Binary and Two's Complement Quiz

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Questions and Answers

What is the decimal value of the binary number 100 in two's complement representation?

-4 (correct)

-3

Which of the following ranges represents the values that can be expressed using 3 bits in two's complement?

-4 to 4

-4 to 3 (correct)

-2 to 2

-3 to 3

What does the most significant bit in a two's complement binary number indicate?

The magnitude of the number

The number of bits used

The base of the number system

Whether the number is negative or positive (correct)

What is true about the representation of zero in signed binary numbers using two's complement?

It has a unique representation. Signup and view all the answers

Which mathematical expression correctly defines the range of values in two's complement for n bits?

-(2^{n -1}) ≤ N ≤ +(2^{n -1} - 1) Signup and view all the answers

What is the biased exponent for the number (0.015) 8 when represented in the given floating point format?

0011 Signup and view all the answers

When performing the subtraction N2 - N1 with N1=(-0.014)8 and N2=(0.14)8, what must be done first?

Adjust the exponents to be equal. Signup and view all the answers

What is the range of values that can be represented on 3 bits?

-3 to +3 Signup and view all the answers

What operation should be performed to calculate N1 + N2 when e1 and e2 are different?

Raise to the largest exponent and normalize the result. Signup and view all the answers

Which formula correctly represents the addition of a number and its corresponding two's complement?

a + 2^n = a modulo 2^n Signup and view all the answers

What is the two's complement of 01000101 on 8 bits?

10111011 Signup and view all the answers

The most significant bit (MSB) in a binary representation indicates what?

The sign of the number Signup and view all the answers

How can subtraction of two n-bit integers a and b be performed using addition?

a - b = a + (-b) Signup and view all the answers

What is the value of 2's complement of a binary number N if n = 3?

1's complement of N plus 1 Signup and view all the answers

What does the equation 2's C(2's C(N)) equal?

N Signup and view all the answers

Which of the following expressions equals to a - b using the two's complement?

a + 2's C(b) Signup and view all the answers

Which of the following represents the Excess-3 encoding for the decimal number 7?

1010 Signup and view all the answers

What is the binary representation of the decimal number 2 in BCD?

0010 Signup and view all the answers

In Gray code, how many bits change between two successive encodings?

One bit Signup and view all the answers

How many characters can be represented in standard ASCII encoding using 8 bits?

256 Signup and view all the answers

Which binary code corresponds to the character 'A' in ASCII?

01000001 Signup and view all the answers

What is the primary use of the most significant bit in sign/absolute value representation?

Indicate the sign of the number Signup and view all the answers

Which of the following represents a limitation of sign/absolute value representation?

It requires more complex circuits for arithmetic operations Signup and view all the answers

How many total representations of zero exist in sign/absolute value representation?

Two Signup and view all the answers

What is the formula for determining the range of values that can be represented using n bits in sign/absolute value representation?

$-(2^{(n-1)} - 1) ext{ to } +(2^{(n-1)} - 1)$ Signup and view all the answers

What techniques are used for representing negative numbers in binary?

Sign/absolute value, 1's complement, and 2's complement Signup and view all the answers

What is the primary advantage of the sign/absolute value representation?

It is simple and straightforward to understand Signup and view all the answers

What is a potential consequence of having two representations for zero in the sign/absolute value format?

Ambiguity leading to errors in arithmetic operations Signup and view all the answers

What is the typical range of values that can be represented on 3 bits using sign/absolute value representation?

-3 to +3 Signup and view all the answers

What is the biased exponent of N1 after calculation?

11 Signup and view all the answers

What is the hidden bit in the IEEE 754 format?

The leading 1 in the mantissa Signup and view all the answers

What is the value of the bias in the IEEE 754 single precision format?

127 Signup and view all the answers

How is the true exponent derived from the biased exponent?

By subtracting the bias from the biased exponent Signup and view all the answers

When converting from decimal to binary using BCD coding, how is each decimal digit treated?

Each digit is represented as a 4-bit binary value Signup and view all the answers

What is the result of N2 - N1 with the true exponents calculated?

