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Questions and Answers
What is the binary representation of the decimal number 8.75?
What is the binary representation of the decimal number 8.75?
What is the proper normalized format of the binary number 1000.11?
What is the proper normalized format of the binary number 1000.11?
How is the exponent calculated for the number 8.75 in excess 127 notation?
How is the exponent calculated for the number 8.75 in excess 127 notation?
What is the mantissa for the normalized binary number 1.00011?
What is the mantissa for the normalized binary number 1.00011?
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If the number is positive, what will the sign bit be in IEEE 754 format?
If the number is positive, what will the sign bit be in IEEE 754 format?
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What is the final hexadecimal representation of the IEEE 754 single precision format for the number 8.75?
What is the final hexadecimal representation of the IEEE 754 single precision format for the number 8.75?
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How is the exponent represented for the binary number 0.5 in excess 127 notation?
How is the exponent represented for the binary number 0.5 in excess 127 notation?
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What value is assigned to the sign bit for the number -0.5?
What value is assigned to the sign bit for the number -0.5?
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What is the purpose of the sign bit in the IEEE-754 standard?
What is the purpose of the sign bit in the IEEE-754 standard?
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How many bits are allocated for the exponent in the IEEE-754 Single Precision standard?
How many bits are allocated for the exponent in the IEEE-754 Single Precision standard?
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Which step is NOT part of converting a decimal number to IEEE 754 Single Precision?
Which step is NOT part of converting a decimal number to IEEE 754 Single Precision?
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What is the value of K when calculating the exponent for IEEE 754?
What is the value of K when calculating the exponent for IEEE 754?
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What is the maximum number of significant decimal digits that can be accurately represented using IEEE-754 Single Precision?
What is the maximum number of significant decimal digits that can be accurately represented using IEEE-754 Single Precision?
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What is the 'hidden bit' in IEEE-754 floating point representation?
What is the 'hidden bit' in IEEE-754 floating point representation?
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In the IEEE-754 representation of floating point numbers, how are the mantissa bits structured?
In the IEEE-754 representation of floating point numbers, how are the mantissa bits structured?
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When converting a decimal value to IEEE 754 format, how should the binary fractional number be normalized?
When converting a decimal value to IEEE 754 format, how should the binary fractional number be normalized?
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What is the sign bit for the number -0.510?
What is the sign bit for the number -0.510?
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What is the exponent in binary form for the number -1.0625 after normalization?
What is the exponent in binary form for the number -1.0625 after normalization?
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How is the mantissa formatted for the number -1.0625 in IEEE 754 format?
How is the mantissa formatted for the number -1.0625 in IEEE 754 format?
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What is the hexadecimal representation of the final IEEE 754 format for the number -1.0625?
What is the hexadecimal representation of the final IEEE 754 format for the number -1.0625?
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Which of the following represents the correct process for converting a decimal to binary?
Which of the following represents the correct process for converting a decimal to binary?
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What value does the exponent 0111 1110 represent in decimal?
What value does the exponent 0111 1110 represent in decimal?
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What is the purpose of the sign bit in the IEEE 754 format?
What is the purpose of the sign bit in the IEEE 754 format?
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When converting a fractional number to binary, what step follows the normalization process?
When converting a fractional number to binary, what step follows the normalization process?
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Study Notes
IEEE 754 Single Precision Format
- The IEEE 754-2008 standard defines the 32-bit base 2 format, also known as Binary32.
- Older computer formats for 4-byte floating-point numbers existed before this standard.
- Fortran was one of the first programming languages to include single and double-precision floating-point data types.
- Single precision is denoted as 'float' in programming languages such as C, C++, C#, and Java.
- IEEE 754 single-precision format uses an 8-bit exponent and a 23-bit significand.
Terms to Know
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Range: The range of values representable by the format.
- Approximately 1.2 × 10-38 to 3.4 × 1038.
- Despite this wide range, infinitely many numbers fall outside it.
-
Accuracy: How close a represented number is to its true value.
- Example: 0.1 cannot be precisely represented; however, an approximation within a range close to 0.1 exists.
-
Precision: The amount of detail or information used to represent a value.
- Higher precision often allows a number to be more accurate. Example: 1.666 has 4 decimal digits of precision, and 1.6660 has 5, but this does not make the latter more accurate.
IEEE 754 Storage
- Floating-point numbers are stored on byte boundaries.
- The IEEE-754 single-precision standard can be represented with a sign bit (1 bit), an exponent (8 bits), and a mantissa (23 bits).
- This format allows for numbers with 6 to 9 significant decimal digits. Converting to and from this format should maintain this accuracy.
Converting Decimal to IEEE 754
- Step 1: Convert the decimal number to binary.
- Step 2: Normalize the binary fractional number. Shift the decimal point until there is a "1" to the left of the decimal.
- Step 3: Convert the exponent to 8-bit excess 127 notation.
- Step 4: Convert the significant/mantissa to a "hidden bit format." Remove the initial "1" to the left of the decimal point.
- Step 5: Determine the sign bit. Positive numbers have 0, negative ones 1.
- Step 6: Assemble the three components into the final format.
- Step 7: Often useful to convert to hexadecimal format for better readability.
Example #1
- Convert 8.7510 to IEEE 754 Single-Precision format.
- Binary: 1000.112
- Normalized: 1.00011 x 23
- Exponent (excess 127): 130 (binary: 10000010)
- Mantissa: 00011
- Final IEEE 754 format: 0 10000010 00011000000000000000000
- Hexadecimal equivalent: 410C000016
Example #2
- Convert -0.510 to IEEE 754 Single-Precision format.
- Binary: 0.12
- Normalized: 1.0 x 2-1
- Exponent (excess 127): 126 (binary: 01111110)
- Mantissa: 0
- Final IEEE 754 format: 1 01111110 00000000000000000000000
- Hexadecimal equivalent: BF00000016
Example #3
- Convert -1.062510 to IEEE 754 Single-Precision format.
- Binary: 1.00012 x 2-0
- Normalized: 1.0001 x 20
- Exponent (excess 127): 127 (binary: 01111111)
- Mantissa: 0001
- Final IEEE 754 format: 1 01111111 00010000000000000000000
- Hexadecimal equivalent: BF88000016
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Description
Test your knowledge on the IEEE 754-2008 standard for single precision format. This quiz covers the basics of Binary32, its historical context, and key terms like range, accuracy, and precision. Prepare to demonstrate your understanding of floating-point representation in programming languages such as C and Java.
