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Questions and Answers
What is the result of the expression $(1 + 1 × 10^{20}) − (1 × 10^{20})$?
What is the result of the expression $(1 + 1 × 10^{20}) − (1 × 10^{20})$?
What is the binary representation of -4 in sign magnitude format?
What is the binary representation of -4 in sign magnitude format?
Which term is used for the '1' in the formula (−1)sign × (1 + mantissa) × 2expo.−bias in IEEE754 format?
Which term is used for the '1' in the formula (−1)sign × (1 + mantissa) × 2expo.−bias in IEEE754 format?
What is the range of integers represented by an n bit two's complement number system?
What is the range of integers represented by an n bit two's complement number system?
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What is the biased representation for an exponent of 3 in a 4-bit system?
What is the biased representation for an exponent of 3 in a 4-bit system?
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What is the maximum difference between two successive real numbers representable in a 16-bit floating point system?
What is the maximum difference between two successive real numbers representable in a 16-bit floating point system?
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Which of the following accurately describes the relationship between boundary numbers −2n−1 and +2n−1 − 1?
Which of the following accurately describes the relationship between boundary numbers −2n−1 and +2n−1 − 1?
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How is overflow detected in binary addition without using the carry bit?
How is overflow detected in binary addition without using the carry bit?
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How many bits are used for the significand in IEEE754 double precision format?
How many bits are used for the significand in IEEE754 double precision format?
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What is a key drawback of signed magnitude representation?
What is a key drawback of signed magnitude representation?
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In IEEE754 format, what is the role of the 'sign bit'?
In IEEE754 format, what is the role of the 'sign bit'?
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Which number representation format typically requires simpler hardware?
Which number representation format typically requires simpler hardware?
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In two's complement representation, what is the range for an n-bit binary number?
In two's complement representation, what is the range for an n-bit binary number?
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What does the term 'mantissa' represent in floating-point representation?
What does the term 'mantissa' represent in floating-point representation?
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Which of the following is the correct two's complement representation for -3?
Which of the following is the correct two's complement representation for -3?
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How is the actual magnitude of a negative result obtained in two’s complement addition or subtraction?
How is the actual magnitude of a negative result obtained in two’s complement addition or subtraction?
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What is the primary reason for the preference of two's complement over signed magnitude?
What is the primary reason for the preference of two's complement over signed magnitude?
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What is a necessary consideration when selecting a number representation format?
What is a necessary consideration when selecting a number representation format?
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How is a negative number represented in signed magnitude representation?
How is a negative number represented in signed magnitude representation?
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What does an 8-bit binary number represent in terms of non-negative integers?
What does an 8-bit binary number represent in terms of non-negative integers?
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What is the unbiased exponent for a 32-bit number represented as 10110101110100010000000000000000?
What is the unbiased exponent for a 32-bit number represented as 10110101110100010000000000000000?
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Which of the following is true regarding the representation of zero in IEEE754 format?
Which of the following is true regarding the representation of zero in IEEE754 format?
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In IEEE754 floating-point representation, which condition indicates a NaN (Not a Number)?
In IEEE754 floating-point representation, which condition indicates a NaN (Not a Number)?
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During floating point multiplication, what is the first step in the algorithm?
During floating point multiplication, what is the first step in the algorithm?
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How many bits are used for the fraction in a single precision floating point representation?
How many bits are used for the fraction in a single precision floating point representation?
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What happens during the normalization step in floating point addition?
What happens during the normalization step in floating point addition?
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What is the implication of an exponent of all 1's in the IEEE754 representation?
What is the implication of an exponent of all 1's in the IEEE754 representation?
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What is the range of representable values for single precision floating-point numbers?
What is the range of representable values for single precision floating-point numbers?
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What is a characteristic of nonrestoring division?
What is a characteristic of nonrestoring division?
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In floating-point arithmetic, what operation is performed when $X + Y$ with $XE ≤ YE$?
