Lecture 2 | Programming Paradigms (Stanford) | Summary and Q&A

Summary

This presentation discusses the low-level memory mechanics of data representation in memory, particularly focusing on characters, integers, and floating-point numbers in C and C++ programming languages. The speaker explains how different data types are represented in memory using binary digits (bits) and how they can distinguish between different values. The presentation also covers the concept of sign and magnitude representation, two's complement representation for negative numbers, and the representation of fractional parts in floating-point numbers.

Questions & Answers

Q: How are characters represented in memory?

Characters are usually represented by a single byte in memory. Each character is backed by a number, typically from 0 to 255, which represents its ASCII value. The bit pattern of the number is stored in memory, using binary digits. For example, the character 'A' is represented by the number 65, which is stored as the bit pattern 01000001.

Q: How are integers represented in memory?

Integers can be represented using different data types, such as short, int, and long. The bit patterns used to represent integers depend on the data type. The number of bits allocated for an integer can vary, with short typically using 2 bytes and int and long using 4 bytes. The bit pattern represents the magnitude of the number, with the leftmost bit commonly used as the sign bit to indicate whether the number is positive or negative. Two's complement representation is often used for negative numbers in order to simplify addition and subtraction.

Q: What is two's complement representation?

In two's complement representation, the leftmost bit of a binary number represents the sign (0 for positive, 1 for negative) and the remaining bits represent the magnitude. To obtain the negative representation of a number, the bits are inverted (0 becomes 1, 1 becomes 0) and 1 is added to the result. For example, the bit pattern 0111 represents the number 7, and its two's complement representation would be 1001 (which represents -7).

Q: How are floating-point numbers represented in memory?

Floating-point numbers are represented using a combination of sign, exponent, and fraction. The sign bit indicates the sign (0 for positive, 1 for negative), the exponent represents a power of 2 by which the fraction should be multiplied or divided, and the fraction represents the significant digits of the number. The bit pattern is used to store these components in memory according to a specific format, such as IEEE 754.

Q: What happens when an integer is assigned to a float?

When an integer is assigned to a float, the computer evaluates the integer value and converts it to a floating-point representation. The integer's bit pattern is used to initialize the sign, exponent, and fraction components of the floating-point number. The resulting floating-point number may not have the same decimal representation as the original integer due to the differences in how they are represented in memory.

Q: How are negative floating-point numbers represented?

Negative floating-point numbers are represented by setting the sign bit to 1, similar to negative integers. The rest of the floating-point representation, including the exponent and fraction, is calculated based on the binary pattern of the negative number.

Q: Why is two's complement representation used for negative integers?

Two's complement representation is used for negative integers because it simplifies addition and subtraction operations. By using two's complement, the same addition algorithm can be applied to both positive and negative numbers, with the sign propagation occurring naturally in the calculations.

Q: Can all real numbers be represented accurately using floating-point numbers?

No, due to the finite number of bits used to represent floating-point numbers, not all real numbers can be represented accurately. The precision and range of floating-point numbers depend on the number of bits allocated for the sign, exponent, and fraction. This means that some numbers may be rounded or approximated when stored and manipulated as floating-point numbers.

Q: Are there any limitations or trade-offs in using floating-point numbers?

Yes, there are limitations and trade-offs in using floating-point numbers. The precision of floating-point numbers decreases as the number becomes larger or smaller, leading to potential rounding errors. Additionally, operations involving floating-point numbers may be slower than those involving integers due to the extra overhead required to handle the sign, exponent, and fraction components.

Q: How can I convert a float to an integer?

Converting a float to an integer usually involves discarding the fractional part of the float and, optionally, rounding the result. This can be achieved by using casting or rounding functions available in the programming language. However, it is important to consider the potential loss of information when converting between different data types.