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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?
- 64 bits
- 16 bits
- 32 bits (correct)
- 128 bits
Which statement about significant digits is correct?
Which statement about significant digits is correct?
- Zeros are always considered significant figures.
- The number of significant figures can vary based on the measurement instrument. (correct)
- The estimated digit must always be an integer.
- All digits in a number are significant.
How many significant figures does the speedometer reading of 48.5 have?
How many significant figures does the speedometer reading of 48.5 have?
- Five
- Two
- Three (correct)
- Four
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?
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?
Which of the following digits is typically not considered significant?
Which of the following digits is typically not considered significant?
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?
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?
What is the primary advantage of using scientific notation?
What is the primary advantage of using scientific notation?
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'?
What does IEEE 754 floating-point standard primarily address?
What does IEEE 754 floating-point standard primarily address?
Which of the following is a characteristic of the binary system?
Which of the following is a characteristic of the binary system?
How would the decimal number 1.1 be represented in binary?
How would the decimal number 1.1 be represented in binary?
What distinguishes single precision from double precision in floating-point representation?
What distinguishes single precision from double precision in floating-point representation?
What is the purpose of the signed exponent in floating-point representation?
What is the purpose of the signed exponent in floating-point representation?
What is the decimal equivalent of the binary number (1.101)2?
What is the decimal equivalent of the binary number (1.101)2?
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?
What does precision refer to in the context of measurements?
What does precision refer to in the context of measurements?
What is the result when a number is chopped to k digits?
What is the result when a number is chopped to k digits?
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?
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?
When is a value considered accurate?
When is a value considered accurate?
What happens if the true value is not known regarding error estimation?
What happens if the true value is not known regarding error estimation?
Which of the following numbers has four significant figures?
Which of the following numbers has four significant figures?
Flashcards
Decimal System
Decimal System
A way to represent numbers using ten digits (0-9) and place value. The position of a digit determines its value multiplied by powers of 10.
Binary System
Binary System
A way to represent numbers using two digits (0 and 1) and place value, where each position represents a power of two.
Scientific Notation
Scientific Notation
A standard way of writing a number using a single non-zero digit before the decimal point, a decimal part, and a power of 10.
Normalized Floating Point Representation
Normalized Floating Point Representation
A way of representing real numbers in a computer using three parts: a sign, a mantissa (fractional part), and an exponent. The mantissa is normalized to have a single non-zero digit before the decimal point.
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IEEE 754 Floating-Point Standard
IEEE 754 Floating-Point Standard
A standard for representing real numbers in computers, using specific formats for single and double precision. It determines how numbers are stored and manipulated in computers.
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Mantissa
Mantissa
The portion of a floating point number that represents the actual digits of the number, usually a fraction between 0 and 1.
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Exponent
Exponent
A power of 10 (or 2 in binary) that specifies the magnitude of the floating point number. It indicates how many places the decimal point is shifted.
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Single Precision and Double Precision
Single Precision and Double Precision
The two main types of floating point representations in IEEE 754: single precision uses 32 bits, while double precision uses 64 bits. Double precision offers higher accuracy and a larger range.
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Single Precision
Single Precision
A numerical representation that uses 32 bits to store a value. It provides a balance between accuracy and storage efficiency.
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Double Precision
Double Precision
A numerical representation that uses 64 bits to store a value. It offers high precision and a wider range of representable numbers.
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Significant Digits
Significant Digits
The digits in a number that are considered reliable and contribute to its overall accuracy. They represent the level of certainty in a measurement.
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Scale Division
Scale Division
The smallest division on a measurement instrument determines the precision of the measurement. It limits the number of significant digits.
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Placeholding Zeros
Placeholding Zeros
Zeros used solely to position the decimal point in a number. They do not contribute to its accuracy and are not considered significant digits.
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Significant Figure Determination
Significant Figure Determination
The process of determining and representing a number with only a certain number of significant digits. It involves identifying the reliable digits and estimating the last significant digit.
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Reliability of Numerical Value
Reliability of Numerical Value
The reliability of a numerical value is determined by the number of significant digits. The more significant digits, the more confident we are in its accuracy.
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Significant Figure Count
Significant Figure Count
The number of certain digits in a measurement plus one estimated digit. This provides a comprehensive measure of the value's accuracy.
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Accuracy
Accuracy
The closeness of a measured or calculated value to the true value.
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Precision
Precision
The closeness of several measured or calculated values to each other.
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Chopping
Chopping
A way to represent a real number as a floating-point number by discarding all digits after a specified digit.
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Rounding
Rounding
A way to represent a real number as a floating-point number by rounding the last digit based on the next digit.
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True Error
True Error
The difference between the true value of a number and its approximation.
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Absolute Percent Relative Error
Absolute Percent Relative Error
The true error expressed as a percentage of the true value.
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Estimated Error
Estimated Error
The difference between the current and previous estimates of a value.
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Estimated Absolute Percent Relative Error
Estimated Absolute Percent Relative Error
The estimated error expressed as a percentage of the current estimate.
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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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