In floating-point arithmetic, what operation is performed when $X + Y$ with $XE ≤ YE$?
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What does the ASCII encoding standard provide for text representation?
What does the ASCII encoding standard provide for text representation?
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What is the maximum representation error, e, when using 8 significant bits after the decimal point?
What is the maximum representation error, e, when using 8 significant bits after the decimal point?
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In the operation $X ÷ Y$, which representation is used for the numbers?
In the operation $X ÷ Y$, which representation is used for the numbers?
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Which characteristic defines EBCDIC encoding?
Which characteristic defines EBCDIC encoding?
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When adding two normalized floating-point numbers, what is assumed about the mantissa?
When adding two normalized floating-point numbers, what is assumed about the mantissa?
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Among the options presented, which number is divisible by 11111011 under 2’s complement representation?
Among the options presented, which number is divisible by 11111011 under 2’s complement representation?
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Study Notes
Number Representation Factors
- Selection factors for number representation include types of numbers (integers, reals, complex), range of values, required precision, and hardware costs.
- Fixed-point format offers a small range and simpler hardware, whereas floating-point format allows for a larger range but with complex hardware.
Binary Number Basics
- Binary representation: -1101.0101 in binary equals -13.3125 in decimal.
- Non-negative integers are represented with 8 bits, covering values from 0 to 255.
- Signed magnitude representation uses the first bit for the sign; 0 indicates positive while 1 indicates negative.
Negative Number Representation
- Signed magnitude representation has drawbacks like double representation for zero (+0 and -0).
- Two’s complement method resolves these issues, using the most significant bit as the sign and offering a single representation for zero.
- Range for an n-bit binary number using two’s complement is from -2^(n-1) to + (2^(n-1) - 1).
Two’s Complement vs Sign Magnitude
- Two's complement provides a more efficient representation compared to sign-magnitude.
- Example comparisons show equivalent values for positive numbers but different representations for negative values.
IEEE754 Floating-Point Format
- Decimal numbers can be represented using floating point: 9760000 can be represented as 0.97 × 10^7.
- Components include:
- Sign bit
- Exponent (biased)
- Significand (or mantissa)
- Single precision includes 1 sign bit, 8 exponent bits, and 23 significand bits.
- Double precision consists of 1 sign bit, 11 exponent bits, and 52 significand bits.
Biased Representation
- A bias is introduced to facilitate easier sorting of exponents in floating-point representation.
- Single precision has a bias of 127; double precision has a bias of 1023.
- Range of single precision: from -2 × 10^(-38) to 2 × 10^(38), double precision: from -2 × 10^(-308) to 2 × 10^(308).
Floating-Point Arithmetic Operations
-
Addition/Subtraction:
- Align numbers by shifting the smaller number's exponent until they match.
- Normalize the sum and handle overflow/underflow.
-
Multiplication:
- Add exponents after adjusting for bias and multiply significands, normalizing if necessary.
-
Division:
- Utilize a fast division technique through precomputed lookup tables and skip unnecessary additions.
ASCII and EBCDIC
- ASCII encodes 128 characters into 7-bit binary integers, covering numbers, letters, and control symbols.
- EBCDIC, an 8-bit character encoding, is primarily used in IBM systems.
Practice Exercises Overview
- Exercises include converting decimal numbers to signed binary, calculating maximum representation error, performing biased floating-point arithmetic, and analyzing overflow detection in two's complement representation.
Important Concepts in Floating-Point Representation
- Denormal numbers indicate subnormal values.
- Special cases in floating-point include representations for zero, infinity, and NaN (Not a Number).
- Exception handling is critical in the implementation of floating-point operations.
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Description
This quiz focuses on the concepts of number representation and arithmetic in computer organization. It covers various types of numbers, including integers and real numbers, and discusses factors influencing their representation. Test your understanding of number formats as explored in the context of computer science.
