Podcast
Questions and Answers
What base does the decimal number system utilize?
What base does the decimal number system utilize?
- 10 (correct)
- 8
- 16
- 2
In a decimal number, what does the position of a digit to the left of the decimal point represent?
In a decimal number, what does the position of a digit to the left of the decimal point represent?
- An integer power of 10 (correct)
- An integer power of 2
- A fractional power of 10
- A negative power of 10
Which of the following is the primary reason the binary number system is used in digital electronics?
Which of the following is the primary reason the binary number system is used in digital electronics?
- It directly corresponds to the on/off states of electronic switches (correct)
- It is easier for humans to read
- It uses fewer digits than the decimal system
- It is more efficient mathematically
When converting the decimal number 27 to binary using the sum-of-weights method, which binary weights would be used?
When converting the decimal number 27 to binary using the sum-of-weights method, which binary weights would be used?
During the repeated division-by-2 method for decimal-to-binary conversion, when does the conversion proccess stop?
During the repeated division-by-2 method for decimal-to-binary conversion, when does the conversion proccess stop?
If a decimal fraction like 0.625 is converted to binary using repeated multiplication by 2, what triggers the process to end?
If a decimal fraction like 0.625 is converted to binary using repeated multiplication by 2, what triggers the process to end?
What is the sum of 1 + 1 in binary arithmetic?
What is the sum of 1 + 1 in binary arithmetic?
When performing binary subtraction, what action is taken when a borrow is necessary from the next column to the left?
When performing binary subtraction, what action is taken when a borrow is necessary from the next column to the left?
Which operation is equivalent to taking the 1's complement of a binary number?
Which operation is equivalent to taking the 1's complement of a binary number?
To find the 2's complement of a binary number, what two steps are performed?
To find the 2's complement of a binary number, what two steps are performed?
Why are 1's and 2's complements important in digital systems?
Why are 1's and 2's complements important in digital systems?
In signed number representation, what does the sign bit indicate?
In signed number representation, what does the sign bit indicate?
Which of the following is a drawback of the sign-magnitude representation?
Which of the following is a drawback of the sign-magnitude representation?
In 2's complement representation with n bits, what is the range of values that can be represented?
In 2's complement representation with n bits, what is the range of values that can be represented?
How is subtraction performed using 2's complement arithmetic?
How is subtraction performed using 2's complement arithmetic?
What condition indicates an overflow in 2's complement addition?
What condition indicates an overflow in 2's complement addition?
To perform the operation (+A) - (+B) in 2's complement arithmetic, what steps would you take?
To perform the operation (+A) - (+B) in 2's complement arithmetic, what steps would you take?
What is the result of adding two negative numbers in 2's complement form, and a carry occurs out of the sign bit?
What is the result of adding two negative numbers in 2's complement form, and a carry occurs out of the sign bit?
When performing arithmetic operations with signed binary numbers, which representation automatically provides the correct result in its complement form for negative numbers without additional conversion steps post-operation?
When performing arithmetic operations with signed binary numbers, which representation automatically provides the correct result in its complement form for negative numbers without additional conversion steps post-operation?
Given two n-bit binary numbers in 2’s complement form, what condition unequivocally signals an overflow during addition?
Given two n-bit binary numbers in 2’s complement form, what condition unequivocally signals an overflow during addition?
What is the largest decimal number that can be represented using an 8-bit unsigned binary number?
What is the largest decimal number that can be represented using an 8-bit unsigned binary number?
Using an 8-bit signed representation, what is the binary representation of -128 in 2's complement?
Using an 8-bit signed representation, what is the binary representation of -128 in 2's complement?
When converting the decimal number 65 to its binary equivalent, what is the resulting binary number?
When converting the decimal number 65 to its binary equivalent, what is the resulting binary number?
If you have the binary number 11001100, what is its decimal equivalent if interpreted as an unsigned integer?
If you have the binary number 11001100, what is its decimal equivalent if interpreted as an unsigned integer?
