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Questions and Answers
What is the representation size of Single Precision in bits?
What is the representation size of Single Precision in bits?
Which statement about significant digits is correct?
Which statement about significant digits is correct?
How many significant figures does the speedometer reading of 48.5 have?
How many significant figures does the speedometer reading of 48.5 have?
If a car's odometer shows a reading of 87,324.5 km, how many significant figures does this reading have?
If a car's odometer shows a reading of 87,324.5 km, how many significant figures does this reading have?
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Why is it unreasonable to claim the speed of a car as 48.8642138 km/h based on a speedometer reading?
Why is it unreasonable to claim the speed of a car as 48.8642138 km/h based on a speedometer reading?
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Which of the following digits is typically not considered significant?
Which of the following digits is typically not considered significant?
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What is the estimated digit in the speedometer reading of 48.5 km/h?
What is the estimated digit in the speedometer reading of 48.5 km/h?
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What concept was developed to formally designate the reliability of a numerical value?
What concept was developed to formally designate the reliability of a numerical value?
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What is the primary advantage of using scientific notation?
What is the primary advantage of using scientific notation?
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In normalized floating point representation, what is the requirement for the digit 'd'?
In normalized floating point representation, what is the requirement for the digit 'd'?
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What does IEEE 754 floating-point standard primarily address?
What does IEEE 754 floating-point standard primarily address?
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Which of the following is a characteristic of the binary system?
Which of the following is a characteristic of the binary system?
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How would the decimal number 1.1 be represented in binary?
How would the decimal number 1.1 be represented in binary?
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What distinguishes single precision from double precision in floating-point representation?
What distinguishes single precision from double precision in floating-point representation?
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What is the purpose of the signed exponent in floating-point representation?
What is the purpose of the signed exponent in floating-point representation?
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What is the decimal equivalent of the binary number (1.101)2?
What is the decimal equivalent of the binary number (1.101)2?
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How can the uncertainty in the number of significant figures in a large number be resolved?
How can the uncertainty in the number of significant figures in a large number be resolved?
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What does precision refer to in the context of measurements?
What does precision refer to in the context of measurements?
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What is the result when a number is chopped to k digits?
What is the result when a number is chopped to k digits?
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In rounding a number to k digits, when is the k-th digit increased by one?
In rounding a number to k digits, when is the k-th digit increased by one?
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What is the formula for Absolute Percent Relative Error when the true value is known?
What is the formula for Absolute Percent Relative Error when the true value is known?
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When is a value considered accurate?
When is a value considered accurate?
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What happens if the true value is not known regarding error estimation?
What happens if the true value is not known regarding error estimation?
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Which of the following numbers has four significant figures?
Which of the following numbers has four significant figures?
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Study Notes
Computational Methods in Physics I: PHY405 - Lesson 2
- Number Representation and Accuracy
- Textbook Chapters 3 and 4 cover this material.
Representing Real Numbers
-
Decimal System:
- Base 10
- Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
- Example: 312.45 = (3 × 10²) + (1 × 10¹) + (2 × 10⁰) + (4 × 10⁻¹) + (5 × 10⁻²)
-
Standard Representation:
- ± (sign) 312 (integral part) . 45 (fractional part)
Decimal System (Continued)
-
Normalized Floating-Point Representation:
- ± d. f₁ f₂ f₃ f₄ × 10±n
- d ≠ 0
- ±n: signed exponent
- d represents the mantissa (the digits before the decimal point)
- f₁, f₂, f₃, f₄ represent the digits after the decimal point
- Scientific notation: Exactly one non-zero digit before decimal point
- Efficient for very small or very large numbers
Binary System
-
Binary System:
- Base 2
- Digits 0, 1
- Example: (1.101)₂ = (1 × 2⁰) + (1 × 2⁻¹) + (0 × 2⁻²) + (1 × 2⁻³) = 1.625₁₀
- Representation in Binary: ± 1. f₁ f₂ f₃ f₄ × 2±n
Note
- Numbers with finite expansions in one system may have infinite expansions in another.
- Example: 1.1₁₀ = 1.000110011001100...(binary)
- Exact representation of 1.1 is not possible in binary.
IEEE 754 Floating-Point Standard
- A technical standard for floating-point computations established in 1985 by IEEE.
- Addresses problems in diverse implementations of floating-point numbers.
- Most common representation for real numbers on computers (Intel-based PCs, Macs, Unix).
- Divided into single and double precision based on the number of bits for each component (sign, exponent, mantissa).
Single Precision (32-bit)
- Sign: 1 bit
- Exponent: 8 bits
- Mantissa: 23 bits
Double Precision (64-bit)
- Sign: 1 bit
- Exponent: 11 bits
- Mantissa: 52 bits
Significant Digits
- Confidence in a number used in computations.
- Visual inspection of a speedometer, for example, might show a car's speed is between 48 and 49 km/h.
- Reasonable individuals will estimate the speed similarly.
- Significant digits represent the certain digits plus one estimated digit.
- Example: Speedometer reading 48.5 km/h has three significant figures and odometer reading 87,324.45 km has seven significant figures. Zeros that are necessary for place value are not necessarily significant.
Accuracy and Precision
- Accuracy: How closely a computed or measured value agrees with the true value.
- Precision: How closely individual computed or measured values agree with each other.
Rounding and Chopping
- Two ways to represent a real number (x) as a floating-point machine number (fl(x)).
- Chopping: Discard digits after kth position.
- Rounding: Choose nearest value, to the value of k digits. If the (k+1)th digit is greater than or equal to 5, add 1 to the last digit.
Error Definitions
- True Error: If the true value is known, it is the absolute difference between the true value and the approximation.
- Absolute Percent Relative Error: The absolute error divided by the true value, expressed as a percentage.
- Estimated Error: If the true value is not known, the estimated error is the absolute difference between the current estimate and the previous estimate.
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Description
Explore the concepts of number representation and accuracy in computational methods. This lesson focuses on the decimal and binary systems, including normalized floating-point representations. Review key topics from Chapters 3 and 4 of the textbook to deepen your understanding.
