Lecture 9 Multiplication and Intro to Floating Point Numbers Flashcards
What is binary multiplication?
Multiplication using logical ANDs to obtain partial products and summing them.
How many bits are needed for the result of multiplying two binary numbers?
If the multiplicand is \(X\) bits and the multiplier is \(Y\) bits, the result needs \(X+Y\) bits.
Why do modern processors use fixed operand sizes?
To reduce complexity, ensuring simpler and more efficient hardware design.
What is an example of binary multiplication?
Multiply 011₂ by 010₂:<br></br>Partial products:<br></br>000000<br></br>000011<br></br>+ 000000<br></br>= 000110 (6 in decimal).
What is the difference between RISC and CISC processors?
RISC processors have fewer instructions (simpler, more efficient), while CISC processors support complex operations.
How does two’s complement affect multiplication?
Sign extension must be considered to ensure the correct result for negative numbers.
What is an example of multiplying two’s complement numbers?
Multiply -9 (10111₂) by 3 (00011₂), extend to 10 bits, and apply binary multiplication.
What is division in binary?
Successive subtraction of the divisor from the dividend, requiring registers and comparators.
Why is division costly in hardware?
It requires additional circuitry, making some processors rely on software for division.
What are fixed-point numbers?
Numbers with a fixed position for the decimal point, such as 132.56 in decimal.
What is scientific notation?
Expressing numbers as \( M \times 10^E \), e.g., \( 543.12 = 5.4312 \times 10^2 \).
What are floating-point numbers?
Numbers represented with a mantissa, exponent, and base, allowing flexibility in precision.
What is an example of floating-point representation?
\( 543.12 \) can be represented as \( 5.4312 \times 10^2 \) or \( 0.54312 \times 10^3 \).
How do you convert fractions to binary?
Multiply by 2 repeatedly, taking the integer parts as binary digits.
What is an example of converting fractions to binary?
Convert 0.75 to binary:<br></br>0.75 × 2 = 1.5 → 1<br></br>0.5 × 2 = 1.0 → 1<br></br>Result: 0b0.11.
Why do some fractions have repeating binary representations?
Some decimal fractions cannot be exactly represented in binary, leading to rounding errors.
What is an example of a repeating fraction in binary?
0.1 in decimal becomes a repeating sequence: \( 0.0001100110011… \) in binary.
What is an example of converting a real number to binary?
Convert 17.625:<br></br>17 → 0b10001<br></br>0.625 → 0b0.11<br></br>Result: 0b10001.11.
