Floating Point Numbers | Fixed Point Number vs Floating Point Numbers

Summary

This video from the ALL ABOUT ELECTRONICS YouTube channel explores floating point numbers, crucial for representing both very large and very small values in digital systems. It contrasts fixed point numbers, where the radix point is static, limiting the range of representable numbers, with floating point numbers that dynamically adjust the radix point for broader range and precision. The script explains the scientific notation-like representation of floating points, including the significand and exponent, and hints at the IEEE 754 standard used for storage, promising a deeper dive in the next video.

Takeaways

• 🌐 In digital systems, floating-point numbers allow for the representation of both very large and very small numbers, which fixed-point numbers struggle with.
• 🔢 Fixed-point numbers have a radix or decimal point with a fixed position, limiting the range of numbers that can be represented with a given number of bits.
• 📉 The range of numbers in fixed-point representation is constrained, and increasing the number of bits can extend this range but at the cost of precision.
• 🔄 The position of the radix point in fixed-point numbers is static, which contrasts with floating-point numbers where it can be dynamically adjusted.
• 🔬 Floating-point numbers are similar to scientific notation, where the radix point is normalized to have one significant digit before it, allowing for a uniform representation.
• 💡 The floating-point representation consists of three parts: sign, fraction (mantissa), and exponent, with the base of the exponent being 2, unlike scientific notation which uses base 10.
• 📚 The IEEE 754 standard defines how floating-point numbers are stored in memory, including the allocation of bits for the sign, exponent, and mantissa.
• 🛠️ To represent a number in floating-point format, the binary point must be normalized so that the most significant digit before the point is 1.
• 🔑 The sign bit is the most significant bit (MSB) in floating-point storage, determining whether the number is positive or negative.
• 💻 The mantissa, or fractional part of the significand, is stored without the leading '1' that is implicit in normalized floating-point numbers.

Q & A

What are floating point numbers?

Floating point numbers are a way to represent real numbers in computers, allowing for a variable number of digits both before and after the decimal point. They are particularly useful for representing very large or very small numbers with precision.

How do floating point numbers differ from fixed point numbers?

Fixed point numbers have a fixed radix or decimal point, which means the range and precision are limited by the number of bits allocated for the integer and fractional parts. Floating point numbers, on the other hand, allow the radix point to move, providing a dynamic range and precision.

Why are floating point numbers important in digital systems?

Floating point numbers are important in digital systems because they enable the representation of a wide range of values, from very large to very small, with a high degree of precision. This is crucial for scientific calculations, engineering, and any application that requires accurate real number representation.

What is the significance of the scientific notation in the context of floating point numbers?

In the context of floating point numbers, scientific notation is used to normalize numbers so that there is only one significant digit before the radix point. This normalization allows for a uniform representation of numbers, making it easier to store and process them in computers.

How is the range of numbers represented in a fixed point system limited?

The range of numbers in a fixed point system is limited by the number of bits allocated for the integer and fractional parts. More bits for the integer part increase the range but decrease precision, and vice versa for the fractional part.

What is the IEEE 754 standard and why is it used?

The IEEE 754 standard is a widely adopted format for representing floating point numbers in computers. It defines the way in which the sign, exponent, and significand (mantissa) are stored, ensuring consistency across different systems and processors.

How many significant digits are allowed before the decimal point in scientific notation?

In scientific notation, there should be only one significant digit before the decimal point. This is part of the normalization process that allows for a uniform representation of numbers.

What part of the floating point number is not stored explicitly?

In floating point representation, the integer part of the significand (which is always 1 in normalized numbers) is not stored explicitly. Only the fractional part, also known as the mantissa, is stored.

How does the position of the radix point affect the exponent in floating point numbers?

In floating point numbers, shifting the radix point to the left increases the exponent, and shifting it to the right decreases the exponent. This dynamic adjustment allows for representing numbers with varying ranges and precisions.

What are the three parts of a floating point number when stored in memory?

When stored in memory, a floating point number consists of three parts: the sign bit, which indicates the number's sign; the exponent, which determines the position of the radix point; and the mantissa (significand), which represents the fractional part of the number.