subtrahend, or more precisely, finding the diminished radix complement of the subtrahend.
The one’s-complements of binary numbers are computed the same way as for natural numbers, except that the weight of the MSB is $- 2^{n - 1} + 1$ instead of $+ 2^{n - 1}$ .
For a binary number, its one’s-complement is obtained by subtracting from all ones. For example, the one’s-complement of $010 1_{2}$ is $1111 - 0101 = 101 0_{2}$ .
Complement notation provides a method to represent negative numbers. The idea is that a negative number needs to be converted to its complement, which should have a ‘1’ in the MSB. Positive numbers, which have a ‘0’ in the MSB are used as is, i.e., they are not converted to their complements.
- - To subtract $10$ from $24$ , one needs first to express the subtrahend $(10)$ in one’s-complement and then add it to the minuend $(24)$ . This effectively adds $- 10$ to $24$ . The MSB will have a ' $0$ ' or a ' $1$ ' carry, which needs to be added to the LSB of the sum. This operation is designated end carry-around and results from the use of the diminished radix complement, in this case the one’s-complement.

Given a numeric value $V$ in base $r$ having $n$ digits, the radix complement of $V$ is defined to be $r^{n} - V$ for $V \neq = 0$ . With three decimal digits, the ten’s-complement of 43 is 957 $(1 0^{3} - 43)$ .
The decimal value for a two’s-complement number is computed the same way as for a natural number, except that the weight of the MSB is $- 2^{n - 1}$ instead of $+ 2^{n - 1}$ .
Two’s-complement number are added and subtracted by the same methods as unsigned numbers with the same number of bits, so the same hardware can handle numbers in both
Its biggest drawback is the assymmetry in the range of values that can be represented.

In a excess-b representation, an n-bit pattern, whose unsigned integer value is $V$ $(0 \leq V \leq 2^{n})$ represents the signed integer $V - b$ , where b is the bias (or offset) of the natural system.
The representable numeric values range from from $- b$ to $2^{n} - 1 - b$ .
The main advantage lies in the fact that the all-zero pattern corresponds to the minimal negative value and the all-one pattern to the maximal positive value.
It also permits to represent unbalanced sets of consecutuve negative and positive numbers (e.g., from -10 to +245)
From a mathematical point of view, the representation of natural numbers is an excess-0 representation.

Quais são as melhores afirmações erradas que consegues verbalizar para as matérias abordadas hoje?