Given that you are using a full adder circuit, which of the following inputs will produce a 'Sum' output of 1 and a 'Carry out' of 1?
Given that you are using a full adder circuit, which of the following inputs will produce a 'Sum' output of 1 and a 'Carry out' of 1?
What is the output 'Difference' and 'Borrow' of a half-subtractor when A = 0 and B = 1 are applied?
What is the output 'Difference' and 'Borrow' of a half-subtractor when A = 0 and B = 1 are applied?
What describes overflow condition with signed numbers using 2's complement arithmetic?
What describes overflow condition with signed numbers using 2's complement arithmetic?
Given two binary numbers $A = 0101$ and $B = 0011$, what is the result of $A \times B$?
Given two binary numbers $A = 0101$ and $B = 0011$, what is the result of $A \times B$?
Given the binary numbers $A = 110$ and $B = 10$, what is A / B?
Given the binary numbers $A = 110$ and $B = 10$, what is A / B?
What determines whether the leftmost bit in a signed binary number indicates positive or negative?
What determines whether the leftmost bit in a signed binary number indicates positive or negative?
For a 1's complement signed representation of "1000", what is the decimal?
For a 1's complement signed representation of "1000", what is the decimal?
If you take the decimal number +39 and turn it into a 8-bit binary number what is the magnitude?
If you take the decimal number +39 and turn it into a 8-bit binary number what is the magnitude?
To find final decimal number of negative 1’s complement, what is adding one to the result analogous to?
To find final decimal number of negative 1’s complement, what is adding one to the result analogous to?
What are you looking for when using the Sum-of-Weights Method of Binary Number 9=8+1
What are you looking for when using the Sum-of-Weights Method of Binary Number 9=8+1
If there is a carry in the result of number using 2’s Complement rules what do you do?
If there is a carry in the result of number using 2’s Complement rules what do you do?
Which expression helps provide more flexibility to perform arithmetic operations to addition using with 2's compliment:
Which expression helps provide more flexibility to perform arithmetic operations to addition using with 2's compliment:
In 2s compliment the overflow helps ensure numbers have sufficient:
In 2s compliment the overflow helps ensure numbers have sufficient:
To find value as decimal number -21 sign is bit1, so:
To find value as decimal number -21 sign is bit1, so:
Acarry bit is
Acarry bit is
Flashcards
Binary Number System
Binary Number System
A number system that uses only two digits, 0 and 1, to represent values.
Decimal Number System
Decimal Number System
A number system that uses ten digits, 0 through 9, to represent values.
Decimal to Binary Conversion
Decimal to Binary Conversion
Process of converting a base-10 number into its equivalent base-2 representation.
Binary Arithmetic
Binary Arithmetic
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Complements of Binary Numbers
Complements of Binary Numbers
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Signed Numbers
Signed Numbers
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Arithmetic Operations with Signed Numbers
Arithmetic Operations with Signed Numbers
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Hexadecimal Numbers
Hexadecimal Numbers
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Octal Numbers
Octal Numbers
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Binary Coded Decimal (BCD)
Binary Coded Decimal (BCD)
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Digital Codes
Digital Codes
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Least Significant Bit (LSB)
Least Significant Bit (LSB)
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Most Significant Bit (MSB)
Most Significant Bit (MSB)
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Binary to Decimal Conversion
Binary to Decimal Conversion
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Half Adder
Half Adder
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Full Adder
Full Adder
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Half Subtractor
Half Subtractor
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Full Subtractor
Full Subtractor
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1's Complement