The same as calculated without biased exponent Signup and view all the answers

Which of the following represents the IEEE 754 64-bit double-precision format?

1 bit for sign, 11 bits for exponent, 52 bits for mantissa Signup and view all the answers

Study Notes

Chapter 3: Representation and Coding of Information in the Machine

The various types of information stored in central memory include instructions and data.
Information types include numerical and non-numerical data, which further break down into integers, real numbers, and characters.
Numerical data includes positive and negative integers, fractions, and fixed-point/floating-point real numbers, all requiring different representation methods.
Non-numerical data concerns different types of characters.
Methods for representing negative numbers include sign/absolute value, 1's complement, and 2's complement.

1. Representation of Integer Numbers

Integers have two subtypes: unsigned (positive only) and signed (positive/negative).
Different methods exist to indicate the sign of an integer to the machine:
- Sign/absolute value
- 1's complement
- 2's complement

1.1. Sign/Absolute Value Representation (S/VA)

The most significant bit signifies the sign (1 for negative, 0 for positive).
Remaining bits represent the absolute value of the number.
Example: on 4 bits, 1001 represents -1, and 0001 represents +1.
On n bits, representable values range from -(2^(n-1) - 1) to +(2^(n-1) - 1).

Advantages and Disadvantages of S/VA

Simple representation
Zero has two representations (+0 and -0), causing arithmetic complications.
Requires separate circuits for addition and subtraction, inefficient compared to a single circuit.

1.2. One's Complement Representation (Restricted Complement)

The one's complement of a number N (N’) satisfies N + N’ = 2^n – 1, where n is the number of bits.
To find the one's complement: invert all the bits.
Example: On 4 bits, the one's complement of 1010 is 0101.

1.3. Two's Complement Representation (True Complement)

A number a expressed in n-bits, using modulo 2^n satisfies a + 2^n = a
To find the two's complement (2's C): Find the one's complement and add 1.
Example: On 4 bits, the two's complement of 1001 is 0111 + 1 = 1000.
This method handles negative numbers more directly, enabling a unified addition/subtraction circuit.

Arithmetic Operations in 2's Complement

The result of addition can cause overflow (result falls outside representable range)
Overflow arises when the sum of two positive numbers is negative or the sum of two negative numbers is positive.
An overflow does not occur if the operands have different signs.

2. The Representation of Real Numbers

Real numbers comprise an integer and a fractional part.
Methods to represent real numbers include fixed-point and floating-point.

2.1 Fixed-Point Representation

The position of the decimal or comma is fixed.
Integer part and fractional part are encoded separately.
Limited precision.
Fewer bits lead to less precision.

2.2 Floating-Point Representation

The machine represents a number as M * b^e.
M denotes mantissa, b denotes base (usually 2), and e denotes exponent.
The mantissa and exponent are encoded separately.
Normalize mantissa to ensure proper decimal representation.

Encoding of Real Numbers’ Exponents

Employing two's complement for storing the exponent.
Use shifted (biased) exponents to ensure positive values.
The biased exponent helps to normalize arithmetic operations by ensuring all exponents are positive allowing more efficient calculations.

IEEE 754 Standard

Defines standard format for floating-point numbers in 32 or 64-bit formats.
Normalize the mantissa to enhance precision.

3. BCD Coding (Binary Coded Decimal)

Each decimal digit is converted to its 4-bit binary representation
Combinations exceeding 9 are invalid.

Excess 3 Encoding (BCD+3)

Adds 3 to each decimal digit.
Transforms each decimal into 4-bit binary representation.

Reflected Binary Code (Gray Code)

Only one bit changes between successive codes.

4. Character Encoding

ASCII (American Standard Code for Information Interchange) assigns 8-bit codes to characters.
Characters include letters, numbers, and symbols.
Unicode is a 16-bit encoding scheme that can represent a larger set of characters.

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Description

Test your knowledge of binary numbers and two's complement representation with this quiz. It covers topics such as binary to decimal conversion, ranges of values in two's complement, and operations involving signed numbers. Perfect for students studying computer science or digital logic.