ECAP268 - U01L04 - Fixed point and floating point representation

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20 Mar 202211:20

Summary

TLDRThis lecture delves into the representation of integers in computers, focusing on fixed and floating point representations. It explains how positive and negative integers are encoded using bits, with the sign bit determining the value's positivity or negativity. The video introduces three methods for representing negative numbers: sign magnitude, one's complement, and two's complement. It also covers the concept of mantissa and exponent in floating point representation, demonstrating how to encode numbers like 6132.789 as a combination of a fixed point number and an exponent.

Takeaways

🖥️ Computers represent integers using binary numbers, with different methods for positive and negative values.
🔢 Positive integers and zero are represented as unsigned numbers, while negative integers require a notation to denote their sign.
🔣 In computer systems, the sign of an integer is represented by a bit in the leftmost position, with '0' for positive and '1' for negative.
📍 The position of the decimal or binary point is crucial for representing fractions, integers, or mixed numbers in computers.
🔁 There are two main types of fixed point representation: one with the binary point fixed at the leftmost position (fraction) and the other at the rightmost position (integer).
🔑 The sign of a positive integer is represented by a '0', and its magnitude by a positive binary number.
📌 Negative integers can be represented in three ways: sign-magnitude, one's complement, and two's complement.
🔄 In sign-magnitude representation, the sign is indicated by '1' for negative, followed by the binary representation of the number.
🔄 One's complement representation is found by taking the complement of the binary number representing the positive value.
🔄 Two's complement representation is obtained by adding one to the one's complement of the positive number.
🔄 Floating point representation uses two registers: one for the mantissa (fixed point number) and another for the exponent (position of the decimal point).

Q & A

What are the two main types of integer representation discussed in the lecture?
-The two main types of integer representation discussed are fixed point representation and floating point representation.
How are positive integers and zero represented in computers?
-Positive integers and zero are represented as unsigned numbers in computers, using only zeros and ones without any additional notation.

Outlines

00:00

📚 Introduction to Integer Representation in Computers

This paragraph introduces the topic of how integers, including positive, negative, and zero, are represented in computers. It explains that unsigned numbers can represent positive integers and zero, while a separate notation is needed for negative integers. The hardware limitations of computers require that signs be represented using bits, with the most significant bit typically indicating the sign (0 for positive, 1 for negative). The paragraph also touches on the representation of decimal or binary points in fixed and floating point representations, setting the stage for a deeper dive into these concepts.

05:01

🔢 Fixed Point and Floating Point Representations Explained

This paragraph delves into the specifics of fixed point representation, where the binary point's position is fixed, either at the leftmost or rightmost position of the register, to indicate whether the number is a fraction or an integer. It also introduces the concept of floating point representation, where a second register is used to store the position of the decimal point, allowing for more flexibility in representing numbers. The paragraph outlines the two main components of floating point numbers: the mantissa, a fixed point number, and the exponent, which indicates the decimal point's position.

10:03

👨‍🏫 Methods of Representing Negative Numbers in Computers

The third paragraph focuses on the different methods used to represent negative numbers in computers due to hardware limitations that prevent the use of plus or minus signs. It discusses three primary methods: sign-magnitude representation, where the most significant bit indicates the sign and the rest represent the magnitude; one's complement representation, which involves taking the complement of the positive number; and two's complement representation, where one is added to the one's complement of the positive number. The paragraph provides examples of how to represent the number 14 in both positive and negative forms using these methods.

Mindmap

Keywords

💡Representation

In the context of the video, 'representation' refers to the method by which integers, including positive, negative, and zero, are encoded in a computer system. It is central to the theme as it explains how computers understand and process numbers. For instance, the script discusses different representations such as unsigned numbers for positive integers and zero, and various notations for negative integers.

💡Unsigned Numbers

Unsigned numbers are integers that do not include negative values and are represented using binary digits without any specific notation for negativity. They are integral to the video's narrative as it explains that positive integers and zero can be represented as unsigned numbers in computers, which simplifies the process of encoding these values.

💡Negative Integers

The term 'negative integers' is used to describe a subset of integers that have a minus sign in front of them. The video emphasizes the need for special notation to represent these in computers, given that hardware limitations require the use of binary digits only, hence the introduction of sign bits to indicate negativity.

💡Sign Bit

A 'sign bit' is the most significant bit in a binary number used to represent whether the number is positive or negative. The video explains the convention of using '0' for positive numbers and '1' for negative numbers in the sign bit position, which is crucial for understanding how computers encode the sign of an integer.

💡Fixed Point Representation

Fixed point representation is a method where the position of the binary point (decimal point in binary form) is set and does not change. The video discusses this concept as one of the two primary ways to represent numbers in computers, with the binary point either at the extreme left to represent fractions or at the extreme right to represent integers.

💡Floating Point Representation

Floating point representation is an alternative method to fixed point representation, where the position of the decimal point is not fixed but rather stored in a separate register. This concept is significant in the video as it allows for a more flexible representation of numbers, especially those with varying scales.

💡Mantissa

In the context of floating point representation, the 'mantissa' is the part of the number that represents the significant digits. The video explains that the mantissa is a fixed point number and is one of the two components of a floating point number, alongside the exponent.

💡Exponent

The 'exponent' in floating point representation indicates the power of 10 that the mantissa should be multiplied by to get the original number. The video uses the example of the number 6132.789, where the exponent is 4, to illustrate how the position of the decimal point is encoded in the exponent.

💡Sign-Magnitude Representation

Sign-magnitude representation is a method for encoding negative numbers where the most significant bit is used to represent the sign (0 for positive, 1 for negative), and the remaining bits represent the magnitude of the number. The video provides this as the first of three ways to represent the magnitude of negative numbers in computers.

💡One's Complement Representation

One's complement representation is a method where the negative of a number is found by inverting all the bits of its positive binary representation. The video explains this concept as the second way to represent negative numbers, using the example of the number 14, where the positive binary is inverted to represent negative 14.

💡Two's Complement Representation

Two's complement representation is a widely used method for encoding negative numbers, where the negative of a number is found by adding one to its one's complement. The video describes this as the third and final method for representing negative numbers, providing the example of how to convert the binary representation of +14 into -14 using two's complement.

Highlights

Lecture focuses on different representations of integers in computers.

Fixed point and floating point representations are discussed for integer representation.

Positive integers and zero can be represented as unsigned numbers.

Negative integers require a notation for representation.

Computers represent positive and negative numbers using only zeros and ones.

Sign bit is placed in the leftmost position to indicate the sign of the number.

Decimal or binary point is needed to represent fractions or mixed numbers.

Fixed point representation assumes a fixed position for the binary point.

Two common fixed point positions are the leftmost for fractions and rightmost for integers.

Sign magnitude, signed one's complement, and signed two's complement are methods for negative number representation.

Sign magnitude representation uses 0 for positive and 1 for negative signs.

Signed one's complement representation involves taking the complement of the positive number.

Signed two's complement is found by adding one to the one's complement.

Floating point representation uses two registers: one for mantissa and another for exponent.

Mantissa is a fixed point number, and exponent indicates the position of the decimal point.

Floating point representation allows for the storage of large and small numbers efficiently.

Example given for representing the number 6132.0789 using floating point representation.

The lecture concludes with a summary of the discussed representations and their applications.