1's Complement
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2's Complement
2's Complement
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Sign Bit
Sign Bit
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Sign-Magnitude Form
Sign-Magnitude Form
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Study Notes
Digital Electronics: Number Systems, Operations, and Codes
- Covers binary numbers, decimal numbers, conversions, arithmetic, complements, signed numbers, hexadecimal, octal, BCD, and digital codes
Decimal Numbers
- The decimal system has ten digits and a base of 10
- In the number 23:
- The digit 2 has a weight of 10, contributing 20 (2 x 10)
- The digit 3 has a weight of 1, contributing 3 (3 x 1)
Binary Numbers
- The binary system has two digits (bits) and a base of 2
- Has a largest decimal number = 2^n - 1
- With five bits, one can count from zero to thirty-one: 2^5 - 1 = 32 - 1 = 31
- With six bits, one can count from zero to sixty-three: 2^6 - 1 = 64 - 1 = 63
- LSB means least significant bit
- MSB means most significant bit
Binary-to-Decimal Conversion
- Example: Converting the binary whole number 1101101 to decimal
- 1101101 = (1 x 2^6) + (1 x 2^5) + (0 x 2^4) + (1 x 2^3) + (1 x 2^2) + (0 x 2^1) + (1 x 2^0)
- = 64 + 32 + 0 + 8 + 4 + 0 + 1 = 109
- Example: Converting the fractional binary number 0.1011 to decimal
- 0.1011 = (1 x 2^-1) + (0 x 2^-2) + (1 x 2^-3) + (1 x 2^-4)
- = 0.5 + 0 + 0.125 + 0.0625 = 0.6875
Decimal-to-Binary Conversion
- Using the Sum-of-Weights Method.
- The binary number for decimal 9 is 1001
- 9 = 8 + 1 = 2^3 + 2^0
Decimal-to-Binary Conversion: Repeated Division-by-2 Method
- Stop dividing when the whole-number quotient is 0
Converting Decimal Fractions to Binary: Repeated Multiplication by 2
- Continue to the desired number of decimal places
- Stop when the fractional part is all zeros
Binary Arithmetic
- 0 + 0 = 0 = Sum of 0 with a carry of 0
- 0 + 1 = 1 = Sum of 1 with a carry of 0
- 1 + 0 = 1 = Sum of 1 with a carry of 0
- 1 + 1 = 10 = Sum of 0 with a carry of 1
- Binary Subtraction example
- 10 - 1 = 1 = 0 - 1 with a borrow of 1
- In binary 10 - 1 = 1, not 9
Half Adder and Full Adder
- They are digital circuits used for binary addition, differing in the number of inputs they take
Half Subtractor and Full Subtractor
- They are digital circuits used for binary subtraction
Binary Multiplication and Division
- Presents the four basic rules for multiplying bits
- Shows examples of binary multiplication and division
Complements of Binary Numbers
- The 1's complement and the 2's complement of a binary number are important because they permit the representation of negative numbers.
- The method of 2's complement arithmetic is commonly used in computers to handle negative numbers.
Finding the 1's Complement
- Found by changing all 1's to 0's and all 0's to 1's
Finding the 2's Complement
- The 2's complement is found by adding 1 to the LSB of the 1's complement
Alternative Method for Finding the 2's Complement
- Start at the right LSB and write the bits as they are up to and including the first 1
- Take the 1's complements of the remaining bits
Signed Numbers
- Two main types: unsigned and signed representations
- Signed numbers can be represented in sign-magnitude, 1's complement, or 2's complement form
- The leftmost bit in a signed binary number is the sign bit; 0 means positive, and 1 means negative
Sign-Magnitude Form
- The decimal number +25 is expressed as 00011001 (8-bit signed binary)
- The decimal number -25 is expressed as 10011001 (8-bit signed binary)
Range of Signed Integer Numbers
- In 2's-complement, there are eight positive numbers (including one zero) and eight negative numbers.
- In signed-magnitude and 1's-complement, there are eight positive/negative numbers (including two zeros)
The Decimal Value of Signed Numbers
- Shows how to determine this for numbers expressed in sign-magnitude or 1's complement
Arithmetic Operations with Binary Signed Numbers (2's Complement)
- Two numbers in addition: addend and augend, the result is the sum
- In 2'S Compliment form: Both Positive or Negative numbers must discard carry
- There is both arithmetic addition with signed-2's-complement subtraction and subtraction using 2's complement
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