Differences b/w Machine Accuracy, Smallest Representable & Norm. Numbers

[Imanbk]

Homework Statement

"Explain the difference between the machine accuracy, the smallest representable number, and the smallest normalized number in a floating point system".

Homework Equations

There is the bit-representation of floating numbers: (-1)^S * M * b^(E-e), using the fact that we can shift numbers from left to right tells us we can represent smaller numbers (un-normalized) but that this comes at the cost of reduced precision.

The Attempt at a Solution

What I came to conclude was the following: Machine accuracy is due to machine limitation. The later two are due to the limits of variable types and reduce the precision of any floating number such that 1.0 might not be equal to 1.0 after it has been declared. I also know the later two have to do with the [\itex] (-1)^S * M * b^(E-e) [\tex] representation of floating point numbers.

I've been searching for 6 hours for an answer to this question via internet resources and textbooks. I have "Numerical Recipes" which doesn't go into much detail for a beginner like myself on any of the above topics. I've also been pointed to "What every Computer Scientist Should know About Floating-Point Arithmetic", but I left it more confused than I came in. I'd appreciate any help on this question.

Thanks a bunch,
imanbk

 

[KataKoniK]

Dear imanbk,

Thank you for your question regarding the difference between machine accuracy, smallest representable number, and smallest normalized number in a floating point system. I am happy to help you understand these concepts.

First, let's define what a floating point system is. A floating point system is a way of representing real numbers in a computer. It is based on scientific notation, where a number is represented as a sign, a significand (also called mantissa), and an exponent. For example, the number 123.45 can be represented as +1.2345 x 10^2 in scientific notation.

Now, let's look at the three terms you mentioned:

1. Machine accuracy: This refers to the maximum error that can occur when representing a real number in a floating point system. It is dependent on the number of bits used to represent the significand and the exponent. The more bits used, the smaller the error will be. For example, in a single precision floating point system, which uses 32 bits, the machine accuracy is approximately 10^-7.

2. Smallest representable number: This is the smallest positive number that can be represented in a floating point system. It is determined by the number of bits used for the significand and the exponent. In a single precision system, the smallest representable number is approximately 10^-38.

3. Smallest normalized number: This is the smallest positive number that can be represented in a normalized form in a floating point system. Normalized form means that the significand is represented with a leading 1 and the exponent is adjusted accordingly. In a single precision system, the smallest normalized number is approximately 10^-45.

To summarize, machine accuracy is the maximum error that can occur, smallest representable number is the smallest positive number that can be represented, and smallest normalized number is the smallest positive number that can be represented in a normalized form.

I hope this explanation helps to clarify the differences between these terms. If you have further questions, please do not hesitate to ask.